WSGR Docket No.55436-729.601 SYSTEMS AND METHODS FOR FOUR-PHOTON SINGLE-QUBIT GATES FOR METASTABLE QUBITS CROSS-REFERENCE [0001] This application claims the benefit of U.S. Provisional Application No.63/650,830, filed on May 22, 2024, which application is incorporated herein by reference for all purposes. BACKGROUND [0002] Quantum computers typically make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers may be different from digital electronic computers based on transistors. For instance, whereas digital computers require data to be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses quantum bits (qubits), which can be in superpositions of states. [0003] In neutral-atom quantum computers or simulation devices, qubits may be encoded in optically trapped atoms. The qubit can be represented by a linear superposition of its two orthonormal basis states. The two orthonormal basis states are usually denoted as |0^ =
(the “zero state”) and |1^ =
(the “one state”). The two orthonormal basis states, {|0^, |1^}, together called the computational basis, span the two-dimensional linear vector (Hilbert) space of the qubit. The basis states can also be combined to form product basis states, e.g., |00^, |01^, |10^, |11^, each called a quantum register. Generally, n qubits are represented by a superposition state vector in 2n dimensional Hilbert space. [0004] In quantum computing, a quantum logic gate may be a basic quantum circuit operating on a small number of qubits (e.g., one qubit, two qubits, three qubits, etc.). Like classical logic gates for conventional digital circuits, quantum logic gates are the building blocks of quantum circuits. However, unlike many classical logic gates, quantum logic gates may be reversible. It may be possible to perform at least some or all operations performed by classical circuits using quantum circuits. Quantum gates may be unitary operators and may be described as unitary matrices relative to some basis. In some cases, quantum computing may implement the computational basis of orthogonal basis vectors of |0^, |1^, … |d − 1^ for a d-level quantum system. In some cases, a quantum gate that acts on n qubits may be represented by a 2n × 2n unitary matrix.
WSGR Docket No.55436-729.601 SUMMARY [0005] In an aspect, the present disclosure provides systems and methods for single-qubit and two-qubit gates comprising multi-photon transitions within a metastable manifold of alkaline earth, or alkaline earth-like atoms. [0006] In one aspect, a method, comprises: implementing a qubit gate on a qubit of an array of qubits, wherein qubit states of said qubit are within a metastable manifold, wherein said qubit states are nuclear spin states, and wherein said qubit gate comprises a multi-photon transition through an intermediate metastable state. [0007] In some embodiments, the multi-photon transition comprises a four-photon transition from said metastable manifold to said intermediate metastable state via an intermediate excited state. In some embodiments, said metastable manifold is a 3P0 manifold. In some embodiments, said intermediate metastable state is a 3P2 state. In some embodiments, said intermediate excited is a 3S1 state. In some embodiments, said excited state is a 3D1 state. In some embodiments, said implementing comprises (i) applying a first electromagnetic energy from a first source and (ii) applying a second electromagnetic energy from a second source to implement said multi-photon transition. In some embodiments, (i) comprises implementing a first two-photon transition and (ii) comprises implementing a second two-photon transition. In some embodiments, (i) comprises, with said first two-photon transition, coupling said metastable manifold off- resonantly to an intermediate excited state with a first detuning, and wherein (ii) comprises, with said second two-photon transition, coupling said intermediate excited state off-resonantly to said intermediate metastable state with a second detuning. In some embodiments said first electromagnetic energy and said second electromagnetic energy are in an optical frequency range. In some embodiments, said first electromagnetic energy or said second electromagnetic energy are directed by a pair of crossed acousto-optic deflectors (AODs). In some embodiments, said first electromagnetic energy or said second electromagnetic energy is global. In some embodiments, said first electromagnetic energy or said second electromagnetic energy is configured to address globally said array of qubits. In some embodiments, said first electromagnetic energy or said second electromagnetic energy is site-specific. In some embodiments, said first electromagnetic energy or said second electromagnetic energy is configured to address site-specifically said array of qubits. In some embodiments, the gate is a single qubit gate or a two-qubit gate. In some embodiments, said single qubit gate is a Pauli Z gate. In some embodiments, said single qubit gate is a Pauli X gate. In some embodiments, said Pauli X gate is configured to also impart an azimuthal phase. In some embodiments, the multi- photon transition is a four-photon transition. In some embodiments, the multi-photon transition is Doppler insensitive. In some embodiments, said qubit is an alkaline earth or alkaline earth-
WSGR Docket No.55436-729.601 like atom. In some embodiments, said alkaline earth-like atom comprises a closed s-shell. In some embodiments, said alkaline earth-like atom is Ytterbium. In some embodiments, said alkaline earth-like atom is Ytterbium-171. In some embodiments, said alkaline earth atom is Strontium. In some embodiments, said implementing couples a lower-lying state of said qubit to a higher-lying state, wherein a detuning of said lower-lying state is δ and a detuning of said higher-lying state is Δ, and wherein Δ is greater in magnitude that δ. In some embodiments, said lower-lying state is a 3P2 state. In some embodiments, said higher-lying state is a 3S1 state. In some embodiments, said higher-lying state is a 3D1 state. In some embodiments, δ is smaller in magnitude than both hyperfine splitting and Zeeman splitting between states in said lower-lying state. In some embodiments, the method further comprises tuning Δ to suppress scattering from said higher-lying state. In some embodiments, the method further comprises (iii) applying a third electromagnetic energy from a third source. In some embodiments, (iii) comprises implementing a third two-photon transition. In some embodiments, (iii) comprises, with said third two-photon transition, coupling said intermediate metastable state with a high-lying Rydberg state. In some embodiments, the method further comprises preparing a qubit of an array of qubits, wherein preparing said qubit comprises a single-photon transition between a ground-state and said metastable manifold. [0008] In another aspect, a system, comprises: a qubit of an array of qubits, wherein qubit states of said qubit are within a metastable manifold, and wherein said qubit states are nuclear spin states, and a source of electromagnetic energy, wherein said source is configured to induce a multi-photon transition through an intermediate metastable state of said qubit, wherein said multi-photon transition is configured to drive a qubit gate on said qubit of said array of qubits. [0009] In some embodiments, the four-photon transition is from said metastable manifold to said intermediate metastable state via an intermediate excited state. In some embodiments, said metastable manifold is a 3P0 manifold. In some embodiments, said intermediate metastable state is a 3P2 state. In some embodiments, said excited state is a 3S1 state. In some embodiments, said excited state is a 3D1 state. In some embodiments, said source comprises a first source configured to generate a first electromagnetic energy and a second source configured to generate a second electromagnetic energy, wherein said first source and said second source are configured to collectively implement said multi-photon transition. In some embodiments, said first source is configured to implement a first two-photon transition, and said second source is configured to implement a second two-photon transition. In some embodiments, said first two- photon transition is configured to couple said metastable manifold off-resonantly to an intermediate excited state with a first detuning, and wherein said second two-photon transition is configured to couple said intermediate excited state off-resonantly to said intermediate
WSGR Docket No.55436-729.601 metastable state with a second detuning. In some embodiments, said first electromagnetic energy and said second electromagnetic energy are in an optical frequency range. In some embodiments, the system further comprises a pair of crossed AODs, wherein said first electromagnetic energy or said second electromagnetic energy are directed by said pair of crossed AODs. In some embodiments, said first electromagnetic energy or said second electromagnetic energy is global. In some embodiments, said first electromagnetic energy or said second electromagnetic energy is configured to address globally said array of qubits. In some embodiments, said first electromagnetic energy or said second electromagnetic energy is site-specific. In some embodiments, said first electromagnetic energy or said second electromagnetic energy is configured to address site-specifically said array of qubits. In some embodiments, the gate is a single qubit gate or a two-qubit gate. In some embodiments, said single qubit gate is a Pauli Z gate. In some embodiments, said single qubit gate is a Pauli X gate. In some embodiments, said Pauli X gate is configured to also impart an azimuthal phase. In some embodiments, the multi-photon transition is a four-photon transition. In some embodiments, the multi-photon transition is Doppler insensitive. In some embodiments, said qubit is an alkaline earth or alkaline earth-like atom. In some embodiments, said alkaline earth- like atom comprises a closed s-shell. In some embodiments, said alkaline earth-like atom is Ytterbium. In some embodiments, said alkaline earth-like atom is Ytterbium-171. In some embodiments, said alkaline earth atom is Strontium. In some embodiments, said implementing couples a lower-lying state of said qubit to a higher-lying state, wherein a detuning of said lower-lying state is δ and a detuning of said higher-lying state is Δ, and wherein Δ is greater in magnitude that δ. In some embodiments, said lower-lying state is a 3P2 state. In some embodiments, said higher-lying state is a 3S1 state. In some embodiments, said higher-lying state is a 3D1 state. In some embodiments, δ is smaller in magnitude than both hyperfine splitting and Zeeman splitting between states in said lower-lying state. In some embodiments, Δ is configured to suppress scattering from said higher-lying state. In some embodiments, the system further comprises an array of spatially distinct optical traps, wherein said array of spatially distinct optical traps comprise said array of qubits. In some embodiments, said source comprises a third source configured to generate a third electromagnetic energy, wherein said first source, said second source, and said third source are configured to collectively implement said multi-photon transition. In some embodiments, said third source is configured to implement a third two-photon transition. In some embodiments, said third two-photon transition is configured to couple said intermediate metastable state to a high-lying Rydberg state. In some embodiments, the system further comprises a fourth source configured to generate a fourth
WSGR Docket No.55436-729.601 electromagnetic energy, wherein said fourth electromagnetic energy is configured to implement a single photon transition between a ground-state and said metastable manifold. [0010] Another aspect of the present disclosure provides a method of implementing a single- qubit gate for non-classical computing, comprising: addressing a qubit for an array of qubits within a metastable manifold with a light source, thereby coupling a lower-lying state of the qubit to a higher-lying state, wherein a detuning of the lower-lying state is δ and a detuning of the higher-lying state is Δ, and wherein Δ is greater in magnitude that δ. In some embodiments, the qubit is an alkaline earth or alkaline earth-like atom. In some embodiments, the lower-lying state is a 3P0 state. In some embodiments, the higher-lying state is a 3S1 state. In some embodiments, the higher-lying state is a 3D1 state. In some embodiments, δ is smaller in magnitude than both hyperfine splitting and Zeeman splitting between states in the lower-lying state. In some embodiments, Δ is configured to suppress scattering from the higher-lying state. [0011] Another aspect of the present disclosure provides a system configured to implement the method of any one of the above aspects. Another aspect of the present disclosure provides computer-readable media configured to implement the method of any one of the above aspects. [0012] Additional aspects and advantages of the present disclosure will become readily apparent to those skilled in this art from the following detailed description, wherein only illustrative embodiments of the present disclosure are shown and described. As will be realized, the present disclosure is capable of other and different embodiments, and its several details are capable of modifications in various obvious respects, all without departing from the disclosure. Accordingly, the drawings and description are to be regarded as illustrative in nature, and not as restrictive. INCORPORATION BY REFERENCE [0013] All publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent publications and patents or patent applications incorporated by reference contradict the disclosure contained in the specification, the specification is intended to supersede or take precedence over any such contradictory material. BRIEF DESCRIPTION OF THE DRAWINGS [0014] The novel features of the invention are set forth with particularity in the appended claims. A better understanding of the features and advantages of the present invention will be obtained by reference to the following detailed description that sets forth illustrative
WSGR Docket No.55436-729.601 embodiments, in which the principles of the invention are utilized, and the accompanying drawings (also “Figure” and “FIG.” herein), of which: [0015] FIG.1 provides a non-limiting example of level diagrams for realizing multi-photon single-gate operations about various axes, in accordance with some embodiments. [0016] FIG.2 provides a non-limiting example of a method for performing multi-photon gates, in accordance with some embodiments. [0017] FIG.3 provides a non-limiting example of a system for performing a non-classical computation, in accordance with some embodiments. [0018] FIG.4A provides a non-limiting example of an optical trapping unit, in accordance with some embodiments. [0019] FIG.4B provides a non-limiting example of a plurality of optical trapping sites, in accordance with some embodiments. [0020] FIG.4C provides a non-limiting example of an optical trapping unit that is partially filled with atoms, in accordance with some embodiments. [0021] FIG.4D provides a non-limiting example of an optical trapping unit that is completely filled with atoms, in accordance with some embodiments. [0022] FIG.5 provides a non-limiting example of an electromagnetic delivery unit, in accordance with some embodiments. [0023] FIG.6 provides a non-limiting example of a state preparation unit, in accordance with some embodiments. [0024] FIG.7 provides a non-limiting example of a method for error corrected non-classical computation, in accordance with some embodiments. [0025] FIG.8 provides a non-limiting example of a system for error corrected non-classical computing that is programmed or otherwise configured to implement methods and systems provided herein, in accordance with some embodiments. [0026] FIG.9 provides a non-limiting example process for performing continuous, non- classical computation, in accordance with some embodiments. [0027] FIG.10A provides a non-limiting example of a plurality of mirrors configured to provide a plurality of optical cavities, in accordance with some embodiments. [0028] FIG.10B provides a non-limiting example of a spacer with a complicated set of cavities to be aligned, including example views and images of Rayleigh scattered light taken from each view, in accordance with some embodiments. [0029] FIG.11 provides a non-limiting example of a repeated loading sequence for continuous non-classical computation, in accordance with some embodiments.
WSGR Docket No.55436-729.601 [0030] FIG.12 provides a non-limiting example of a computer control system that is programmed or otherwise configured to implement the methods and systems provided herein, in accordance with some embodiments. DETAILED DESCRIPTION [0031] While various embodiments of the invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions may occur to those skilled in the art without departing from the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed. [0032] Fast, high-fidelity, single-qubit and two-qubit gates may be important for advances in the efficiency and reliability of non-classical computing. In many cases, however, other approaches may lead to less fast and/or lower-fidelity gates. [0033] For example, to perform the two-photon single qubit gate using a qubit encoded in the nuclear spin states of an electronic ground state of a neutral atom, there may be at least two ways to get to an intermediate state, e.g., 3P1 or 3P1, for example, directly populating the intermediate state or using a Raman transition which is near detuned from a state near the intermediate state. The Raman process may be a two-photon process which excites to a virtual state, so the intermediate state is not actually populated with amplitude near unity. [0034] For two qubit gates involving a Rydberg state and a qubit encoded in the nuclear spin states of the electronic ground state manifold, the two-qubit gate may comprise a two-operation process including excitation to an intermediate state, such as 3P0 or 3P1, before excitation to a Rydberg state. The first leg of the transition may comprise a two-photon single qubit gate. The second leg of the transition is from the intermediate state to the Rydberg states. The two legs can be performed sequentially if the intermediate state is long-lived (e.g., 3P0) or using a two-photon transition from the ground state to the Rydberg state detuned from the intermediate state for either short-lived or long-lived states. The detuned two-photon transition may be useful to suppress scattering from the intermediate state if it is not long-lived, as is the case for 3P1. [0035] However, there may be some unfavorable scattering during the first leg of the two-qubit gate or during the single qubit gate when using a transition from 1S0 to a short-lived state such as 3P1. Scattering during these processes can lead to gate errors in single and two-qubit gates. Further, the transition may be weak, leading to less than 100% population transfer. Further, this transition may in some cases be slow, leading to gates on the 100 microsecond timescale. In some cases, the relatively weak transfer from the ground-state to an intermediate metastable state, such as the 3P1 manifold, can result in far less than 100 percent transfer of the population.
WSGR Docket No.55436-729.601 This has the effect of degrading the fidelity of the gate, which can also lead to higher sensitivity to additional effects such as laser phase noise effects. [0036] The relatively weak transfer may relate to the selection rules for the transition. For example, the direct transition between 1S0 to 3P0 or 3P1 may be forbidden due to selection rules against singlet-to-triplet transitions. The second leg of the transition to the Rydberg may be on the few hundred nanosecond timescale. Slower gates may also be more sensitive to phase noise. [0037] Rather than encoding the qubit in the ground state, qubits formed within the metastable states of an atom, such as the nuclear spin states of 3P0 in alkaline earth atoms have certain advantages for quantum computing including single-photon transitions to Rydberg states for high-fidelity two-qubit gates, the possibility to detect errors associated with decay from the qubit subspace, and opportunities for mid-circuit measurement. [0038] However, fast high-fidelity single-qubit gates have also remained elusive in metastable qubits. Two existing approaches include driving transitions between the nuclear spin states with oscillating magnetic fields at the frequency corresponding to the energy difference between the qubit states, and the use of two-photon Raman transitions. However, the oscillating magnetic fields lead to slow gates, while the two-photon Raman transitions may suffer from levels of Raman scattering that may reduce the gate fidelities below a desired level. Multi-Photon Gates [0039] Systems and methods of the present disclosure address at least some of the above identified drawbacks. Qubits formed within metastable states of an atom, such as the nuclear spin states of 3P0 in alkaline earth atoms, have certain advantages for quantum computing. Starting from the ground state, the qubit states may be populated with a first pulse, and if the metastable state is relatively long lived, then operations may be performed from the metastable qubit state. For example, if the qubit state is 3P0, the coherence lifetimes on the state may exceed 10’s of seconds. Other metastable intermediate states with shorter or longer lifetimes may also be useful. Even if the transition to the metastable state is lossy, this operation may only need to be performed at the start of the calculation where the effect of errors or less than 100% population transfer may be smaller. [0040] Another advantage may be reducing sources of error in two-qubit gates, which involve excitations to a high-lying Rydberg state. This high-lying Rydberg state is far in energy from the ground-state, 1S0, so two-qubit gates with qubit states in the ground-state manifold must first promote to a metastable state or other intermediate state before going to the Rydberg state. This two-photon transition can lead to unfavorable scattering characteristics.
WSGR Docket No.55436-729.601 [0041] Instead of the two-photon singled qubit gate described above, a four-photon single-qubit gate may be favorable in this instance, because the linewidth for the higher-lying 3S1 or 3D1 states are relatively short lived. This means in the time it takes for the magnetic field of the nucleus to change the state of the nuclear spin qubit, there may be high chance of decay to an undesirable state. Instead, the higher-lying 3S1 or 3D1 states may be used as an excited state, and another intermediate metastable state with a more favorable linewidth may be used to change the state of the nuclear spin qubit. In a non-limiting example, the 3P2 state, which has a favorable linewidth to the higher-lying 3S1 or 3D1 states, may be used to change the state of the nuclear spin qubit. Four photons are required for this single-qubit gate, because two are required to get into 3P2, and two are required to get back out. Single-photon transfers from 3P0 to 3P2 are dipole forbidden. [0042] A non-limiting example of a method for performing qubits gates is provided in FIG.2, in accordance with some embodiments. In the illustrated example, the method 200 comprises operation 210. Operation 210 may comprise implementing a qubit gate on an array of qubits. The qubit states of the qubit are within a metastable manifold. The qubit states may be nuclear spin states. The qubit gate may comprise a multi-photon transition through an intermediate metastable state. [0043] The qubit can be represented by a linear superposition of its two orthonormal basis states. The two orthonormal basis states are usually denoted as |0^ =
(the “zero state”) and |1^ = [0] (the “one state”). In quantum computing, a quantum logic gate may be a basic 1 quantum circuit operating on a small number of qubits (e.g., one qubit, two qubits, three qubits, etc.). Like classical logic gates for conventional digital circuits, quantum logic gates are the building blocks of quantum circuits. In some cases, the qubit gate is a single qubit gate. In some cases, the qubit gate is a multi-qubit gate, such as a 2, 3, 4, 5, 6, or more qubit gate. A qubit gate may comprise a quantum logic gate that couples at least two states, e.g., first and second qubit states. In some cases, the first and second qubit states are atom states. [0044] The qubit states may comprise a first qubit state, e.g., the zero state, and a second qubit state, e.g., the one state. In some cases, the qubit states herein may be nuclear spin states. In some cases, the qubit states comprise nuclear spin states of an alkaline earth atom. In some cases, the qubit states comprise nuclear spin states of an alkaline earth-like atom. In some cases, the qubit states comprise nuclear spin states of an alkaline earth-like atom comprising a ground state with a closed s-shell. In some cases, the qubit states comprise nuclear spin states of ytterbium. In some cases, the qubit states comprise nuclear spin states of ytterbium-171. In some cases, the qubit states comprise nuclear spin states of strontium. In some cases, the qubit
WSGR Docket No.55436-729.601 states comprise nuclear spin states of strontium-87. In some cases, the qubit spin states may be Stark shifted nuclear spin states. A Stark shift may be driven optically. An optical Stark shift may be driven off resonance with any, all, or a combination of a single-qubit transition, a two- qubit transition, a shelving transition, an imaging transition, etc. [0045] The first atomic state may comprise a first nuclear spin state and the second atomic state may comprise a second nuclear spin state that is different from the first nuclear spin state. The first and second atomic states may comprise first and second nuclear spin states, respectively, of a quadrupolar nucleus. The first and second atomic states may comprise first and second nuclear spin states, respectively, of a spin-1, spin-3/2, spin-2, spin-5/2, spin-3, spin-7/2, spin-4, or spin- 9/2 nucleus. The first and second atomic states may comprise first and second nuclear spin states, respectively, of any atom described herein, such as first and second spin states of any alkaline earth or alkaline earth-like atom comprising a ground state with a closed s-shell. [0046] For first and second nuclear spin states associated with a nucleus comprising a spin greater than 1/2 (such as a spin-1, spin-3/2, spin-2, spin-5/2, spin-3, spin-7/2, spin-4, or spin-9/2 nucleus), transitions between the first and second nuclear spin states may be accompanied by transitions between other spin states on the nuclear spin manifold. For instance, for a spin-9/2 nucleus in the presence of a uniform magnetic field, all of the nuclear spin levels may be separated by equal energy. Thus, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9/2 spin state to an mN = 7/2 spin state, may also drive mN = 7/2 to mN = 5/2, mN = 5/2 to mN = 3/2, mN = 3/2 to mN = 1/2, mN = 1/2 to mN = -1/2, mN = -1/2 to mN = -3/2, mN = -3/2 to mN = -5/2, mN = -5/2 to mN = -7/2, and mN = -7/2 to mN = -9/2, where mN is the nuclear spin state. Similarly, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9/2 spin state to an mN = 5/2 spin state, may also drive mN = 7/2 to mN = 3/2, mN = 5/2 to mN = 1/2, mN = 3/2 to mN = -1/2, mN = 1/2 to mN = -3/2, mN = -1/2 to mN = -5/2, mN = -3/2 to mN = -7/2, and mN = -5/2 to mN = -9/2. Such a transition may thus not be selective for inducing transitions between particular spin states on the nuclear spin manifold. [0047] In some cases, the qubit states comprise nuclear spin states of an alkaline earth atom. In some cases, the qubit states comprise nuclear spin states of an alkaline earth-like atom. In some cases, the qubit states comprise nuclear spin states of an alkaline earth-like atom comprising a ground state with a closed s-shell. In some cases, the qubit states comprise nuclear spin states of ytterbium. In some cases, the qubit states comprise nuclear spin states of ytterbium-171. In some cases, the qubit states comprise nuclear spin states of strontium. In some cases, the qubit states comprise nuclear spin states of strontium-87.
WSGR Docket No.55436-729.601 [0048] In some cases, the qubit states comprise qubit states of neutral atoms. The neutral atoms may be uncharged atoms. The neutral atoms may comprise alkali atoms. The neutral atoms may comprise lithium (Li) atoms, sodium (Na) atoms, potassium (K) atoms, rubidium (Rb) atoms, or cesium (Cs) atoms. One or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium- 23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, or caesium-133 atoms. One or more atoms may comprise alkaline earth atoms. One or more atoms may comprise beryllium (Be) atoms, magnesium (Mg) atoms, calcium (Ca) atoms, strontium (Sr) atoms, or barium (Ba) atoms. One or more atoms may comprise beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, or barium-138 atoms. One or more atoms may comprise rare earth atoms. One or more atoms may comprise scandium (Sc) atoms, yttrium (Y) atoms, lanthanum (La) atoms, cerium (Ce) atoms, praseodymium (Pr) atoms, neodymium (Nd) atoms, samarium (Sm) atoms, europium (Eu) atoms, gadolinium (Gd) atoms, terbium (Tb) atoms, dysprosium (Dy) atoms, holmium (Ho) atoms, erbium (Er) atoms, thulium (Tm) atoms, ytterbium (Yb) atoms, or lutetium (Lu) atoms. One or more atoms may comprise scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium- 140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium- 143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium- 144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms. [0049] The atoms of the atomic qubits may comprise a single element selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a mixture of elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of
WSGR Docket No.55436-729.601 atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. atoms may comprise rare earth atoms. For instance, the plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium- 42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum- 139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136
WSGR Docket No.55436-729.601 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum- 139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum- 139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176
WSGR Docket No.55436-729.601 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance that is within a range defined by any two of the preceding values. [0050] The qubit states may be within a metastable manifold. A manifold may describe a state having a larger band of sub-levels, such as nuclear spin states, hyperfine states, etc. A metastable manifold may be a state with a finite lifetime. For example, a metastable state may not be a bound state over all coordinates. For example, a metastable state may have a local may have a local maximum on some coordinate or set of coordinates. The qubit states as described herein may be metastable. The intermediate states as described herein may be metastable. [0051] The qubit states as described herein may be metastable. A metastable state may not be a ground state. A metastable state may be an excited state. In an example, the method 200 optionally comprises operation 240. Operation 240 may comprise preparing a qubit of an array of qubits, wherein preparing said qubit comprises a single-photon transition between a ground- state and said metastable manifold. In some cases, the ground state is 1S0. In some cases, the metastable manifold is a 3P0 manifold. As described herein, the 3P0 manifold may be relatively long lived and may offer various advantages; however, other metastable states with excited state lifetimes that exceed times for error correction may be used herein. [0052] The intermediate states as described herein may be metastable. The intermediate metastable state may be an excited state above the qubit state. In some embodiments, said intermediate metastable state is a 3P1 state. [0053] In some cases, the multi-photon transition is Doppler insensitive. An advantage of systems and methods disclosed herein is that each of the qubit operations disclosed herein are Doppler insensitive. Each of the paths disclosed herein involve absorbing and emitting into each of the two beams. Single-Qubit Gates [0054] In some cases, qubit gate is a single qubit gate. A single qubit gate may operate on a single qubit. Single qubit gates may comprise various gates such as, the Pauli X, Pauli Y, Pauli Z, NOT, bit flip, phase flip, Hadamard, etc. A single qubit gate may change an azimuthal phase without changing the relative probabilities of the first and second qubit states. A single qubit gate may change the relative probabilities of the first and second qubit states without changing the azimuthal phase. A single qubit gate may change both the relative probabilities of the qubit states and the azimuthal phase. In some cases, the single qubit gate is a Pauli Z gate. In some cases, the single qubit gate is a Pauli X gate. In some cases, the Pauli X gate is configured to also impart an azimuthal phase.
WSGR Docket No.55436-729.601 [0055] FIG.1 shows non-limiting examples level diagrams 101, 102, and 103 for realizing multi-photon single-qubit gates about various axes. Specifically, the diagram 101 illustrates a fast gate about the x-axis (e.g., an X gate), with a fixed phase. The diagram 102 illustrates a slow gate about the x-axis or the y-axis, with an arbitrary phase. The diagram 103 illustrates a gate about the z-axis (e.g., a Z gate). [0056] Each of the diagrams 101, 102, and 103 comprise levels corresponding to a metastable manifold, 104, an intermediate excited state, 106, and an intermediate metastable state, 105. In some cases, the metastable manifold is a 3P0 manifold. In some case, the intermediate excited state is a 3S1 state. In some cases, the intermediate excited state is a 3D1 state. In some cases, the intermediate metastable state is a 3P2 state. In some cases, the metastable manifold 104, intermediate excited state 106, and intermediate metastable state 105, are electronic states of an alkaline earth atom. In some cases, the metastable manifold 104, intermediate excited state 106, and intermediate metastable state 105, are electronic states of an alkaline earth-like atom. In some cases, the metastable manifold 104, intermediate excited state 106, and intermediate metastable state 105, are electronic states of an alkaline earth-like atom comprising a closed s- shell ground state. In some cases, the metastable manifold 104, intermediate excited state 106, and intermediate metastable state 105, are electronic states of an ytterbium atom. In some cases, the metastable manifold 104, intermediate excited state 106, and intermediate metastable state 105, are electronic states of an ytterbium-171 atom. In some cases, the metastable manifold 104, intermediate excited state 106, and intermediate metastable state 105, are electronic states of a strontium atom. In some cases, the metastable manifold 104, intermediate excited state 106, and intermediate metastable state 105, are electronic states of a strontium-87 atom. [0057] In some cases, to realize a gate that changes population in the qubit states (such as the X gates 101 and 102), the first electromagnetic energy 107 used to couple the metastable manifold 104 to the intermediate excited state 106 may have two orthogonal polarization components. In some cases, the second electromagnetic energy 108 used to couple the intermediate metastable state 105 to the excited state 106 has a single polarization component. In some cases, the first electromagnetic energy 107 comprises a wavelength of 649 nm. In some cases, the second electromagnetic energy 108 comprises a wavelength of 770 nm. [0058] In a non-limiting example, the qubit states may be encoded in two nuclear spin states of 3P0 of an alkaline earth or alkaline-earth-like atom. In this non-limiting example, the four-photon transition for of a single-qubit gates 101, 102, and 103 may couple off-resonantly to an intermediate metastable state 1053P2 with a detuning δ 109 and a higher-lying intermediate excited state 106, such as a 3S1 state or 3D1 state with a detuning ∆ 110. In some cases, the relatively long lifetime of the intermediate metastable state may allow the detuning δ 109 to be
WSGR Docket No.55436-729.601 configured as relatively small, e.g., compared to both the hyperfine splitting and the Zeeman splitting between relevant states in 3P2, without causing excessive scattering. Meanwhile, ∆ 110 may be configured as relatively large to suppress scattering from the relatively short-lived higher-lying excited state (e.g., the 3S1 state or 3D1 state). [0059] In some cases, the gate may be operated in a regime where the four-photon Rabi frequency, Ω, may exceed the qubit energy splitting, as shown in the fast X gate 101. In this case, the azimuthal phase and Rabi frequency of the gate may be set by the phase difference between the two polarization components of the 3P2 beam to the higher-lying excited state (3S1, as illustrated) beam. [0060] In some cases, this example illustrated in the diagram 101 may be used with a fixed phase. Alternatively, in some cases, the four-photon Rabi frequency may be configured as relatively small compared to the qubit frequency, in which case the relative phase of the 3P2 polarization component to the higher-lying state (3S1, as illustrated) polarization components can be used to control the azimuthal phase of the gate without coupling to the Rabi frequency. This alternative example, which is illustrated in the slow X gate 102, may be used to implement gates with arbitrary azimuthal phase, without tuning the pulse area separately for different phases. [0061] In some cases, a gate that changes the qubit phase but not populations gate 103, may be implemented by operating with a single polarization component in the 3P2 beam to the higher- lying state (3S1, as illustrated) beam. In some cases, this may be considered a four-photon gate, as the qubit phase shift results from a four-photon AC Stark shift. An example advantage of the multi-photon Z gate 103 is this gate may be performed without requiring circular polarization with respect to the quantization axis. This may allow for the Z gate 103 to be driven by light incident perpendicular to the magnetic field. [0062] This multi-photon gates, illustrated in the diagrams 101,102, and 103, may have certain advantages. In one example advantage, the four-photon transition may include absorbing a photon from each laser beam and emitting a photon into each beam. Accordingly, this four- photon transition may be Doppler-free (e.g., up to a very long wavelength associated with the qubit frequency) with respect to each beam. Further, this four-photon transition may be insensitive to phase noise on the lasers used to generate each beam. [0063] A simple estimate of the gate error rate for the multi-photon gates illustrated in 101,102, and 103, may be performed by assuming all single-photon transitions involved in the gate have the same Rabi rate, Ω. In such a case, the Rabi rate (or four-photon Stark shift) goes as: Ω4~Ω4/Δ2. The scattering rate goes as: Γ^^^^~Ω2Γ/Δ2, so the error goes as: Γδ/Ω2. [0064] Returning to FIG.2, the method 200 optionally comprises operation 220 implementing a first two-photon transition by applying a first electromagnetic energy from a first source,
WSGR Docket No.55436-729.601 coupling the metastable manifold off-resonantly to an intermediate excited state with a first detuning, and implementing a second two-photon transition by applying a second electromagnetic energy from a second source, coupling the intermediate excited state with an intermediate metastable state with a second detuning. In some cases, the first two-photon transition may be operation 107 of FIG.1. In some cases, the second two photon transition may be operation 108 of FIG.1. [0065] In some cases, the multi-photon transition comprises a four-photon transition from the metastable manifold to an intermediate metastable state via an intermediate excited state. An example four-photon transition is shown in the example of FIG.1. In some cases, the first two- photon transition may be operation 107 of FIG.1. In some cases, the second two photon transition may be operation 108 of FIG.1. [0066] In some cases, the metastable manifold is a 3P0 manifold. In some cases, the intermediate metastable state is a 3P2 state; however, other metastable states may be used. In some cases, the intermediate excited state is a 3S1 state. In some cases, the intermediate excited state is a 3D1 state. [0067] In some cases, operation 220 of the method 200 comprises (i) applying a first electromagnetic energy from a first source and (ii) applying a second electromagnetic energy from a second source to implement the multi-photon transition. A first source may be an electromagnetic delivery unit, such as any electromagnetic delivery unit herein, for example, as described with respect to FIG.3 and FIG.5 herein. The first and second electromagnetic delivery units may be the same. The first and second electromagnetic delivery units may be different. The second electromagnetic delivery units may be configured to apply a third electromagnetic energy to the one or more multi-qubit units, such as the third electromagnetic energy of operation 230 of the method 200. The third electromagnetic energy may comprise one or more pulse sequences. The first and second electromagnetic energies of operation 220 of the method 200 may precede, be simultaneous with, or follow the third electromagnetic energy of method 230 of the method 200. [0068] In some cases, the first electromagnetic energy and the second electromagnetic energy are in an optical frequency range. An optical frequency range may comprise a visible frequency range. The first and second sources may emit electromagnetic energy comprising one or more wavelengths in the visible region of the optical spectrum (e.g., within a range from 400 nm to 800 nm or from 650 nm to 700 nm). The light may be at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm,
WSGR Docket No.55436-729.601 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, or more. The light may be at most about 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. In some cases, the optical source may comprise one or more lasers, such as any of the lasers described herein. [0069] In some cases, the first electromagnetic energy or the second electromagnetic energy is global. In some cases, the first electromagnetic energy or the second electromagnetic energy is configured to address globally the array of qubits. In some cases, the first electromagnetic energy or the second electromagnetic energy is site-specific. In some cases, the first electromagnetic energy or the second electromagnetic energy is configured to address site- specifically the array of qubits. [0070] In some cases, the first electromagnetic energy or the second electromagnetic energy are directed by a pair of crossed acousto-optic deflectors (AODs). In some embodiments, the first AOD or the second AOD comprises a two-dimensional (2D) AOD. In some embodiments, the first AOD or the second AOD itself comprises a pair of crossed one-dimensional (1D) AODs. [0071] A site-specific excitation may be implemented by having either or both of the first electromagnetic energy or the second electromagnetic energy be directed by a pair of crossed AODs. In some cases, only one of the first electromagnetic energy or the second electromagnetic energy is directed by a pair of crossed AODs. [0072] AODs may be used to generate beams that can be steered to different sites in the qubit array by driving the AOD at different frequencies. This may introduce a position-dependent frequency and phase matching condition. This complication may be overcome by using identical or near identical AOD paths for the two beams such that, while the intermediate-state detuning changes, the driven two-photon process remains resonant. Put another way, the four AOD frequencies are fully constrained by selecting a specific site to address. Two frequencies select the position of the first beam and the frequency matching conditions enforce that the two frequencies for the second beam are the same, up to an offset of the qubit frequency (the splitting between the two nuclear spin states, which is around 150 kHz depending on the applied field). [0073] Using AODs to generate the beams for single-qubit operations allows for arbitrary addressing of atoms in a single row (or column) at any given time. This may be useful in order to maintain full control over the amplitude and phase of each. Using AODs also allows full phase control over each beam. This allows tracking of not only the phase of each qubit (allowing application of all rotations in the local qubit frame) but also can also be used to perform more
WSGR Docket No.55436-729.601 complex pulse sequences on each qubit. By controlling the amplitude of the RF for each qubit, the pulse area of each qubit operation can be locally scaled. Combining both phase and amplitude of the RF allows full control of the operation performed on each qubit during a single pulse from the EOM. [0074] For single-photon operations, a single driving beam may be generated with a single 2D AOD system. Alternatively or in addition, the transition may be sufficiently off-resonant to be ignored. The use of a single 2D AOD system generates an array of spots whose spacing can be tuned by adjusting the frequency difference of the RF tones driving the acousto-optic crystal, and whose phase can be tuned by adjusting the RF drive phases. By configuring the AODs in a “crossed” configuration (e.g., the first AOD deflects into the +1 order and the second AOD deflects into the -1 order), lines of deflections are created that have the same absolute frequency (such as along the diagonal created with respect to the axes of deflection of the two AODs). [0075] As an illustrative example, consider the case where the light into the 2D AOD is resonant with a transition of interest. Then, for any RF frequency into the first AOD, if the second AOD deflects with the same frequency, the optical frequency will be brought back into resonance. The final optical phase of the light driving the transition can be controlled by tuning the relative RF phase of the tones into the two AODs. To parallelize addressing, multiple frequencies can be added to both AODs and the diagonal where the corresponding frequencies are deflected will all be resonant. The remaining spots that are deflected will be off-resonant and can be filtered out, but in many cases (e.g., for driving ultranarrow “clock” transitions), the extra spots will be so far off-resonant that this may be unnecessary. [0076] There are at least two modes of operation for addressing atoms in a square array. In a first example, the AODs may be aligned with the trap array. In such case, all or substantially spots may be aligned to a spot in the array, but those along the resonant diagonal will be driven. If the detuning is insufficient, a DMD in an image plane of the optical system may be used to dynamically filter out the other undesired spots. In a second example, the AODs may be aligned at 45 degrees with respect to the atom array, such that the diagonal row of resonant spots aligns to a single row or column of the qubit array. In this case, many of the other spots will miss qubits. However, the remaining spots can be filtered out if desired. [0077] In some cases, applying a first electromagnetic energy from a first source comprises implementing a first two-photon transition and applying a second electromagnetic energy from a second source to implement the multi-photon transition comprises implementing a second two- photon transition. The two-photon transition may be a Raman or Raman-like transition. The two-photon transition may be via a virtual state. The virtual state may be near detuned from another state such as the intermediate excited state or the intermediate metastable state. A
WSGR Docket No.55436-729.601 virtual state may be a very short-lived, otherwise unobservable quantum state. In some cases, the two-photon transitions described herein may populate the intermediate excited state or the intermediate meta stable state. In such cases, some amount of population may be transferred from the ground to the upper state. [0078] In some cases, applying a first electromagnetic energy from a first source comprises, with the first two-photon transition, coupling the metastable manifold off-resonantly to an intermediate excited state with a first detuning, and applying a second electromagnetic energy from a second source to implement the multi-photon transition comprises, with the second two- photon transition, coupling the intermediate excited state off-resonantly to the intermediate metastable state with a second detuning. [0079] In the example of FIG.1, the detunings Δ and δ represent accessing a virtual state near detuned from the intermediate excited state or the intermediate metastable state. In some cases, these detunings may be zero. In some cases, the implementing couples a lower-lying state of the qubit to a higher-lying state, wherein a detuning of the lower-lying state is δ and a detuning of the higher-lying state is Δ, and wherein Δ is greater in magnitude that δ. In some cases, the lower-lying state is a 3P2 state. In some cases, the higher-lying state is a 3S1 state. In some cases, the higher-lying state is a 3D1 state. In some cases, δ is smaller in magnitude than both hyperfine splitting and Zeeman splitting between states in the lower-lying state. In some cases, the method further comprises tuning Δ to suppress scattering from the higher-lying state. Two-Qubit Gates [0080] In some cases, the qubit gate of operation 210 is a two-qubit gate. A two-qubit gate may a qubit gate that operates on two qubits. In some cases, operation 210 is a multi-qubit gate. A multi-qubit gate may a qubit gate that operates on a plurality of qubits. For example, a multi- qubit gate may operate on n qubits, where n may be 5, 6, 7, 8, 9, 10, or more. Examples of two- qubit gates include CNOT, XOR, CX, anti-CX, controlled[0]-not, XNOR, CZ, CPF, CSIGN, SPHASE, DCNOT, SWAP, iSWAP, etc. [0081] A two or more qubit gate may comprise quantum mechanically entangling a first qubit with at least another qubit. An entangling operation may be performed by one or more entangling units disclosed herein. The entanglement units may be configured to quantum mechanically entangle at least a subset of the atoms with at least another atom to form one or more multi-qubit units. The multi-qubit units may comprise two-qubit units, three-qubit units, four-qubit units, or n-qubit units, where n may be 5, 6, 7, 8, 9, 10, or more. For instance, a two- qubit unit may comprise a first atom quantum mechanically entangled with a second atom, a three-qubit unit may comprise a first atom quantum mechanically entangled with a second and
WSGR Docket No.55436-729.601 third atom, a four-qubit unit may comprise a first atom quantum mechanically entangled with a second, third, and fourth atom, and so forth. The first, second, third, or fourth atom may be in a superposition state at the time of quantum mechanical entanglement. Alternatively or in addition, the first, second, third, or fourth atom may not be in a superposition state at the time of quantum mechanical entanglement. The first, second, third, and fourth atom may be quantum mechanically entangled through one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions. [0082] In some cases, quantum mechanically entangling a first qubit with at least another qubit may comprise exiting one or more atoms to a Rydberg state. State selective excitation of atoms to Rydberg levels may enable the implementation of multi-qubit operations. The multi-qubit operations may comprise two-qubit operations, three-qubit operations, or n-qubit operations, where n is 4, 5, 6, 7, 8, 9, 10, or more. Multi-photon transitions may be used to excite atoms from a metastable manifold to a Rydberg state (such as an n3S1 state, wherein n is a principal quantum number described herein). [0083] Multi-qubit operations (such as two-qubit operations, three-qubit operations, four-qubit operations, and so forth) may rely on energy shifts of levels due to van der Waals interactions described herein. Such shifts may either prevent the excitation of one atom conditional on the state of the other or change the coherent dynamics of excitation of the two-atom system to enact a two-qubit operation. In some cases, “dressed states” may be generated under continuous driving to enact two-qubit operations without requiring full excitation to a Rydberg level (for instance, as described in www.arxiv.org/abs/1605.05207, which is incorporated herein by reference in its entirety for all purposes). In some cases, excitation to the Rydberg state allows for the use of existing Rydberg-mediated gate techniques, such as the Blockade gates described in Maller, K. M., et al. “Rydberg-blockade controlled-not gate and entanglement in a two- dimensional array of neutral-atom qubits.” Physical Review A 92.2 (2015): 022336, which is incorporated by reference herein for all purposes, or the time-optimal or symmetric phase gates described in Levine, Harry, et al. “Parallel implementation of high-fidelity multiqubit gates with neutral atoms.” Physical Review Letters, 123.17 (2019): 170503, which is incorporated by reference herein for all purposes. [0084] An entangling unit described herein may comprise a Rydberg unit. A two or more qubit gate herein may be implemented by one or more Rydberg units such as described herein with respect to FIG.3. [0085] In some cases, operation 230 comprises implementing a third two-photon transition by applying a third electromagnetic energy from a third source, coupling the intermediate metastable state with a high-lying Rydberg state. In some cases, the method 200 further
WSGR Docket No.55436-729.601 comprises at an operation 230 applying a third electromagnetic energy from a third source. In some cases, at an operation 230 comprises implementing a third two-photon transition. [0086] In some cases, a single-photon transition may be made from the metastable manifold to a high-lying Rydberg state for performing two-qubit gates. In some cases, this single photon transition may comprise one or more UV electromagnetic energy sources. In other cases, it may be advantageous to perform a six-photon two-qubit gate, using a three-photon transition from the metastable manifold to the high-lying Rydberg state. One potential advantage of this three- photon transition is the ability to control the gate using lasers with comparatively convenient visible wavelengths, described herein, compared to the ultraviolet light used for the single- photon transition. In some cases, the three-photon transition may be implemented using visible wavelength light sources, while ultraviolet light is applied globally. In some cases, the electromagnetic source used in the six-photon two-qubit gate comprise wavelengths of one or more of 649 nm, 770 nm, 1388 nm, or 2093 nm. Preparing Qubits for Multi-Photon Gates [0087] A non-limiting example of preparing a qubit in an array of qubits for multi-photon gates is provided in 240 of FIG.2, wherein preparing the qubits comprises a single-photon transition between a ground-state and the metastable manifold. Method 200 may comprise one or more other state preparation states prior to operation 240. For example, method 200 may comprise one or more state preparation operation. In some cases, preparing a qubit comprises one or more state preparation units comprising one or more optical pumping units, such as those described herein and in FIG.6. In some cases, state preparation may comprise generating a plurality of spatially distinct optical traps. Trapping one or more atoms in those traps. Trapping the one or more atoms may comprise optically cooling atoms in those traps. Examples of Systems for Performing a Non-Classical Computation [0088] FIG.3 shows an example of a system 300 for performing a non-classical computation. The non-classical computation may comprise a quantum computation. The quantum computation may comprise a gate-model quantum computation. The quantum computation may comprise one or more multi-photon gates, described herein. [0089] The system 300 may comprise one or more trapping units 310. The trapping units may comprise one or more optical trapping units. The optical trapping units may comprise any optical trapping unit described herein, such as an optical trapping unit described herein with respect to FIG.4A. The optical trapping units may be configured to generate a plurality of optical trapping sites. The optical trapping units may be configured to generate a plurality of spatially distinct optical trapping sites. For instance, the optical trapping units may be configured
WSGR Docket No.55436-729.601 to generate at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more optical trapping sites. The optical trapping units may be configured to generate at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer optical trapping sites. The optical trapping units may be configured to trap a number of optical trapping sites that is within a range defined by any two of the preceding values. [0090] The optical trapping units may be configured to trap a plurality of atoms. For instance, the optical trapping units may be configured to trap at least about 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, 2,000, 3,000, 4,000, 5,000, 6,000, 7,000, 8,000, 9,000, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000, 1,000,000, or more atoms. The optical trapping units may be configured to trap at most about 1,000,000, 900,000, 800,000, 700,000, 600,000, 500,000, 400,000, 300,000, 200,000, 100,000, 90,000, 80,000, 70,000, 60,000, 50,000, 40,000, 30,000, 20,000, 10,000, 9,000, 8,000, 7,000, 6,000, 5,000, 4,000, 3,000, 2,000, 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, or fewer atoms. The optical trapping units may be configured to trap a number of atoms that is within a range defined by any two of the preceding values. [0091] Each optical trapping site of the optical trapping units may be configured to trap at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atoms. Each optical trapping site may be configured to trap at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, or fewer atoms. Each optical trapping site may be configured to trap a number of atoms that is within a range defined by any two of the preceding values. Each optical trapping site may be configured to trap a single atom. [0092] One or more atoms of the plurality of atoms may comprise qubits, as described herein (for instance, with respect to FIG.5). Two or more atoms may be quantum mechanically entangled. Two or more atoms may be quantum mechanically entangled with a coherence lifetime of at least about 1 microsecond (µs), 2 µs, 3 µs, 4 µs, 5 µs, 6 µs, 7 µs, 8 µs, 9 µs, 10 µs, 20 µs, 30 µs, 40 µs, 50 µs, 60 µs, 70 µs, 80 µs, 90 µs, 100 µs, 200 µs, 300 µs, 400 µs, 500 µs, 600 µs, 700 µs, 800 µs, 900 µs, 1 millisecond (ms), 2 ms, 3 ms, 4 ms, 5 ms, 6 ms, 7 ms, 8 ms, 9 ms, 10 ms, 20 ms, 30 ms, 40 ms, 50 ms, 60 ms, 70 ms, 80 ms, 90 ms, 100 ms, 200 ms, 300 ms, 400 ms, 500 ms, 600 ms, 700 ms, 800 ms, 900 ms, 1 second (s), 2 s, 3 s, 4 s, 5 s, 6 s, 7 s, 8 s, 9 s, 10 s, or more. Two or more atoms may be quantum mechanically entangled with a coherence
WSGR Docket No.55436-729.601 lifetime of at most about 10 s, 9 s, 8 s, 7 s, 6 s, 5 s, 4 s, 3 s, 2 s, 1 s, 900 ms, 800 ms, 700 ms, 600 ms, 500 ms, 400 ms, 300 ms, 200 ms, 100 ms, 90 ms, 80 ms, 70 ms, 60 ms, 50 ms, 40 ms, 30 ms, 20 ms, 10 ms, 9 ms, 8 ms, 7 ms, 6 ms, 5 ms, 4 ms, 3 ms, 2 ms, 1 ms, 900 µs, 800 µs, 700 µs, 600 µs, 500 µs, 400 µs, 300 µs, 200 µs, 100 µs, 90 µs, 80 µs, 70 µs, 60 µs, 50 µs, 40 µs, 30 µs, 20 µs, 10 µs, 9 µs, 8 µs, 7 µs, 6 µs, 5 µs, 4 µs, 3 µs, 2 µs, 1 µs, or less. Two or more atoms may be quantum mechanically entangled with a coherence lifetime that is within a range defined by any two of the preceding values. One or more atoms may comprise neutral atoms. One or more atoms may comprise uncharged atoms. [0093] One or more atoms may comprise alkali atoms. One or more atoms may comprise lithium (Li) atoms, sodium (Na) atoms, potassium (K) atoms, rubidium (Rb) atoms, or cesium (Cs) atoms. One or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, or caesium-133 atoms. One or more atoms may comprise alkaline earth atoms. One or more atoms may comprise beryllium (Be) atoms, magnesium (Mg) atoms, calcium (Ca) atoms, strontium (Sr) atoms, or barium (Ba) atoms. One or more atoms may comprise beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, or barium-138 atoms. One or more atoms may comprise rare earth atoms. One or more atoms may comprise scandium (Sc) atoms, yttrium (Y) atoms, lanthanum (La) atoms, cerium (Ce) atoms, praseodymium (Pr) atoms, neodymium (Nd) atoms, samarium (Sm) atoms, europium (Eu) atoms, gadolinium (Gd) atoms, terbium (Tb) atoms, dysprosium (Dy) atoms, holmium (Ho) atoms, erbium (Er) atoms, thulium (Tm) atoms, ytterbium (Yb) atoms, or lutetium (Lu) atoms. One or more atoms may comprise scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium- 140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium- 143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium- 144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms,
WSGR Docket No.55436-729.601 ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms. [0094] The plurality of atoms may comprise a single element selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a mixture of elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The plurality of atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The plurality of atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. atoms may comprise rare earth atoms. For instance, the plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium- 42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum- 139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%,
WSGR Docket No.55436-729.601 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum- 139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The plurality of atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-133 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum- 139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms,
WSGR Docket No.55436-729.601 gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium-166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium- 171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance that is within a range defined by any two of the preceding values. [0095] The system 300 may comprise one or more first electromagnetic delivery units 320. The first electromagnetic delivery units may comprise any electromagnetic delivery unit described herein, such as an electromagnetic delivery unit described herein with respect to FIG.5. The first electromagnetic delivery units may be configured to apply first electromagnetic energy to one or more atoms of the plurality of atoms. Applying the first electromagnetic energy may induce the atoms to adopt one or more superposition states of a first atomic state and a second atomic state that is different from the first atomic state. The first electromagnetic delivery units may be configured to supply the first electromagnetic energy 107 from the first source of operation 220 of the method 200. The first electromagnetic delivery units may be configured to supply the second electromagnetic energy 108 from said second source of operation 220 of the method 200. The first electromagnetic delivery units may be configured to supply said third electromagnetic energy from said third source of the method 230 of the method 200. [0096] The first atomic state may comprise a first single-qubit state. The second atomic state may comprise a second single-qubit state. The first atomic state or second atomic state may be elevated in energy with respect to a ground atomic state of the atoms. The first atomic state or second atomic state may be equal in energy with respect to the ground atomic state of the atoms. [0097] In some cases, the qubit states comprise nuclear spin states of an atom. The first atomic state may comprise a first nuclear spin state and the second atomic state may comprise a second nuclear spin state that is different from the first nuclear spin state. For instance, the first and second atomic states may comprise first and second nuclear spin states on a multiplet manifold, such as a triplet manifold. The first and second atomic states may comprise first and second nuclear spin states, respectively, on a 3P0 or 3P2 manifold. The first and second atomic states may comprise first and second nuclear spin states, respectively, on a 3P0 or 3P2 manifold of any atom described herein, such as a strontium-873P0 manifold or a strontium-873P2 manifold or the ytterbium-1213P0 manifold or an ytterbium-121. The first and second atomic states may comprise first and second nuclear spin states, respectively, on a metastable manifold, described herein. The first and second atomic states may comprise first and second nuclear spin states,
WSGR Docket No.55436-729.601 respectively, of an intermediate metastable state, described herein. The first and second atomic states may comprise first and second nuclear spin states, respectively, of an excited state, described herein. The first and second atomic states may comprise first and second nuclear spin states, respectively, of a metastable 3P0 manifold. The first and second atomic states may comprise first and second nuclear spin states, respectively, of an excited 3P2 manifold. [0098] In some cases, the first and second atomic states are first and second nuclear spin states of a first electronic state. Optical excitation may be applied between a first electronic state and a second electronic state. In some cases, moving between a first electronic state and a second electronic state comprises one or more intermediate metastable states, described herein. The optical excitation may excite the first nuclear spin state and/or the second nuclear spin state to the second electronic state. The optical excitation may excite the first nuclear spin state and/or the second nuclear spin state to the second electronic state by first moving to one or more intermediate electronic states. A single-qubit transition may comprise a four-photon transition between two nuclear spin states within the first electronic state using two intermediate metastable states. To drive a four-photon single-qubit transition, a first electromagnetic energy may be applied from a first source to implement a first two-photon transition, wherein said first two-photon transition couples said first electronic state off-resonantly to an intermediate excited state, and a second electromagnetic energy may be applied from a second source to implement a second two-photon transition, wherein said second two-photon transition couples said intermediate excited state off-resonantly to said intermediate metastable state. In some cases, the first and second nuclear spin states are nuclear spin states of the metastable manifold, described herein. [0099] In some cases, the qubit states comprise nuclear spin states of an alkaline earth atom. In some cases, the qubit states comprise nuclear spin states of an alkaline earth-like atom. In some cases, the qubit states comprise nuclear spin states of an alkaline earth-like atom comprising a ground state with a closed s-shell. In some cases, the qubit states comprise nuclear spin states of ytterbium. In some cases, the qubit states comprise nuclear spin states of ytterbium-171. In some cases, the qubit states comprise nuclear spin states of strontium. In some cases, the qubit states comprise nuclear spin states of strontium-87. In some cases, the nuclear spin states may be Stark shifted nuclear spin states. A Stark shift may be driven optically. An optical Stark shift may be driven off resonance with any, all, or a combination of a single-qubit transition, a two- qubit transition, a shelving transition, an imaging transition, etc. [0100] The first atomic state may comprise a first nuclear spin state and the second atomic state may comprise a second nuclear spin state that is different from the first nuclear spin state. The first and second atomic states may comprise first and second nuclear spin states, respectively, of
WSGR Docket No.55436-729.601 a quadrupolar nucleus. The first and second atomic states may comprise first and second nuclear spin states, respectively, of a spin-1, spin-3/2, spin-2, spin-5/2, spin-3, spin-7/2, spin-4, or spin- 9/2 nucleus. The first and second atomic states may comprise first and second nuclear spin states, respectively, of any atom described herein, such as first and second spin states of any alkaline earth or alkaline earth-like atom comprising a ground state with a closed s-shell. [0101] For first and second nuclear spin states associated with a nucleus comprising a spin greater than 1/2 (such as a spin-1, spin-3/2, spin-2, spin-5/2, spin-3, spin-7/2, spin-4, or spin-9/2 nucleus), transitions between the first and second nuclear spin states may be accompanied by transitions between other spin states on the nuclear spin manifold. For instance, for a spin-9/2 nucleus in the presence of a uniform magnetic field, all of the nuclear spin levels may be separated by equal energy. Thus, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9/2 spin state to an mN = 7/2 spin state, may also drive mN = 7/2 to mN = 5/2, mN = 5/2 to mN = 3/2, mN = 3/2 to mN = 1/2, mN = 1/2 to mN = -1/2, mN = -1/2 to mN = -3/2, mN = -3/2 to mN = -5/2, mN = -5/2 to mN = -7/2, and mN = -7/2 to mN = -9/2, where mN is the nuclear spin state. Similarly, a transition (such as a Raman transition) designed to transfer atoms from, for instance, an mN = 9/2 spin state to an mN = 5/2 spin state, may also drive mN = 7/2 to mN = 3/2, mN = 5/2 to mN = 1/2, mN = 3/2 to mN = -1/2, mN = 1/2 to mN = -3/2, mN = -1/2 to mN = -5/2, mN = -3/2 to mN = -7/2, and mN = -5/2 to mN = -9/2. Such a transition may thus not be selective for inducing transitions between particular spin states on the nuclear spin manifold. [0102] It may be desirable to instead implement selective transitions between particular first and second spins states on the nuclear spin manifold. This may be accomplished by providing light from a light source that provides an AC Stark shift and pushes neighboring nuclear spin states out of resonance with a transition between the desired transition between the first and second nuclear spin states. For instance, if a transition from first and second nuclear spin states having mN = -9/2 and mN = -7/2 is desired, the light may provide an AC Stark shift to the mN = -5/2 spin state, thereby greatly reducing transitions between the mN = -7/2 and mN = -5/2 states. Similarly, if a transition from first and second nuclear spin states having mN = -9/2 and mN = -5/2 is desired, the light may provide an AC Stark shift to the mN = -1/2 spin state, thereby greatly reducing transitions between the mN = -5/2 and mN = -1/2 states. This may effectively create a two-level subsystem within the nuclear spin manifold that is decoupled from the remainder of the nuclear spin manifold, greatly simplifying the dynamics of the qubit systems. It may be advantageous to use nuclear spin states near the edge of the nuclear spin manifold (e.g., mN = - 9/2 and mN = -7/2, mN = 7/2 and mN = 9/2, mN = -9/2 and mN = -5/2, or mN = 5/2 and mN = 9/2 for a spin-9/2 nucleus) such that only one AC Stark shift is required. Alternatively, nuclear spin
WSGR Docket No.55436-729.601 states farther from the edge of the nuclear spin manifold (e.g., mN = -5/2 and mN = -3/2 or mN = - 5/2 and mN = -1/2) may be used and two AC Stark shifts may be implemented (e.g., at mN = -7/2 and mN = -1/2 or mN = -9/2 and mN = 3/2). [0103] Stark shifting of the nuclear spin manifold may shift neighboring nuclear spin states out of resonance with the desired transition between the first and second nuclear spin states and a second electronic state or a state detuned therefrom. Stark shifting may decrease leakage from the first and second nuclear spin state to other states in the nuclear spin manifold. Starks shifts may be achievable up to 100s of kHz for less than 10 mW beam powers. Upper state frequency selectivity may decrease scattering from imperfect polarization control. Separation of different angular momentum states in the metastable manifold, excited state manifold, or intermediate metastable state manifold may be many gigahertz from the single and two-qubit gate light. Leakage to other states in the nuclear spin manifold may lead to decoherence. The Rabi frequency for two-qubit transitions (e.g., how quickly the transition can be driven) may be faster than the decoherence rate. Scattering from the intermediate state in the two-qubit transition may be a source of decoherence. Detuning from the intermediate state may improve fidelity of two- qubit transitions. [0104] Qubits based on nuclear spin states in long-lived metastable manifolds may allow exploitation of long-lived excited electronic states or the ground-state for qubit storage. Atoms may be selectively transferred into such a state to reduce cross-talk or to improve gate or detection fidelity. Such a storage or shelving process may be atom-selective using the SLMs or AODs described herein. A shelving transition may comprise a transition between the 3P0 metastable manifold to the 1S0 ground-state, or from the 3P0 metastable manifold to another long- lived excited state. [0105] The clock transition (also a “shelving transition” or a “storage transition” herein) may be qubit-state selective. The upper state of the clock transition may have a very long natural lifetime, e.g., greater than 1 second. The linewidth of the clock transition may be much narrower than the qubit energy spacing. This may allow direct spectral resolution. Population may be transferred from one of the qubit states into the clock state. This may allow individual qubit states to be read out separately, by first transferring population from one qubit state into the clock state, performing imaging on the qubits, then transferring the population back into the ground state from the clock state and imaging again. In some cases, a magic wavelength transition is used to drive the clock transition. [0106] The clock light for shelving can be atom-selective or not atom-selective. In some cases, the clock transition is globally applied (e.g., not atom selective). A globally applied clock transition may include directing the light without passing through a microscope objective or
WSGR Docket No.55436-729.601 structuring the light. In some cases, the clock transition is atom-selective. Clock transition which are atom-selective may potentially allow us to improve gate fidelities by minimizing cross-talk. For example, to reduce cross talk in an atom, the atom may be shelved in the clock state where it may not be affected by the light. This may reduce cross-talk between neighboring qubits undergoing transitions. To implement atom-selective clock transitions, the light may pass through one or more microscope objectives and/or may be structured on one or more of a spatial light modulator, digital micromirror device, crossed acousto-optic deflectors, etc. [0107] The system 300 may comprise one or more readout units 330. The readout units may comprise one or more readout optical units. The readout optical units may be configured to perform one or more measurements of the one or more superposition states to obtain the non- classical computation. The readout optical units may comprise one or more optical detectors. The detectors may comprise one or more photomultiplier tubes (PMTs), photodiodes, avalanche diodes, single-photon avalanche diodes, single-photon avalanche diode arrays, phototransistors, reverse-biased light emitting diodes (LEDs), charge coupled devices (CCDs), or complementary metal oxide semiconductor (CMOS) cameras. The optical detectors may comprise one or more fluorescence detectors. The readout optical unit may comprise one or more objectives, such as one or more objective having a numerical aperture (NA) of at least about 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1, or more. The objective may have an NA of at most about 1, 0.95, 0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.15, 0.1, or less. The objective may have an NA that is within a range defined by any two of the preceding values. [0108] The one or more readout optical units 330 may make measurements, such as projective measurements, by applying light resonant with an imaging transition. The imaging transition may cause fluorescence. An imaging transition may comprise a transition between the metastable manifold to an excited state. An imaging transition may comprise a multi-photon transition between the metastable manifold and an intermediate metastable state via an intermediate excited state. An imaging transition may comprise a transition between the metastable manifold to an intermediate metastable state. Any imaging transition may comprise fluorescence. The lower state of the qubit transition may comprise two nuclear spin states in the metastable manifold. The one or more states may be resonant with the imaging transition. A measurement may comprise two or more excitations. In a first excitation, one of the two lower states may be excited to or from a shelving state. In a second excitation, the imaging transition may be excited. The first transition may reduce cross-talk between neighboring atoms during computation. Fluorescence generated from the imaging transition may be collected on one or more readout optical units 330.
WSGR Docket No.55436-729.601 [0109] The imaging units may be used to determine if one or more atoms were lost from the trap. The imaging units may be used to observe the arrangement of atoms in the trap. [0110] The system 300 may comprise one or more vacuum units 340. The one or more vacuum units may comprise one or more vacuum pumps. The vacuum units may comprise one or more roughing vacuum pumps, such as one or more rotary pumps, rotary vane pumps, rotary piston pumps, diaphragm pumps, piston pumps, reciprocating piston pumps, scroll pumps, or screw pumps. The one or more roughing vacuum pumps may comprise one or more wet (for instance, oil-sealed) or dry roughing vacuum pumps. The vacuum units may comprise one or more high- vacuum pumps, such as one or more cryosorption pumps, diffusion pumps, turbomolecular pumps, molecular drag pumps, turbo-drag hybrid pumps, cryogenic pumps, ions pumps, or getter pumps. [0111] The vacuum units may comprise any combination of vacuum pumps described herein. For instance, the vacuum units may comprise one or more roughing pumps (such as a scroll pump) configured to provide a first stage of rough vacuum pumping. The roughing vacuum pumps may be configured to pump gases out of the system 300 to achieve a low vacuum pressure condition. For instance, the roughing pumps may be configured to pump gases out of the system 300 to achieve a low vacuum pressure of at most about 103 Pascals (Pa). The vacuum units may further comprise one or more high-vacuum pumps (such as one or more ion pumps, getter pumps, or both) configured to provide a second stage of high vacuum pumping or ultra- high vacuum pumping. The high-vacuum pumps may be configured to pump gases out of the system 300 to achieve a high vacuum pressure of at most about 10-3 Pa or an ultra-high vacuum pressure of at most about 10-6 Pa once the system 300 has reached the low vacuum pressure condition provided by the one or more roughing pumps. [0112] The vacuum units may be configured to maintain the system 300 at a pressure of at most about 10-6 Pa, 9 x 10-7 Pa, 8 x 10-7 Pa, 7 x 10-7 Pa, 6 x 10-7 Pa, 5 x 10-7 Pa, 4 x 10-7 Pa, 3 x 10-7 Pa, 2 x 10-7 Pa, 10-7 Pa, 9 x 10-8 Pa, 8 x 10-8 Pa, 7 x 10-8 Pa, 6 x 10-8 Pa, 5 x 10-8 Pa, 4 x 10-8 Pa, 3 x 10-8 Pa, 2 x 10-8 Pa, 10-8 Pa, 9 x 10-9 Pa, 8 x 10-9 Pa, 7 x 10-9 Pa, 6 x 10-9 Pa, 5 x 10-9 Pa, 4 x 10-9 Pa, 3 x 10-9 Pa, 2 x 10-9 Pa, 10-9 Pa, 9 x 10-10 Pa, 8 x 10-10 Pa, 7 x 10-10 Pa, 6 x 10-10 Pa, 5 x 10-10 Pa, 4 x 10-10 Pa, 3 x 10-10 Pa, 2 x 10-10 Pa, 10-10 Pa, 9 x 10-11 Pa, 8 x 10-11 Pa, 7 x 10-11 Pa, 6 x10-11 Pa, 5 x 10-11 Pa, 4 x 10-11 Pa, 3 x 10-11 Pa, 2 x 10-11 Pa, 10-11 Pa, 9 x 10-12 Pa, 8 x 10-12 Pa, 7 x 10-12 Pa, 6 x 10-12 Pa, 5 x 10-12 Pa, 4 x 10-12 Pa, 3 x 10-12 Pa, 2 x 10-12 Pa, 10-12 Pa, or lower. The vacuum units may be configured to maintain the system 300 at a pressure of at least about 10-12 Pa, 2 x 10-12 Pa, 3 x 10-12 Pa, 4 x 10-12 Pa, 5 x 10-12 Pa, 6 x 10-12 Pa, 7 x 10-12 Pa, 8 x 10-12 Pa, 9 x 10-12 Pa, 10-11 Pa, 2 x 10-11 Pa, 3 x 10-11 Pa, 4 x 10-11 Pa, 5 x 10-11 Pa, 6 x 10-11 Pa, 7 x 10- 11 Pa, 8 x 10-11 Pa, 9 x 10-11 Pa, 10-10 Pa, 2 x 10-10 Pa, 3 x 10-10 Pa, 4 x 10-10 Pa, 5 x 10-10 Pa, 6 x
WSGR Docket No.55436-729.601 10-10 Pa, 7 x 10-10 Pa, 8 x 10-10 Pa, 9 x 10-10 Pa, 10-9 Pa, 2 x 10-9 Pa, 3 x 10-9 Pa, 4 x 10-9 Pa, 5 x 10-9 Pa, 6 x 10-9 Pa, 7 x 10-9 Pa, 8 x 10-9 Pa, 9 x 10-9 Pa, 10-8 Pa, 2 x 10-8 Pa, 3 x 10-8 Pa, 4 x 10-8 Pa, 5 x 10-8 Pa, 6 x 10-8 Pa, 7 x 10-8 Pa, 8 x 10-8 Pa, 9 x 10-8 Pa, 10-7 Pa, 2 x 10-7 Pa, 3 x 10-7 Pa, 4 x 10-7 Pa, 5 x 10-7 Pa, 6 x 10-7 Pa, 7 x 10-7 Pa, 8 x 10-7 Pa, 9 x 10-7 Pa, 10-6 Pa, or higher. The vacuum units may be configured to maintain the system 200 at a pressure that is within a range defined by any two of the preceding values. [0113] The system 300 may comprise one or more state preparation units 350. The state preparation units may comprise any state preparation unit described herein, such as a state preparation unit described herein with respect to FIG.6. The state preparation units may be configured to prepare a state of the plurality of atoms. [0114] The system 300 may comprise one or more atom reservoirs 360. The atom reservoirs may be configured to supply one or more replacement atoms to replace one or more atoms at one or more optical trapping sites upon loss of the atoms from the optical trapping sites. The atom reservoirs may be spatially separated from the optical trapping units. For instance, the atom reservoirs may be located at a distance from the optical trapping units. [0115] Alternatively or in addition, the atom reservoirs may comprise a portion of the optical trapping sites of the optical trapping units. A first subset of the optical trapping sites may be utilized for performing quantum computations and may be referred to as a set of computationally-active optical trapping sites, while a second subset of the optical trapping sites may serve as an atom reservoir. For instance, the first subset of optical trapping sites may comprise an interior array of optical trapping sites, while the second subset of optical trapping sites comprises an exterior array of optical trapping sites surrounding the interior array. The interior array may comprise a rectangular, square, rectangular prism, or cubic array of optical trapping sites. [0116] The system 300 may comprise one or more atom movement units 370. The atom movement units may be configured to move the one or more replacement atoms from the one or more atoms reservoirs to the one or more optical trapping sites. For instance, the one or more atom movement units may comprise one or more electrically tunable lenses, acousto-optic deflectors (AODs), or spatial light modulators (SLMs). [0117] The system 300 may comprise one or more entanglement units 380. The entanglement units may be configured to quantum mechanically entangle at least a first atom of the plurality of atoms with at least a second atom of the plurality of atoms. The first or second atom may be in a superposition state at the time of quantum mechanical entanglement. Alternatively or in addition, the first or second atom may not be in a superposition state at the time of quantum mechanical entanglement. The first atom and the second atom may be quantum mechanically
WSGR Docket No.55436-729.601 entangled through one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions. The entanglement units may be configured to quantum mechanically entangle any number of atoms described herein. [0118] The entanglement units may also be configured to quantum mechanically entangle at least a subset of the atoms with at least another atom to form one or more multi-qubit units. The multi-qubit units may comprise two-qubit units, three-qubit units, four-qubit units, or n-qubit units, where n may be 5, 6, 7, 8, 9, 10, or more. For instance, a two-qubit unit may comprise a first atom quantum mechanically entangled with a second atom, a three-qubit unit may comprise a first atom quantum mechanically entangled with a second and third atom, a four-qubit unit may comprise a first atom quantum mechanically entangled with a second, third, and fourth atom, and so forth. The first, second, third, or fourth atom may be in a superposition state at the time of quantum mechanical entanglement. Alternatively or in addition, the first, second, third, or fourth atom may not be in a superposition state at the time of quantum mechanical entanglement. The first, second, third, and fourth atom may be quantum mechanically entangled through one or more magnetic dipole interactions, induced magnetic dipole interactions, electric dipole interactions, or induced electric dipole interactions. [0119] The entanglement units may comprise one or more Rydberg units. The Rydberg units may be configured to electronically excite the at least first atom to a Rydberg state or to a superposition of a Rydberg state and a lower-energy atomic state, thereby forming one or more Rydberg atoms or dressed Rydberg atoms. The Rydberg units may be configured to induce one or more quantum mechanical entanglements between the Rydberg atoms or dressed Rydberg atoms and the at least second atom. The second atom may be located at a distance of at least about 200 nanometers (nm), 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 micrometer (µm), 2 µm, 3 µm, 4 µm, 5 µm, 6 µm, 7 µm, 8 µm, 9 µm, 10 µm, or more from the Rydberg atoms or dressed Rydberg atoms. The second atom may be located at a distance of at most about 10 µm, 9 µm, 8 µm, 7 µm, 6 µm, 5 µm, 4 µm, 3 µm, 2 µm, 1 µm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less from the Rydberg atoms or dressed Rydberg atoms. The second atom may be located at a distance from the Rydberg atoms or dressed Rydberg atoms that is within a range defined by any two of the preceding values. The Rydberg units may be configured to allow the Rydberg atoms or dressed Rydberg atoms to relax to a lower-energy atomic state, thereby forming one or more two-qubit units. The Rydberg units may be configured to induce the Rydberg atoms or dressed Rydberg atoms to relax to a lower- energy atomic state. The Rydberg units may be configured to drive the Rydberg atoms or dressed Rydberg atoms to a lower-energy atomic state. For instance, the Rydberg units may be
WSGR Docket No.55436-729.601 configured to apply electromagnetic radiation (such as RF radiation or optical radiation) to drive the Rydberg atoms or dressed Rydberg atoms to a lower-energy atomic state. The Rydberg units may be configured to induce any number of quantum mechanical entanglements between any number of atoms of the plurality of atoms. [0120] The Rydberg units may comprise one or more light sources (such as any light source described herein) configured to emit light having one or more ultraviolet (UV) wavelengths. The UV wavelengths may be selected to correspond to a wavelength that forms the Rydberg atoms or dressed Rydberg atoms. For instance, the light may comprise one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, or more. The light may comprise one or more wavelengths of at most about 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 300 nm to 400 nm. [0121] The Rydberg units may be configured to induce a single-photon transition from the metastable manifold to the high-lying Rydberg state to generate an entanglement between two atoms. The Rydberg units may be configured to induce a multi-photon transition from the metastable manifold to the high-lying Rydberg state to generate an entanglement between two atoms. The Rydberg units may be configured to selectively induce a single-photon or multi- photon transition to selectively generate an entanglement between two atoms. For instance, the Rydberg units may be configured to direct electromagnetic energy (such as optical energy) to particular optical trapping sites to selectively induce a multi-photon transition to selectively generate the entanglement between the two atoms. The two atoms may be trapped in nearby optical trapping sites. For instance, the two atoms may be trapped in adjacent optical trapping sites. The multi-photon transition may be induced using a first electromagnetic energy from a first source, a second electromagnetic energy from a second source, and a third electromagnetic energy from a third source, described herein. The first, second, and third sources may each comprise any light source described herein (such as any laser described herein). The first and second source may be the same or similar to the light sources used to perform a multi-photon single-qubit gate, described herein. Alternatively, different light sources may be used to perform a single-qubit operation and to induce multi-photon transition to generate an entanglement. The first and second sources may emit electromagnetic energy comprising one or more wavelengths in the visible region of the optical spectrum (e.g., within a range from 400 nm to 800 nm or from
WSGR Docket No.55436-729.601 650 nm to 700 nm). In the case of a single-photon excitation from the metastable manifold to the high-lying Rydberg state, third source may emit electromagnetic energy comprising one or more wavelengths in the ultraviolet region of the optical spectrum (e.g., within a range from 200 nm to 400 nm or from 300 nm to 350 nm). Alternatively, in the case of a multi-photon excitation from the metastable manifold to the high-lying Rydberg state, the third source may emit electromagnetic energy comprising one or more wavelengths in the visible region of the optical spectrum (e.g., within a range from 400 nm to 800 nm or from 650 nm to 700 nm).The first, second, and third light sources may emit light having substantially equal and opposite spatially- dependent frequency shifts. [0122] The Rydberg atoms or dressed Rydberg atoms may comprise a Rydberg state that may have sufficiently strong interatomic interactions with nearby atoms (such as nearby atoms trapped in nearby optical trapping sites) to enable the implementation of multi-qubit operations. The Rydberg states may comprise a principal quantum number of at least about 50, 60, 70, 80, 90, 100, or more. The Rydberg states may comprise a principal quantum number of at most about 100, 90, 80, 70, 60, 50, or less. The Rydberg states may comprise a principal quantum number that is within a range defined by any two of the preceding values. The Rydberg states may interact with nearby atoms through van der Waals interactions. The van der Waals interactions may shift atomic energy levels of the atoms. [0123] State selective excitation of atoms to Rydberg levels may enable the implementation of multi-qubit operations. The multi-qubit operations may comprise two-qubit operations, three- qubit operations, or n-qubit operations, where n is 4, 5, 6, 7, 8, 9, 10, or more. Multi-photon transitions may be used to excite atoms from a metastable manifold to a Rydberg state (such as an n3S1 state, wherein n is a principal quantum number described herein). State selectivity may be accomplished by a combination of laser polarization and spectral selectivity. The two-photon transitions may be implemented using first and second laser sources, as described herein. The first and second laser sources may emit pi-polarized light, which may not change the projection of atomic angular momentum along a magnetic field. The third laser may emit circularly polarized light, which may change the projection of atomic angular momentum along the magnetic field by one unit. The first and second qubit levels may be excited to Rydberg level using this polarization. However, the Rydberg levels may be more sensitive to magnetic fields than the metastable manifold so that large splittings (for instance, on the order of 100s of MHz) may be readily obtained. This spectral selectivity may allow state selective excitation to Rydberg levels. [0124] Multi-qubit operations (such as two-qubit operations, three-qubit operations, four-qubit operations, and so forth) may rely on energy shifts of levels due to van der Waals interactions
WSGR Docket No.55436-729.601 described herein. Such shifts may either prevent the excitation of one atom conditional on the state of the other or change the coherent dynamics of excitation of the two-atom system to enact a two-qubit operation. In some cases, “dressed states” may be generated under continuous driving to enact two-qubit operations without requiring full excitation to a Rydberg level (for instance, as described in www.arxiv.org/abs/1605.05207, which is incorporated herein by reference in its entirety for all purposes). [0125] The system 300 may comprise one or more second electromagnetic delivery units (not shown in FIG.3). The second electromagnetic delivery units may comprise any electromagnetic delivery unit described herein, such as an electromagnetic delivery unit described herein with respect to FIG.5. The first and second electromagnetic delivery units may be the same. The first and second electromagnetic delivery units may be different. The second electromagnetic delivery units may be configured to apply a third electromagnetic energy to the one or more multi-qubit units, such as the third electromagnetic energy of operation 230 of the method 200. The third electromagnetic energy may comprise one or more pulse sequences. The first and second electromagnetic energies of operation 220 of the method 200 may precede, be simultaneous with, or follow the third electromagnetic energy of method 230 of the method 200. [0126] The pulse sequences may comprise any number of pulses. For instance, the pulse sequences may comprise at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1,000, or more pulses. The pulse sequences may comprise at most about 1,000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 pulses. The pulse sequences may comprise a number of pulses that is within a range defined by any two of the preceding values. Each pulse of the pulse sequence may comprise any pulse shape, such as any pulse shape described herein. [0127] The pulse sequences may be configured to decrease the duration of time required to implement multi-qubit operations, as described herein (for instance, with respect to Example 3). For instance, the pulse sequences may comprise a duration of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (µs), 2 µs, 3 µs, 4 µs, 5 µs, 6 µs, 7 µs, 8 µs, 9 µs, 10 µs, 20 µs, 30 µs, 40 µs, 50 µs, 60 µs, 70 µs, 80 µs, 90 µs, 100 µs, or more. The pulse sequences may comprise a duration of at most about 100 µs, 90 µs, 80 µs, 70 µs, 60 µs, 50 µs, 40 µs, 30 µs, 20 µs, 10 µs, 9 µs, 8 µs, 7 µs, 6 µs, 5 µs, 4 µs, 3 µs, 2 µs, 1 µs, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, or less. The pulse sequences may comprise a duration that is within a range defined by any two of the preceding values.
WSGR Docket No.55436-729.601 [0128] The pulse sequences may be configured to increase the fidelity of multi-qubit operations, as described herein. For instance, the pulse sequences may enable multi-qubit operations with a fidelity of at least about 0.5, 0.6, 0.7, 0.8, 0.9, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 0.991, 0.992, 0.993, 0.994, 0.995, 0.996, 0.997, 0.998, 0.999, 0.9991, 0.9992, 0.9993, 0.9994, 0.9995, 0.9996, 0.9997, 0.9998, 0.9999, 0.99991, 0.99992, 0.99993, 0.99994, 0.99995, 0.99996, 0.99997, 0.99998, 0.99999, 0.999991, 0.999992, 0.999993, 0.999994, 0.999995, 0.999996, 0.999997, 0.999998, 0.999999, or more. The pulse sequences may enable multi-qubit operations with a fidelity of at most about 0.999999, 0.999998, 0.999997, 0.999996, 0.999995, 0.999994, 0.999993, 0.999992, 0.999991, 0.99999, 0.99998, 0.99997, 0.99996, 0.99995, 0.99994, 0.99993, 0.99992, 0.99991, 0.9999, 0.9998, 0.9997, 0.9996, 0.9995, 0.9994, 0.9993, 0.9992, 0.9991, 0.999, 0.998, 0.997, 0.996, 0.995, 0.994, 0.993, 0.992, 0.991, 0.99, 0.98, 0.97, 0.96, 0.95, 0.94, 0.93, 0.92, 0.91, 0.9, 0.8, 0.7, 0.6, 0.5, or less. The pulse sequences may enable multi-qubit operations with a fidelity that is within a range defined by any two of the preceding values. [0129] The pulse sequences may enable the implementation of multi-qubit operations on non- adiabatic timescales while maintaining effectively adiabatic dynamics. For instance, the pulse sequences may comprise one or more of shortcut to adiabaticity (STA) pulse sequences, transitionless quantum driving (TQD) pulse sequences, superadiabatic pulse sequences, counterdiabatic driving pulse sequences, derivative removal by adiabatic gate (DRAG) pulse sequences, and weak anharmonicity with average Hamiltonian (Wah Wah) pulse sequences. For instance, the pulse sequences may be similar to those described in M.V. Berry, “Transitionless Quantum Driving,” Journal of Physics A: Mathematical and Theoretical 42(36), 365303 (2009), www.doi.org/10.1088/1751-8113/42/36/365303; Y.-Y. Jau et al., “Entangling Atomic Spins with a Strong Rydberg-Dressed Interaction,” Nature Physics 12(1), 71-74 (2016); T. Keating et al., “Robust Quantum Logic in Neutral Atoms via Adiabatic Rydberg Dressing,” Physical Review A 91, 012337 (2015); A. Mitra et al., “Robust Mölmer-Sörenson Gate for Neutral Atoms Using Rapid Adiabatic Rydberg Dressing,” www.arxiv.org/abs/1911.04045 (2019); or L.S. Theis et al., “Counteracting Systems of Diabaticities Using DRAG Controls: The Status after 10 Years,” Europhysics Letters 123(6), 60001 (2018), each of which is incorporated herein by reference in its entirety for all purposes. [0130] The pulse sequences may further comprise one or more optimal control pulse sequences. The optimal control pulse sequences may be derived from one or more procedures, including gradient ascent pulse engineering (GRAPE) methods, Krotov’s method, chopped basis methods, chopped random basis (CRAB) methods, Nelder-Mead methods, gradient optimization using parametrization (GROUP) methods, genetic algorithm methods, and gradient optimization of analytic controls (GOAT) methods. For instance, the pulse sequences may be similar to those
WSGR Docket No.55436-729.601 described in N. Khaneja et al., “Optimal Control of Coupled Spin Dynamics: Design of NMR Pulse Sequences by Gradient Ascent Algorithms,” Journal of Magnetic Resonance 172(2), 296- 305 (2005); or J.T. Merrill et al., “Progress in Compensating Pulse Sequences for Quantum Computation,” Advances in Chemical Physics 154, 241-294 (2014), each of which is incorporated by reference in its entirety for all purposes. Example of Cloud Computing [0131] The system 300 may be operatively coupled to a digital computer described herein (such as a digital computer described herein with respect to FIG.12) over a network described herein (such as a network described herein with respect to FIG.12). The network may comprise a cloud computing network. Example of Optical Trapping Units [0132] FIG.4A shows an example of an optical trapping unit 310. The optical trapping unit may be configured to generate a plurality 311 of spatially distinct optical trapping sites, as described herein. For instance, as shown in FIG.4B, the optical trapping unit may be configured to generate a first optical trapping site 311a, second optical trapping site 311b, third optical trapping site 311c, fourth optical trapping site 311d, fifth optical trapping site 311e, sixth optical trapping site 311f, seventh optical trapping site 311g, eighth optical trapping site 311h, and ninth optical trapping site 311i, as depicted in FIG.4A. The plurality of spatially distinct optical trapping sites may be configured to trap a plurality of atoms, such as first atom 312a, second atom 312b, third atom 312c, and fourth atom 312d, as depicted in FIG.4A. As depicted in FIG. 4B, each optical trapping site may be configured to trap a single atom. As depicted in FIG.4B, some of the optical trapping sites may be empty (i.e., not trap an atom). [0133] As shown in FIG.4B, the plurality of optical trapping sites may comprise a two- dimensional (2D) array. The 2D array may be perpendicular to the optical axis of optical components of the optical trapping unit depicted in FIG.4A. Alternatively, the plurality of optical trapping sites may comprise a one-dimensional (1D) array or a three-dimensional (3D) array. [0134] Although depicted as comprising nine optical trapping sites filled by four atoms in FIG. 4B, the optical trapping unit 310 may be configured to generate any number of spatially distinct optical trapping sites described herein and may be configured to trap any number of atoms described herein. [0135] Each optical trapping site of the plurality of optical trapping sites may be spatially separated from each other optical trapping site by a distance of at least about 200 nm, 300 nm, 400 nm, 500 nm, 600 nm, 700 nm, 800 nm, 900 nm, 1 µm, 2 µm, 3 µm, 4 µm, 5 µm, 6 µm, 7
WSGR Docket No.55436-729.601 µm, 8 µm, 9 µm, 10 µm, or more. Each optical trapping site may be spatially separated from each other optical trapping site by a distance of at most about 10 µm, 9 µm, 8 µm, 7 µm, 6 µm, 5 µm, 4 µm, 3 µm, 2 µm, 1 µm, 900 nm, 800 nm, 700 nm, 600 nm, 500 nm, 400 nm, 300 nm, 200 nm, or less. Each optical trapping site maybe spatially separated from each other optical trapping site by a distance that is within a range defined by any two of the preceding values. [0136] The optical trapping sites may comprise one or more optical tweezers. Optical tweezers may comprise one or more focused laser beams to provide an attractive or repulsive force to hold or move the one or more atoms. The beam waist of the focused laser beams may comprise a strong electric field gradient. The atoms may be attracted or repelled along the electric field gradient to the center of the laser beam, which may contain the strongest electric field. The optical trapping sites may comprise one or more optical lattice sites of one or more optical lattices. The optical trapping sites may comprise one or more optical lattice sites of one or more one-dimensional (1D) optical lattices, two-dimensional (2D) optical lattices, or three- dimensional (3D) optical lattices. For instance, the optical trapping sites may comprise one or more optical lattice sites of a 2D optical lattice, as depicted in FIG.4B. [0137] The optical lattices may be generated by interfering counter-propagating light (such as counter-propagating laser light) to generate a standing wave pattern having a periodic succession of intensity minima and maxima along a particular direction. A 1D optical lattice may be generated by interfering a single pair of counter-propagating light beams. A 2D optical lattice may be generated by interfering two pairs of counter-propagating light beams. A 3D optical lattice may be generated by interfering three pairs of counter-propagating lights beams. The light beams may be generated by different light sources or by the same light source. Therefore, an optical lattice may be generated by at least about 1, 2, 3, 4, 5, 6, or more light sources or at most about 6, 5, 4, 3, 2, or 1 light sources. [0138] Returning to the description of FIG.4A, the optical trapping unit may comprise one or more light sources configured to emit light to generate the plurality of optical trapping sites as described herein. For instance, the optical trapping unit may comprise a single light source 313, as depicted in FIG.4A. Though depicted as comprising a single light source in FIG.4A, the optical trapping unit may comprise any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light sources. The light sources may comprise one or more lasers. The lasers may be configured to operate at a resolution limit of the lasers. For example, the lasers can be configured to provide diffraction limited spot sizes for optical trapping. [0139] The lasers may comprise one or more continuous wave lasers. The lasers may comprise one or more pulsed lasers. The lasers may comprise one or more gas lasers, such as one or more
WSGR Docket No.55436-729.601 helium-neon (HeNe) lasers, argon (Ar) lasers, krypton (Kr) lasers, xenon (Xe) ion lasers, nitrogen (N2) lasers, carbon dioxide (CO2) lasers, carbon monoxide (CO) lasers, transversely excited atmospheric (TEA) lasers, or excimer lasers. For instance, the lasers may comprise one or more argon dimer (Ar2) excimer lasers, krypton dimer (Kr2) excimer lasers, fluorine dimer (F2) excimer lasers, xenon dimer (Xe2) excimer lasers, argon fluoride (ArF) excimer lasers, krypton chloride (KrCl) excimer lasers, krypton fluoride (KrF) excimer lasers, xenon bromide (XeBr) excimer lasers, xenon chloride (XeCl) excimer lasers, or xenon fluoride (XeF) excimer lasers. The laser may comprise one or more dye lasers. [0140] The lasers may comprise one or more metal-vapor lasers, such as one or more helium- cadmium (HeCd) metal-vapor lasers, helium-mercury (HeHg) metal-vapor lasers, helium- selenium (HeSe) metal-vapor lasers, helium-silver (HeAg) metal-vapor lasers, strontium (Sr) metal-vapor lasers, neon-copper (NeCu) metal-vapor lasers, copper (Cu) metal-vapor lasers, gold (Au) metal-vapor lasers, manganese (Mn) metal-vapor laser, or manganese chloride (MnCl2) metal-vapor lasers. [0141] The lasers may comprise one or more solid-state lasers, such as one or more ruby lasers, metal-doped crystal lasers, or metal-doped fiber lasers. For instance, the lasers may comprise one or more neodymium-doped yttrium aluminum garnet (Nd:YAG) lasers, neodymium/chromium doped yttrium aluminum garnet (Nd/Cr:YAG) lasers, erbium-doped yttrium aluminum garnet (Er:YAG) lasers, neodymium-doped yttrium lithium fluoride (Nd:YLF) lasers, neodymium-doped yttrium orthovanadate (ND:YVO4) lasers, neodymium- doped yttrium calcium oxoborate (Nd:YCOB) lasers, neodymium glass (Nd:glass) lasers, titanium sapphire (Ti:sapphire) lasers, thulium-doped ytrium aluminum garnet (Tm:YAG) lasers, ytterbium-doped ytrrium aluminum garnet (Yb:YAG) lasers, ytterbium-doped glass (Yt:glass) lasers, holmium ytrrium aluminum garnet (Ho:YAG) lasers, chromium-doped zinc selenide (Cr:ZnSe) lasers, cerium-doped lithium strontium aluminum fluoride (Ce:LiSAF) lasers, cerium-doped lithium calcium aluminum fluoride (Ce:LiCAF) lasers, erbium-doped glass (Er:glass) lasers, erbium-ytterbium-codoped glass (Er/Yt:glass) lasers, uranium-doped calcium fluoride (U:CaF2) lasers, or samarium-doped calcium fluoride (Sm:CaF2) lasers. [0142] The lasers may comprise one or more semiconductor lasers or diode lasers, such as one or more gallium nitride (GaN) lasers, indium gallium nitride (InGaN) lasers, aluminum gallium indium phosphide (AlGaInP) lasers, aluminum gallium arsenide (AlGaAs) lasers, indium gallium arsenic phosphide (InGaAsP) lasers, vertical cavity surface emitting lasers (VCSELs), or quantum cascade lasers. [0143] The lasers may emit continuous wave laser light. The lasers may emit pulsed laser light. The lasers may have a pulse length of at least about 1 femtoseconds (fs), 2 fs, 3 fs, 4 fs, 5 fs, 6
WSGR Docket No.55436-729.601 fs, 7 fs, 8 fs, 9 fs, 10 fs, 20 fs, 30 fs, 40 fs, 50 fs, 60 fs, 70 fs, 80 fs, 90 fs, 100 fs, 200 fs, 300 fs, 400 fs, 500 fs, 600 fs, 700 fs, 800 fs, 900 fs, 1 picosecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, 200 ps, 300 ps, 400 ps, 500 ps, 600 ps, 700 ps, 800 ps, 900 ps, 1 nanosecond (ns), 2 ns, 3 ns, 4 ns, 5 ns, 6 ns, 7 ns, 8 ns, 9 ns, 10 ns, 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1,000 ns, or more. The lasers may have a pulse length of at most about 1,000 ns, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, 9 ns, 8 ns, 7 ns, 6 ns, 5 ns, 4 ns, 3 ns, 2 ns, 1 ns, 900 ps, 800 ps, 700 ps, 600 ps, 500 ps, 400 ps, 300 ps, 200 ps, 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 fs, 800 fs, 700 fs, 600 fs, 500 fs, 400 fs, 300 fs, 200 fs, 100 fs, 90 fs, 80 fs, 70 fs, 60 fs, 50 fs, 40 fs, 30 fs, 20 fs, 10 fs, 9 fs, 8 fs, 7 fs, 6 fs, 5 fs, 4 fs, 3 fs, 2 fs, 1 fs, or less. The lasers may have a pulse length that is within a range defined by any two of the preceding values. [0144] The lasers may have a repetition rate of at least about 1 hertz (Hz), 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 megahertz (MHz), 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1,000 MHz, or more. The lasers may have a repetition rate of at most about 1,000 MHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The lasers may have a repetition rate that is within a range defined by any two of the preceding values. [0145] The lasers may emit light having a pulse energy of at least about 1 nanojoule (nJ), 2 nJ, 3 nJ, 4 nJ, 5 nJ, 6 nJ, 7 nJ, 8 nJ, 9 nJ, 10 nJ, 20 nJ, 30 nJ, 40 nJ, 50 nJ, 60 nJ, 70 nJ, 80 nJ, 90 nJ, 100 nJ, 200 nJ, 300 nJ, 400 nJ, 500 nJ, 600 nJ, 700 nJ, 800 nJ, 900 nJ, 1 microjoule (µJ), 2 µJ, 3 µJ, 4 µJ, 5 µJ, 6 µJ, 7 µJ, 8 µJ, 9 µJ, 10 µJ, 20 µJ, 30 µJ, 40 µJ, 50 µJ, 60 µJ, 70 µJ, 80 µJ, 90
WSGR Docket No.55436-729.601 µJ, 100 µJ, 200 µJ, 300 µJ, 400 µJ, 500 µJ, 600 µJ, 700 µJ, 800 µJ, 900 µJ, a least 1 millijoule (mJ), 2 mJ, 3 mJ, 4 mJ, 5 mJ, 6 mJ, 7 mJ, 8 mJ, 9 mJ, 10 mJ, 20 mJ, 30 mJ, 40 mJ, 50 mJ, 60 mJ, 70 mJ, 80 mJ, 90 mJ, 100 mJ, 200 mJ, 300 mJ, 400 mJ, 500 mJ, 600 mJ, 700 mJ, 800 mJ, 900 mJ, a least 1 Joule (J), or more. The lasers may emit light having a pulse energy of at most about 1 J, 900 mJ, 800 mJ, 700 mJ, 600 mJ, 500 mJ, 400 mJ, 300 mJ, 200 mJ, 100 mJ, 90 mJ, 80 mJ, 70 mJ, 60 mJ, 50 mJ, 40 mJ, 30 mJ, 20 mJ, 10 mJ, 9 mJ, 8 mJ, 7 mJ, 6 mJ, 5 mJ, 4 mJ, 3 mJ, 2 mJ, 1 mJ, 900 µJ, 800 µJ, 700 µJ, 600 µJ, 500 µJ, 400 µJ, 300 µJ, 200 µJ, 100 µJ, 90 µJ, 80 µJ, 70 µJ, 60 µJ, 50 µJ, 40 µJ, 30 µJ, 20 µJ, 10 µJ, 9 µJ, 8 µJ, 7 µJ, 6 µJ, 5 µJ, 4 µJ, 3 µJ, 2 µJ, 1 µJ, 900 nJ, 800 nJ, 700 nJ, 600 nJ, 500 nJ, 400 nJ, 300 nJ, 200 nJ, 100 nJ, 90 nJ, 80 nJ, 70 nJ, 60 nJ, 50 nJ, 40 nJ, 30 nJ, 20 nJ, 10 nJ, 9 nJ, 8 nJ, 7 nJ, 6 nJ, 5 nJ, 4 nJ, 3 nJ, 2 nJ, 1 nJ, or less. The lasers may emit light having a pulse energy that is within a range defined by any two of the preceding values. [0146] The lasers may emit light having an average power of at least about 1 microwatt (µW), 2 µW, 3 µW, 4 µW, 5 µW, 6 µW, 7 µW, 8 µW, 9 µW, 10 µW, 20 µW, 30 µW, 40 µW, 50 µW, 60 µW, 70 µW, 80 µW, 90 µW, 100 µW, 200 µW, 300 µW, 400 µW, 500 µW, 600 µW, 700 µW, 800 µW, 900 µW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. The lasers may emit light having an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 µW, 800 µW, 700 µW, 600 µW, 500 µW, 400 µW, 300 µW, 200 µW, 100 µW, 90 µW, 80 µW, 70 µW, 60 µW, 50 µW, 40 µW, 30 µW, 20 µW, 10 µW, 9 µW, 8 µW, 7 µW, 6 µW, 5 µW, 4 µW, 3 µW, 2 µW, 1 µW, or more. The lasers may emit light having a power that is within a range defined by any two of the preceding values. [0147] The lasers may emit light comprising one or more wavelengths in the ultraviolet (UV), visible, or infrared (IR) portions of the electromagnetic spectrum. The lasers may emit light comprising one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570
WSGR Docket No.55436-729.601 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, 1,010 nm, 1,020 nm, 1,030 nm, 1,040 nm, 1,050 nm, 1,060 nm, 1,070 nm, 1,080 nm, 1,090 nm, 1,100 nm, 1,110 nm, 1,120 nm, 1,130 nm, 1,140 nm, 1,150 nm, 1,160 nm, 1,170 nm, 1,180 nm, 1,190 nm, 1,200 nm, 1,210 nm, 1,220 nm, 1,230 nm, 1,240 nm, 1,250 nm, 1,260 nm, 1,270 nm, 1,280 nm, 1,290 nm, 1,300 nm, 1,310 nm, 1,320 nm, 1,330 nm, 1,340 nm, 1,350 nm, 1,360 nm, 1,370 nm, 1,380 nm, 1,390 nm, 1,400 nm, or more. The lasers may emit light comprising one or more wavelengths of at most about 1,400 nm, 1,390 nm, 1,380 nm, 1,370 n, 1,360 nm, 1,350 nm, 1,340 nm, 1,330 nm, 1,320 nm, 1,310 nm, 1,300 nm, 1,290 nm, 1,280 nm, 1,270 n, 1,260 nm, 1,250 nm, 1,240 nm, 1,230 nm, 1,220 nm, 1,210 nm, 1,200 nm, 1,190 nm, 1,180 nm, 1,170 n, 1,160 nm, 1,150 nm, 1,140 nm, 1,130 nm, 1,120 nm, 1,110 nm, 1,100 nm, 1,090 nm, 1,080 nm, 1,070 n, 1,060 nm, 1,050 nm, 1,040 nm, 1,030 nm, 1,020 nm, 1,010 nm, 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm. The lasers may emit light comprising one or more wavelengths that are within a range defined by any two of the preceding values. [0148] The lasers may emit light having a bandwidth of at least about 1 x 10-15 nm, 2 x 10-15 nm, 3 x 10-15 nm, 4 x 10-15 nm, 5 x 10-15 nm, 6 x 10-15 nm, 7 x 10-15 nm, 8 x 10-15 nm, 9 x 10-15 nm, 1 x 10-14 nm, 2 x 10-14 nm, 3 x 10-14 nm, 4 x 10-14 nm, 5 x 10-14 nm, 6 x 10-14 nm, 7 x 10-14 nm, 8 x 10-14 nm, 9 x 10-14 nm, 1 x 10-13 nm, 2 x 10-13 nm, 3 x 10-13 nm, 4 x 10-13 nm, 5 x 10-13 nm, 6 x 10-13 nm, 7 x 10-13 nm, 8 x 10-13 nm, 9 x 10-13 nm, 1 x 10-12 nm, 2 x 10-12 nm, 3 x 10-12 nm, 4 x 10-12 nm, 5 x 10-12 nm, 6 x 10-12 nm, 7 x 10-12 nm, 8 x 10-12 nm, 9 x 10-12 nm, 1 x 10-11 nm, 2 x 10-11 nm, 3 x 10-11 nm, 4 x 10-11 nm, 5 x 10-11 nm, 6 x 10-11 nm, 7 x 10-11 nm, 8 x 10-11 nm, 9 x 10-11 nm, 1 x 10-10 nm, 2 x 10-10 nm, 3 x 10-10 nm, 4 x 10-10 nm, 5 x 10-10 nm, 6 x 10-10 nm, 7 x 10-10 nm, 8 x 10-10 nm, 9 x 10-10 nm, 1 x 10-9 nm, 2 x 10-9 nm, 3 x 10-9 nm, 4 x 10-9 nm, 5 x 10-9 nm, 6 x 10-9 nm, 7 x 10-9 nm, 8 x 10-9 nm, 9 x 10-9 nm, 1 x 10-8 nm, 2 x 10-8 nm, 3 x 10-8 nm, 4 x 10-8 nm, 5 x 10-8 nm, 6 x 10-8 nm, 7 x 10-8 nm, 8 x 10-8 nm, 9 x 10-8 nm, 1 x 10-7 nm, 2 x 10-7 nm, 3 x 10-7 nm, 4 x 10-7 nm, 5 x 10-7 nm, 6 x 10-7 nm, 7 x 10-7 nm, 8 x 10-7 nm, 9 x 10-7 nm, 1 x
WSGR Docket No.55436-729.601 10-6 nm, 2 x 10-6 nm, 3 x 10-6 nm, 4 x 10-6 nm, 5 x 10-6 nm, 6 x 10-6 nm, 7 x 10-6 nm, 8 x 10-6 nm, 9 x 10-6 nm, 1 x 10-5 nm, 2 x 10-5 nm, 3 x 10-5 nm, 4 x 10-5 nm, 5 x 10-5 nm, 6 x 10-5 nm, 7 x 10-5 nm, 8 x 10-5 nm, 9 x 10-5 nm, 1 x 10-4 nm, 2 x 10-4 nm, 3 x 10-4 nm, 4 x 10-4 nm, 5 x 10-4 nm, 6 x 10-4 nm, 7 x 10-4 nm, 8 x 10-4 nm, 9 x 10-4 nm, 1 x 10-3 nm, or more. The lasers may emit light having a bandwidth of at most about1 x 10-3 nm, 9 x 10-4 nm, 8 x 10-4 nm, 7 x 10-4 nm, 6 x 10-4 nm, 5 x 10-4 nm, 4 x 10-4 nm, 3 x 10-4 nm, 2 x 10-4 nm, 1 x 10-4 nm, 9 x 10-5 nm, 8 x 10-5 nm, 7 x 10-5 nm, 6 x 10-5 nm, 5 x 10-5 nm, 4 x 10-5 nm, 3 x 10-5 nm, 2 x 10-5 nm, 1 x 10-5 nm, 9 x 10-6 nm, 8 x 10-6 nm, 7 x 10-6 nm, 6 x 10-6 nm, 5 x 10-6 nm, 4 x 10-6 nm, 3 x 10-6 nm, 2 x 10-6 nm, 1 x 10-6 nm, 9 x 10-7 nm, 8 x 10-7 nm, 7 x 10-7 nm, 6 x 10-7 nm, 5 x 10-7 nm, 4 x 10-7 nm, 3 x 10-7 nm, 2 x 10-7 nm, 1 x 10-7 nm, 9 x 10-8 nm, 8 x 10-8 nm, 7 x 10-8 nm, 6 x 10-8 nm, 5 x 10-8 nm, 4 x 10-8 nm, 3 x 10-8 nm, 2 x 10-8 nm, 1 x 10-8 nm, 9 x 10-9 nm, 8 x 10-9 nm, 7 x 10-9 nm, 6 x 10-9 nm, 5 x 10-9 nm, 4 x 10-9 nm, 3 x 10-9 nm, 2 x 10-9 nm, 1 x 10-9 nm, 9 x 10-10 nm, 8 x 10-10 nm, 7 x 10-10 nm, 6 x 10-10 nm, 5 x 10-10 nm, 4 x 10-10 nm, 3 x 10-10 nm, 2 x 10-10 nm, 1 x 10-10 nm, 9 x 10-11 nm, 8 x 10-11 nm, 7 x 10-11 nm, 6 x 10-11 nm, 5 x 10-11 nm, 4 x 10-11 nm, 3 x 10-11 nm, 2 x 10-11 nm, 1 x 10-11 nm, 9 x 10-12 nm, 8 x 10-12 nm, 7 x 10-12 nm, 6 x 10-12 nm, 5 x 10-12 nm, 4 x 10-12 nm, 3 x 10-12 nm, 2 x 10-12 nm, 1 x 10-12 nm, 9 x 10-13 nm, 8 x 10-13 nm, 7 x 10-13 nm, 6 x 10-13 nm, 5 x 10-13 nm, 4 x 10-13 nm, 3 x 10-13 nm, 2 x 10-13 nm, 1 x 10-13 nm, 9 x 10-14 nm, 8 x 10-14 nm, 7 x 10-14 nm, 6 x 10-14 nm, 5 x 10-14 nm, 4 x 10-14 nm, 3 x 10-14 nm, 2 x 10-14 nm, 1 x 10-14 nm, 9 x 10-15 nm, 8 x 10-15 nm, 7 x 10-15 nm, 6 x 10-15 nm, 5 x 10-15 nm, 4 x 10-15 nm, 3 x 10-15 nm, 2 x 10-15 nm, 1 x 10-15 nm, or less. The lasers may emit light having a bandwidth that is within a range defined by any two of the preceding values. [0149] The light sources may be configured to emit light tuned to one or more magic wavelengths corresponding to the plurality of atoms. A magic wavelength corresponding to an atom may comprise any wavelength of light that gives rise to equal or nearly equal polarizabilities of the first and second atomic states. The magic wavelengths for a transition between the first and second atomic states may be determined by calculating the wavelength- dependent polarizabilities of the first and second atomic states and finding crossing points. Light tuned to such a magic wavelength may give rise to equal or nearly equal differential light shifts in the first and second atomic states, regardless of the intensity of the light emitted by the light sources. This may effectively decouple the first and second atomic states from motion of the atoms. The magic wavelengths may utilize one or more scalar or tensor light shifts. The scalar or tensor light shifts may depend on magnetic sublevels within the first and second atomic states. [0150] For instance, group III atoms and metastable states of alkaline earth or alkaline earth-like atoms may possess relatively large tensor shifts whose angle relative to an applied magnetic field may be tuned to cause a situation in which scalar and tensor shifts balance and give a zero
WSGR Docket No.55436-729.601 or near zero differential light shift between the first and second atomic states. The angle θ may be tuned by selecting the polarization of the emitted light. For instance, when the emitted light is linearly polarized, the total polarizability α may be written as a sum of the scalar component αscalar and the tensor component αtensor:
[0151] By choosing θ appropriately, the polarizability of the first and second atomic states may be chosen to be equal or nearly equal, corresponding to a zero or near zero differential light shift and the motion of the atoms may be decoupled. [0152] The light sources may be configured to direct light to one or more optical modulators (OMs) configured to generate the plurality of optical trapping sites. For instance, the optical trapping unit may comprise an OM 314 configured to generate the plurality of optical trapping sites. Although depicted as comprising one OM in FIG.4A, the optical trapping unit may comprise any number of OMs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more OMs or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 OMs. The OMs may comprise one or more digital micromirror devices (DMDs). The OMs may comprise one or more liquid crystal devices, such as one or more liquid crystal on silicon (LCoS) devices. The OMs may comprise one or more spatial light modulators (SLMs). The OMs may comprise one or more acousto-optic deflectors (AODs) or acousto-optic modulators (AOMs). The OMs may comprise one or more electro- optic deflectors (EODs) or electro-optic modulators (EOMs). [0153] The OM may be optically coupled to one or more optical element to generate a regular array of optical trapping sites. For instance, the OM may be optically coupled to optical element 319, as shown in FIG.4A. The optical elements may comprise lenses or microscope objectives configured to re-direct light from the OMs to form a regular rectangular grid of optical trapping sites. [0154] For instance, as shown in FIG.4A, the OM may comprise an SLM, DMD, or LCoS device. The SLM, DMD, or LCoS device may be imaged onto the back focal plane of the microscope objectives. This may allow for the generation of an arbitrary configuration of optical trapping sites in two or three dimensions. [0155] Alternatively or in addition, the OMs may comprise first and second AODs. The active regions of the first and second AODs may be imaged onto the back focal plane of the microscope objectives. The output of the first AOD may be optically coupled to the input of the second AOD. In this manner, the second AOD may make a copy of the optical output of the first AOD. This may allow for the generation of optical trapping sites in two or three dimensions. [0156] Alternatively or in addition, the OMs may comprise static optical elements, such as one or more microlens arrays or holographic optical elements. The static optical elements may be
WSGR Docket No.55436-729.601 imaged onto the back focal plane of the microscope objectives. This may allow for the generation of an arbitrary configuration of optical trapping sites in two or three dimensions. [0157] The optical trapping unit may comprise one or more imaging units configured to obtain one or more images of a spatial configuration of the plurality of atoms trapped within the optical trapping sites. For instance, the optical trapping unit may comprise imaging unit 315. Although depicted as comprising a single imaging unit in FIG.4A, the optical trapping unit may comprise any number of imaging units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more imaging units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 imaging units. The imaging units may comprise one or more lens or objectives. The imaging units may comprise one or more PMTs, photodiodes, avalanche photodiodes, phototransistors, reverse-biased LEDs, CCDs, or CMOS cameras. The imaging unit may comprise one or more fluorescence detectors. The images may comprise one or more fluorescence images, single-atom fluorescence images, absorption images, single-atom absorption images, phase contrast images, or single-atom phase contrast images. [0158] The optical trapping unit may comprise one or more spatial configuration artificial intelligence (AI) units configured to perform one or more AI operations to determine the spatial configuration of the plurality of atoms trapped within the optical trapping sites based on the images obtained by the imaging unit. For instance, the optical trapping unit may comprise spatial configuration AI unit 316. Although depicted as comprising a single spatial configuration AI unit in FIG.4A, the optical trapping unit may comprise any number of spatial configuration AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial configuration AI units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial configuration AI units. The AI operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein. [0159] The optical trapping unit may comprise one or more atom rearrangement units configured to impart an altered spatial arrangement of the plurality of atoms trapped with the optical trapping sites based on the one or more images obtained by the imaging unit. For instance, the optical trapping unit may comprise atom rearrangement unit 317. Although depicted as comprising a single atom rearrangement unit in FIG.4A, the optical trapping unit may comprise any number of atom rearrangement units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more atom rearrangement units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 atom rearrangement units. [0160] The optical trapping unit may comprise one or more spatial arrangement artificial intelligence (AI) units configured to perform one or more AI operations to determine the altered spatial arrangement of the plurality of atoms trapped within the optical trapping sites based on the images obtained by the imaging unit. For instance, the optical trapping unit may comprise
WSGR Docket No.55436-729.601 spatial arrangement AI unit 318. Although depicted as comprising a single spatial arrangement AI unit in FIG.4A, the optical trapping unit may comprise any number of spatial arrangement AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more spatial arrangement AI units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 spatial arrangement AI units. The AI operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein. [0161] In some cases, the spatial configuration AI units and the spatial arrangement AI units may be integrated into an integrated AI unit. The optical trapping unit may comprise any number of integrated AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more integrated AI units, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 integrated AI units. [0162] The atom rearrangement unit may be configured to alter the spatial arrangement in order to obtain an increase in a filling factor of the plurality of optical trapping sites. A filling factor may be defined as a ratio of the number of computationally active optical trapping sites occupied by one or more atoms to the total number of computationally active optical trapping sites available in the optical trapping unit or in a portion of the optical trapping unit. For instance, initial loading of atoms within the computationally active optical trapping sites may give rise to a filling factor of less than 100%, 90%, 80%, 70%, 60%, 50%, or less, such that atoms occupy fewer than 100%, 90%, 70%, 60%, 50%, or less of the available computationally active optical trapping sites, respectively. It may be desirable to rearrange the atoms to achieve a filling factor of at least about 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing the imaging information obtained by the imaging unit, the atom rearrangement unit may attain a filling factor of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The atom rearrangement unit may attain a filling factor of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The atom rearrangement unit may attain a filling factor that is within a range defined by any two of the preceding values. [0163] By way of example, FIG.4C shows an example of an optical trapping unit that is partially filled with atoms. As depicted in FIG.4C, initial loading of atoms within the optical trapping sites may give rise to a filling factor of 44.4% (4 atoms filling 9 available optical trapping sites). By moving atoms from different regions of the optical trapping unit (not shown in FIG.4C) to unoccupied optical trapping sites or by moving atoms from an atom reservoir described herein, a much higher filling factor may be obtained, as shown in FIG.4D.
WSGR Docket No.55436-729.601 [0164] FIG.4D shows an example of an optical trapping unit that is completely filled with atoms. As depicted in FIG.4D, fifth atom 312e, sixth atom 312f, seventh atom 312g, eighth atom 312h, and ninth atom 312i may be moved to fill unoccupied optical trapping sites. The fifth, sixth, seventh, eighth, and ninth atoms may be moved from different regions of the optical trapping unit (not shown in FIG.4C) or by moving atoms from an atom reservoir described herein. Thus, the filling factor may be substantially improved following rearrangement of atoms within the optical trapping sites. For instance, a filling factor of up to 100% (such 9 atoms filling 9 available optical trapping sites, as shown in FIG.4D) may be attained. [0165] Atom rearrangement may be performed by (i) acquiring an image of the optical trapping unit, identifying filled and unfilled optical trapping sites, (ii) determining a set of moves to bring atoms from filled optical trapping sites to unfilled optical trapping sites, and (iii) moving the atoms from filled optical trapping sites to unfilled optical trapping sites. Operations (i), (ii), and (iii) may be performed iteratively until a large filling factor is achieved. Operation (iii) may comprise translating the moves identified in operation (ii) to waveforms that may be sent to an arbitrary waveform generator (AWG) and using the AWG to drive AODs to move the atoms. The set of moves may be determined using the Hungarian algorithm described in W. Lee et al, “Defect-Free Atomic Array Formation Using Hungarian Rearrangement Algorithm,” Physical Review A 95, 053424 (2017), which is incorporated herein by reference in its entirety for all purposes. Example of Electromagnetic Delivery Units [0166] FIG.5 shows an example of an electromagnetic delivery unit 320. The electromagnetic delivery unit may be configured to apply electromagnetic energy to one or more atoms of the plurality of atoms, as described herein. The electromagnetic delivery unit may comprise one or more light sources, such as any light source described herein. The electromagnetic energy may comprise optical energy. The optical energy may comprise any repetition rate, pulse energy, average power, wavelength, or bandwidth described herein. [0167] The electromagnetic delivery unit may comprise one or more microwave or radiofrequency (RF) energy sources, such as one or more magnetrons, klystrons, traveling-wave tubes, gyrotrons, field-effect transistors (FETs), tunnel diodes, Gunn diodes, impact ionization avalanche transit-time (IMPATT) diodes, or masers. The electromagnetic energy may comprise microwave energy or RF energy. The RF energy may comprise one or more wavelengths of at least about 1 millimeter (mm), 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, 9 mm, 10 mm, 20 mm, 30 mm, 40 mm, 50 mm, 60 mm, 70 mm, 80 mm, 90 mm, 100 mm, 200 mm, 300 mm, 400 mm, 500 mm, 600 mm, 700 mm, 800 mm, 900 mm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7
WSGR Docket No.55436-729.601 m, 8 m, 9 m, 10 m, 20 m, 30 m, 40 m, 50 m, 60 m, 70 m, 80 m, 90 m, 100 m, 200 m, 300 m, 400 m, 500 m, 600 m, 700 m, 800 m, 900 m, 1 kilometer (km), 2 km, 3 km, 4 km, 5 km, 6 km, 7 km, 8 km, 9 km, 10 km, or more. The RF energy may comprise one or more wavelengths of at most about 10 km, 9 km, 8 km, 7 km, 6 km, 5 km, 4 km, 3 km, 2 km, 1 km, 900 m, 800 m, 700 m, 600 m, 500 m, 400 m, 300 m, 200 m, 100 m, 90 m, 80 m, 70 m, 60 m, 50 m, 40 m, 30 m, 20 m, 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 900 mm, 800 mm, 700 mm, 600 mm, 500 mm, 400 mm, 300 mm, 200 mm, 100 mm, 90 mm, 80 mm, 70 mm, 60 mm, 50 mm, 40 mm, 30 mm, 20 mm, 10 mm, 9 mm, 8 mm, 7 mm, 6 mm, 5 mm, 4 mm, 3 mm, 2 mm, 1 mm, or less. The RF energy may comprise one or more wavelengths that are within a range defined by any two of the preceding values. [0168] The RF energy may comprise an average power of at least about 1 microwatt (µW), 2 µW, 3 µW, 4 µW, 5 µW, 6 µW, 7 µW, 8 µW, 9 µW, 10 µW, 20 µW, 30 µW, 40 µW, 50 µW, 60 µW, 70 µW, 80 µW, 90 µW, 100 µW, 200 µW, 300 µW, 400 µW, 500 µW, 600 µW, 700 µW, 800 µW, 900 µW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 Watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. The RF energy may comprise an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 µW, 800 µW, 700 µW, 600 µW, 500 µW, 400 µW, 300 µW, 200 µW, 100 µW, 90 µW, 80 µW, 70 µW, 60 µW, 50 µW, 40 µW, 30 µW, 20 µW, 10 µW, 9 µW, 8 µW, 7 µW, 6 µW, 5 µW, 4 µW, 3 µW, 2 µW, 1 µW, or less. The RF energy may comprise an average power that is within a range defined by any two of the preceding values. [0169] The electromagnetic delivery unit may comprise one or more light sources, such as any light source described herein. For instance, the electromagnetic delivery unit may comprise light source 221. Although depicted as comprising a single light source in FIG.5, the electromagnetic delivery unit may comprise any number of light sources, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more light sources or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 light sources. [0170] The light sources may be configured to direct light to one or more OMs configured to selectively apply the electromagnetic energy to one or more atoms of the plurality of atoms. For instance, the electromagnetic delivery unit may comprise OM 322. Although depicted as
WSGR Docket No.55436-729.601 comprising a single OM in FIG.5, the electromagnetic delivery unit may comprise any number of OMs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more OMs or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 OMs. The OMs may comprise one or more SLMs, AODs, or AOMs. The OMs may comprise one or more DMDs. The OMs may comprise one or more liquid crystal devices, such as one or more LCoS devices. [0171] The electromagnetic delivery unit may comprise one or more electromagnetic energy artificial intelligence (AI) units configured to perform one or more AI operations to selectively apply the electromagnetic energy to the atoms. For instance, the electromagnetic delivery unit may comprise AI unit 323. Although depicted as comprising a single AI unit in FIG.5, the electromagnetic delivery unit may comprise any number of AI units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more AI units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 AI units. The AI operations may comprise any machine learning (ML) or reinforcement learning (RL) operations described herein. [0172] The electromagnetic delivery unit may be configured to apply one or more single-qubit operations (such as one or more single-qubit gate operations) on the qubits described herein. The electromagnetic delivery unit may be configured to apply one or more two-qubit operations (such as one or more two-qubit gate operations) on the two-qubit units described herein. Each single-qubit or two-qubit operation may comprise a duration of at least about 10 nanoseconds (ns), 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1 microsecond (µs), 2 µs, 3 µs, 4 µs, 5 µs, 6 µs, 7 µs, 8 µs, 9 µs, 10 µs, 20 µs, 30 µs, 40 µs, 50 µs, 60 µs, 70 µs, 80 µs, 90 µs, 100 µs, or more. Each single-qubit or two-qubit operation may comprise a duration of at most about 100 µs, 90 µs, 80 µs, 70 µs, 60 µs, 50 µs, 40 µs, 30 µs, 20 µs, 10 µs, 9 µs, 8 µs, 7 µs, 6 µs, 5 µs, 4 µs, 3 µs, 2 µs, 1 µs, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, or less. Each single-qubit or two-qubit operation may comprise a duration that is within a range defined by any two of the preceding values. The single-qubit or two-qubit operations may be applied with a repetition frequency of at least 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1,000 kHz, or more. The single-qubit or two-qubit operations may be applied with a repetition frequency of at most 1,000 kHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. The single-qubit or two-qubit operations may be applied with a repetition frequency that is within a range defined by any two of the preceding values.
WSGR Docket No.55436-729.601 [0173] The electromagnetic delivery unit may be configured to apply one or more single-qubit operations by inducing one or more Raman transitions between a first qubit state and a second qubit state described herein. The Raman transitions may be detuned from an intermediate metastable state or an intermediate excited state, described herein. For instance, the Raman transitions may be detuned by at least about 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 MHz, 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1 GHz, or more. The Raman transitions may be detuned by at most about 1 GHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, or less. The Raman transitions may be detuned by a value that is within a range defined by any two of the preceding values. [0174] Raman transitions may be induced on individually selected atoms using one or more spatial light modulators (SLMs) or acousto-optic deflectors (AODs) to impart a deflection angle and/or a frequency shift to a light beam based on an applied radiofrequency (RF) signal. The SLM or AOD may be combined with an optical conditioning system that images the SLM or AOD active region onto the back focal plane of a microscope objective. The microscope objective may perform a spatial Fourier transform on the optical field at the position of the SLM or AOD. As such, angle (which may be proportional to RF frequency) may be converted into position. For example, applying a comb of radio frequencies to an AOD may generate a linear array of spots at a focal plane of the objective, with each spot having a finite extent determined by the characteristics of the optical conditioning system (such as the point spread function of the optical conditioning system). [0175] To perform a Raman transition on a single atom with a single SLM or AOD, a pair of frequencies may be applied to the SLM or AOD simultaneously. The two frequencies of the pair may have a frequency difference that matches or nearly matches the splitting energy between the first and second qubit states. For instance, the frequency difference may differ from the splitting energy by at most about 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz,
WSGR Docket No.55436-729.601 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The frequency difference may differ from the splitting energy by at least about 1 Hz, 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 MHz, or more. The frequency difference may differ from the splitting energy by about 0 Hz. The frequency difference may differ from the splitting energy by a value that is within a range defined by any two of the preceding values. The optical system may be configured such that the position spacing corresponding to the frequency difference is not resolved and such that light at both of the two frequencies interacts with a single atom. [0176] The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension of at least about 10 nm, 50 nm, 75 nm, 100 nm, 125 nm, 150 nm, 175 nm, 200 nm, 225 nm, 250 nm, 275 nm, 300 nm, 325 nm, 350 nm, 375 nm, 400 nm, 425 nm, 450 nm, 475 nm, 500 nm, 525 nm, 550 nm, 575 nm, 600 nm, 625 nm, 650 nm, 675 nm, 700 nm, 725 nm, 750 nm, 775 nm, 800 nm, 825 nm, 850 nm, 875 nm, 900 nm, 925 nm, 950 nm, 975 nm, 1 micrometer (µm), 1.5 µm, 2 µm, 2.5 µm 3 µm, 3.5 µm, 4 µm, 4.5 µm, 5 µm, 5.5 µm, 6 µm, 6.5 µm, 7 µm, 7.5 µm, 8 µm, 8.5 µm, 9 µm, 9.5 µm, 10 µm, or more. The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension of at most about 10 µm, 9.5 µm, 9 µm, 8.5 µm, 8 µm, 7.5 µm, 7 µm, 6.5 µm, 6 µm, 5.5 µm, 5 µm, 4.5 µm, 4 µm, 3.5 µm, 3 µm, 2.5 µm, 2 µm, 1.5 µm, 1 µm, 975 nm, 950 nm, 925 nm, 900 nm, 875 nm, 850 nm, 825 nm, 800 nm, 775 nm, 750 nm, 725 nm, 700 nm, 675 nm, 650 nm, 625 nm, 600 nm, 575 nm, 550 nm, 525 nm, 500 nm, 475 nm, 450 nm, 425 nm, 400 nm, 375 nm, 350 nm, 325 nm, 300 nm, 275 nm, 250 nm, 225 nm, 200 nm, 175 nm, 150 nm, 125 nm, 100 nm, 75 nm, 25 nm, 10 nm, or less. The electromagnetic delivery units may be configured to provide a beam with a characteristic dimension as defined by any two of the proceeding values. For example, the beam can have a characteristic dimension of about 1.5 micrometers to about 2.5 micrometers. Examples of characteristic dimensions include, but are not limited to, a Gaussian beam waist, the full width at half maximum (FWHM) of the beam size, the beam diameter, the 1/e2 width, the D4^ width, the D86 width, and the like. For example, the beam may have a Gaussian beam waist of at least about 1.5 micrometers. [0177] The characteristic dimension of the beam may be bounded at the low end by the size of the atomic wavepacket of an optical trapping site. For example, the beam can be formed such that the intensity variation of the beam over the trapping site is sufficiently small as to be
WSGR Docket No.55436-729.601 substantially homogeneous over the trapping site. In this example, the beam homogeneity can improve the fidelity of a qubit in the trapping site. The characteristic dimension of the beam may be bounded at the high end by the spacing between trapping sites. For example, a beam can be formed such that it is small enough that the effect of the beam on a neighboring trapping site/atom is negligible. In this example, the effect may be negligible if the effect can be minimized by techniques such as, for example, composite pulse engineering. The characteristic dimension may be different from a maximum achievable resolution of the system. For example, a system can have a maximum resolution of 700 nm, but the system may be operated at 1.5 micrometers. In this example, the value of the characteristic dimension may be selected to optimize the performance of the system in view of the considerations described elsewhere herein. The characteristic dimension may be invariant for different maximally achievable resolutions. For example, a system with a maximum resolution of 500 nm and a system with a maximum resolution of 2 micrometers may both be configured to operate at a characteristic dimension of 2 micrometers. In this example, 2 micrometers may be the optimal resolution based on the size of the trapping sites. Example of Integrated Optical Trapping Units and Electromagnetic Delivery Units [0178] The optical trapping units and electromagnetic delivery units described herein may be integrated into a single optical system. A microscope objective may be used to deliver electromagnetic radiation generated by an electromagnetic delivery unit described herein and to deliver light for trapping atoms generated by an optical trapping unit described herein. Alternatively or in addition, different objectives may be used to deliver electromagnetic radiation generated by an electromagnetic delivery unit and to deliver light from trapping atoms generated by an optical trapping unit. [0179] A single SLM or AOD may allow the implementation of qubit operations (such as any single-qubit or two-qubit operations described herein) on a linear array of atoms. Alternatively or in addition, two separate SLMs or AODs may be configured to each handle light with orthogonal polarizations. The light with orthogonal polarizations may be overlapped before the microscope objective. In such a scheme, each photon used in a two-photon transition described herein may be passed to the objective by a separate SLM or AOD, which may allow for increased polarization control. Qubit operations may be performed on a two-dimensional arrangement of atoms by bringing light from a first SLM or AOD into a second SLM or AOD that is oriented substantially orthogonally to the first SLM or AOD via an optical relay. Alternatively or in addition, qubit operations may be performed on a two-dimensional arrangement of atoms by using a one-dimensional array of SLMs or AODs.
WSGR Docket No.55436-729.601 [0180] The stability of qubit gate fidelity may be improved by maintaining overlap of light from the various light sources described herein (such as light sources associated with the optical trapping units or electromagnetic delivery units described herein). Such overlap may be maintained by an optical subsystem that measures the direction of light emitted by the various light sources, allowing closed-loop control of the direction of light emission. The optical subsystem may comprise a pickoff mirror located before the microscope objective. The pickoff mirror may be configured to direct a small amount of light to a lens, which may focus a collimated beam and convert angular deviation into position deviation. A position-sensitive optical detector, such as a lateral-effect position sensor or quadrant photodiode, may convert the position deviation into an electronic signal and information about the deviation may be fed into a compensation optic, such as an active mirror. [0181] The stability of qubit gate manipulation may be improved by controlling the intensity of light from the various light sources described herein (such as light sources associated with the optical trapping units or electromagnetic delivery units described herein). Such intensity control may be maintained by an optical subsystem that measures the intensity of light emitted by the various light sources, allowing closed-loop control of the intensity. Each light source may be coupled to an intensity actuator, such as an intensity servo control. The actuator may comprise an acousto-optic modulator (AOM) or electro-optic modulator (EOM). The intensity may be measured using an optical detector, such as a photodiode or any other optical detector described herein. Information about the intensity may be integrated into a feedback loop to stabilize the intensity. Example of State Preparation Units [0182] FIG.6 shows an example of a state preparation unit 350. The state preparation unit may be configured to prepare a state of the plurality of atoms, as described herein. The state preparation unit may be coupled to the optical trapping unit and may direct atoms that have been prepared by the state preparation unit to the optical trapping unit. The state preparation unit may be configured to cool the plurality of atoms. The state preparation unit may be configured to cool the plurality of atoms prior to trapping the plurality of atoms at the plurality of optical trapping sites. The state preparation unit may be configured to promote the plurality of atoms from a ground-state to a metastable manifold, described herein. The state preparation unit may be configured to promote the plurality of atoms from a ground-state to a metastable manifold using one or more light sources, described herein. [0183] The state preparation unit may comprise one or more Zeeman slowers. For instance, the state preparation unit may comprise a Zeeman slower 351. Although depicted as comprising a
WSGR Docket No.55436-729.601 single Zeeman slower in FIG.6, the state preparation may comprise any number of Zeeman slowers, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more Zeeman slowers or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 Zeeman slowers. The Zeeman slowers may be configured to cool one or more atoms of the plurality of atoms from a first velocity or distribution of velocities (such an emission velocity from an of an atom source, room temperature, liquid nitrogen temperature, or any other temperature) to a second velocity that is lower than the first velocity or distribution of velocities. [0184] The first velocity or distribution of velocities may be associated with a temperature of at least about 50 Kelvin (K), 60 K, 70 K, 80 K, 90 K, 100 K, 200 K, 300 K, 400 K, 500 K, 600 K, 700 K, 800 K, 900 K, 1,000 K, or more. The first velocity or distribution of velocities may be associated with a temperature of at most about 1,000 K, 900 K, 800 K, 700 K, 600 K, 500 K, 400 K, 300 K, 200 K, 100 K, 90 K, 80 K, 70 K, 60 K, 50 K, or less. The first velocity or distribution of velocities may be associated with a temperature that is within a range defined by any two of the preceding values. The second velocity may be at least about 1 meter per second (m/s), 2 m/s, 3 m/s, 4 m/s, 5 m/s, 6 m/s, 7 m/s, 8 m/s, 9 m/s, 10 m/s, or more. The second velocity may be at most about 10 m/s, 9 m/s, 8 m/s, 7 m/s, 6 m/s, 5 m/s, 4 m/s, 3 m/s, 2 m/s, 1 m/s, or less. The second velocity may be within a range defined by any two of the preceding values. The Zeeman slowers may comprise 1D Zeeman slowers. [0185] The state preparation unit may comprise a first magneto-optical trap (MOT) 352. The first MOT may be configured to cool the atoms to a first temperature. The first temperature may be at most about 10 millikelvin (mK), 9 mK, 8 mK, 7 mK, 6 mK, 5 mK, 4 mK, 3 mK, 2 mK, 1 mK, 0.9 mK, 0.8 mK, 0.7 mK, 0.6 mK, 0.5 mK, 0.4 mK, 0.3 mK, 0.2 mK, 0.1 mK, or less. The first temperature may be at least about 0.1 mK, 0.2 mK, 0.3 mK, 0.4 mK, 0.5 mK, 0.6 mK, 0.7 mK, 0.8 mK, 0.9 mK, 1 mK, 2 mK, 3 mK, 4 mK, 5 mK, 6 mK, 7 mK, 8 mK, 9 mK, 10 mK, or more. The first temperature may be within a range defined by any two of the preceding values. The first MOT may comprise a 1D, 2D, or 3D MOT. [0186] The first MOT may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960
WSGR Docket No.55436-729.601 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm. [0187] The state preparation unit may comprise a second MOT 353. The second MOT may be configured to cool the atoms from the first temperature to a second temperature that is lower than the first temperature. The second temperature may be at most about 100 microkelvin (µK), 90 µK, 80 µK, 70 µK, 60 µK, 50 µK, 40 µK, 30 µK, 20 µK, 10 µK, 9 µK, 8 µK, 7 µK, 6 µK, 5 µK, 4 µK, 3 µK, 2 µK, 1 µK, 900 nanokelvin (nK), 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300 nK, 200 nK, 100 nK, or less. The second temperature may be at least about 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK, 700 nK, 800 nK, 900 nK, 1 µK, 2 µK, 3 µK, 4 µK, 5 µK, 6 µK, 7 µK, 8 µK, 9 µK, 10 µK, 20 µK, 30 µK, 40 µK, 50 µK, 60 µK, 70 µK, 80 µK, 90 µK, 100 µK, or more. The second temperature may be within a range defined by any two of the preceding values. The second MOT may comprise a 1D, 2D, or 3D MOT. [0188] The second MOT may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range
WSGR Docket No.55436-729.601 defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm. [0189] Although depicted as comprising two MOTs in FIG.6, the state preparation unit may comprise any number of MOTs, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more MOTs or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 MOTs. [0190] The state preparation unit may comprise one or more sideband cooling units or Sisyphus cooling units (such as a sideband cooling unit described in www.arxiv.org/abs/1810.06626 or a Sisyphus cooling unit described in www.arxiv.org/abs/1811.06014, each of which is incorporated herein by reference in its entirety for all purposes). For instance, the state preparation unit may comprise sideband cooling unit or Sisyphus cooling unit 354. Although depicted as comprising a single sideband cooling unit or Sisyphus cooling unit in FIG.6, the state preparation may comprise any number of sideband cooling units or Sisyphus cooling units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more sideband cooling units or Sisyphus cooling units, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 sideband cooling units or Sisyphus cooling units. The sideband cooling units or Sisyphus cooling units may be configured to use sideband cooling to cool the atoms from the second temperature to a third temperature that is lower than the second temperature. The third temperature may be at most about 10 µK, 9 µK, 8 µK, 7 µK, 6 µK, 5 µK, 4 µK, 3 µK, 2 µK, 1 µK, 900 nK, 800 nK, 700 nK, 600 nK, 500 nK, 400 nK, 300 nK, 200 nK, 100 nK, 90 nK, 80 nK, 70 nK, 60 nK, 50 nK, 40 nK, 30 nK, 20 nK, 10 nK, or less. The third temperature may be at most about 10 nK, 20 nK, 30 nK, 40 nK, 50 nK, 60 nK, 70 nK, 80 nK, 90 nK, 100 nK, 200 nK, 300 nK, 400 nK, 500 nK, 600 nK, 700 nK, 800 nK, 900 nK, 1 µK, 2 µK, 3 µK, 4 µK, 5 µK, 6 µK, 7 µK, 8 µK, 9 µK, 10 µK, or more. The third temperature may be within a range defined by any two of the preceding values. [0191] The sideband cooling units or Sisyphus cooling units may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm,
WSGR Docket No.55436-729.601 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm. [0192] The state preparation unit may comprise one or more optical pumping units. For instance, the state preparation unit may comprise optical pumping unit 355. Although depicted as comprising a single optical pumping unit in FIG.6, the state preparation may comprise any number of optical pumping units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more optical pumping units, or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 optical pumping units. The optical pumping units may be configured to emit light to optically pump the atoms from an equilibrium distribution of atomic states to a non-equilibrium atomic state. For instance, the optical pumping units may be configured to emit light to optically pump the atoms from an equilibrium distribution of atomic states to a single pure atomic state. The optical pumping units may be configured to emit light to optically pump the atoms to a ground atomic state or to any other atomic state. The optical pumping units may be configured to emit light to optically pump the atoms from a ground state to a metastable manifold, described herein. The optical pumping units may be configured to optically pump the atoms between any two atomic states. The optical pumping units may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm,
WSGR Docket No.55436-729.601 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm. [0193] The state preparation unit may comprise one or more coherent driving units. For instance, the state preparation unit may comprise coherent driving unit 356. Although depicted as comprising a coherent driving unit in FIG.6, the state preparation may comprise any number of coherent driving units, such as at least about 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or more coherent driving units or at most about 10, 9, 8, 7, 6, 5, 4, 3, 2, or 1 coherent driving units. The coherent driving units may be configured to coherently drive the atoms from the non-equilibrium state to the first or second atomic states described herein. Thus, the atoms may be optically pumped to an atomic state that is convenient to access (for instance, based on availability of light sources that emit particular wavelengths or based on other factors) and then coherently driven to atomic states described herein that are useful for performing quantum computations. The coherent driving units may be configured to induce a single photon transition between the non- equilibrium state and the first or second atomic state. The coherent driving units may be configured to induce a two-photon transition between the non-equilibrium state and the first or second atomic state. The two-photon transition may be induced using light from two light sources described herein (such as two lasers described herein). [0194] The coherent driving units may comprise one or more light sources (such as any light source described herein) configured to emit light. The light may comprise one or more wavelengths of at least about 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, or more. The light may comprise one or more wavelengths of at most about 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm,
WSGR Docket No.55436-729.601 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, or less. The light may comprise one or more wavelengths that are within a range defined by any two of the preceding values. For instance, the light may comprise one or more wavelengths that are within a range from 400 nm to 1,000 nm, 500 nm to 1,000 nm, 600 nm to 1,000 nm, 650 nm to 1,000 nm, 400 nm to 900 nm, 400 nm to 800 nm, 400 nm to 700 nm, 400 nm to 600 nm, 400 nm to 500 nm, 500 nm to 700 nm, or 650 nm to 700 nm. [0195] The coherent driving units may be configured to induce an RF transition between the non-equilibrium state and the first or second atomic state. The coherent driving units may comprise one or more electromagnetic radiation sources configured to emit electromagnetic radiation configured to induce the RF transition. For instance, the coherent driving units may comprise one or more RF sources (such as any RF source described herein) configured to emit RF radiation. The RF radiation may comprise one or more wavelengths of at least about 10 centimeters (cm), 20 cm, 30 cm, 40 cm, 50 cm, 60 cm, 70 cm, 80 cm, 90 cm, 1 meter (m), 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, 8 m, 9 m, 10 m, or more. The RF radiation may comprise one or more wavelengths of at most about 10 m, 9 m, 8 m, 7 m, 6 m, 5 m, 4 m, 3 m, 2 m, 1 m, 90 cm, 80 cm, 70 cm, 60 cm, 50 cm, 40 cm, 30 cm, 20 cm, 10 cm, or less. The RF radiation may comprise one or more wavelengths that are within a range defined by any two of the preceding values. Alternatively or in addition, the coherent driving units may comprise one or more light sources (such as any light sources described herein) configured to induce a two-photon transition corresponding to the RF transition. Example of Controllers [0196] The optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration AI units, spatial arrangement AI units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy AI units, atom reservoirs, atom movement units, or Rydberg excitation units may include one or more circuits or controllers (such as one or more electronic circuits or controllers) that is connected (for instance, by one or more electronic connections) to the optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration AI units, spatial arrangement AI units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy AI units, atom reservoirs, atom movement units, or Rydberg excitation units. The circuits or controllers may be configured to control the optical trapping units, electromagnetic delivery units, entanglement units, readout optical units, vacuum units, imaging units, spatial configuration AI units, spatial arrangement AI
WSGR Docket No.55436-729.601 units, atom rearrangement units, state preparation units, sideband cooling units, optical pumping units, coherent driving units, electromagnetic energy AI units, atom reservoirs, atom movement units, or Rydberg excitation units. Methods for Error Corrected Quantum Computation [0197] Systems and methods disclosed herein may generally relate to qubit loss during error correction. Within qubit loss during error correction, there may be at least two general pieces. Systems and methods described herein may be directed to detecting qubit loss without destroying the data stored on the qubits. Instead of appearing as a gate measurement error, if a qubit is lost, the data that would be there is absent rather than incorrect. Systems and methods disclosed herein may be directed to identifying when an error is caused by a missing qubit. Systems and methods described herein may also be directed to modifying the decoder to handle loss events. For example, the error correcting code may be directed to updating the calculation to address for error. In some cases, knowing about the error may be needed in order to implement error correcting code. However, in other cases, the error correcting code may be modified to account for missing data without explicit knowledge that qubit is lost. [0198] Systems and methods disclosed herein may not generally modify the topology of the underlying surface code. Systems and methods described herein may improve upon methods of detecting atom loss by compressing the underlying protocol. Systems and methods disclosed herein may allow for loss detection between cycles (or possibly less frequently) rather than after every gate. Systems and methods disclosed herein may address un-induced erasure errors in addition to or alternatively to gate induced erasure errors. [0199] Systems and methods of the present disclosure may improve upon other procedures at least because systems and methods of the present disclosure may not comprise or require operations that modify the topology of the underlying surface code. In some cases, the underlying surface code may be unchanged. Instead, a matching graph passed to a decoder algorithm may be updated to account for a predicted probability distribution of a lost qubit. Because the matching graph passed to the decoder is updated, the underlying decoder may also be unchanged. Because the decoder is unchanged, the error correcting code may also not be changed. Accordingly, methods and systems of the present disclosure may be used without changing the underlying surface code. Similarly, because changes to the underlying decoder and surface code are not needed, methods and systems of the present disclosure may be used with a wide variety of decoders and surface codes. [0200] In some cases, tracking syndrome measurements may be used to detect loss events (e.g., defects). See, e.g., Siegel, A. et al., Adaptive Surface Code for Quantum Error Correction in the
WSGR Docket No.55436-729.601 Presence of Temporary or Permanent Defects, arXiv:2211.08468v1 [quant-ph] 15 Nov 2022, available at https://arxiv.org/pdf/2211.08468.pdf, which is incorporated by reference herein in its entirety. [0201] In some cases, gate induced erasure errors may be addressed by error correction. See e.g., Wu, Y., et al., Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays, Nat. Comms. Vol 13, P.4657 (2022), which is incorporated by reference herein in its entirety. In the above, the atoms may still be present. Rather than addressing missing atoms or qubits, the above referenced application may turn gate errors into erasure errors. [0202] In some cases, noise structure in the hardware may be used for error correction. See e.g., Shay, K., et al., High threshold codes for neutral atom qubits with biased erasure errors, arXiv:2302.03063v1 [quant-ph] 6 Feb 2023), which is incorporated by reference herein in its entirety. Similar to Wu, the atoms may still be present. Rather than addressing missing atoms or qubits, the above referenced application may turn gate errors into erasure errors. Error Correction [0203] Quantum error correction is a procedure for encoding quantum information in a distributed manner across many quantum systems in such a way that the information to be stored is protected from localized errors on the constituent systems, provided that these errors are sufficiently sparse. In some cases, the information to be stored and the constituent systems are both two-level quantum systems, or qubits. The information to be protected is encoded across many physical qubits, forming one or more logical qubits. [0204] The process of detecting and correcting errors on the logical qubits amounts to measuring parities of a predetermined set of operators that act on the physical qubits and using the measured parities to diagnose and correct errors. In a simple example, a parity measurement checks the equality of two qubits to return a true or false answer, which can be used to determine whether a correction needs to occur. Additional measurements can be made for a system greater than two qubits. Since the physical qubits cannot be measured directly without collapsing the state of the logical system, these parities are measured using ancillary qubits (i.e., ancilla). Thus, there may be two types of physical qubits: data qubits on which the logical information is stored and ancilla qubits which are used to extract the desired parity checks. [0205] In practice, an error correction cycle consists of a sequence of gates to transfer parity values onto the ancilla qubits followed by measurement of the ancilla qubits. This process is known as syndrome extraction. Errors can occur at any point during the syndrome extraction process, including during readout of the ancilla qubits.
WSGR Docket No.55436-729.601 [0206] One method of error correction (Shor style) uses repeated rounds of syndrome extraction to overcome readout error and build confidence about the state of the system. The extracted syndrome information is then passed to a decoder to determine which errors have occurred and which corrections need to be applied. The decoding problem is commonly represented as a weighted graph or hypergraph. In this setting, each node in the graph corresponds to a collection of syndrome measurements. Such a collection of syndrome measurements is called a detector. Edges or hyperedges in the graph correspond to errors, and the weight of an edge corresponds to the likelihood of that error occurring. The occurrence of a given error may be expected cause the parity of all associated detectors to flip. The decoding problem can then be stated as follows: given a set of detectors (nodes) whose parities differ from those expected in the absence of error, determine the most likely set of physical errors (edges) that could cause the observed detections. Error Correction with Atom Loss [0207] Quantum computers based on trapped atoms may be subject to errors generated by loss of qubit. In trapped atom quantum computers, a qubit may comprise atom in an array. That atom may be a neutral atom or an ion. Error correcting code may generally employ repeated implementations of the circuit implementing the quantum computation. As the circuit is implemented and re-implemented statistics may be generated on what errors occurred. However, error correcting code implemented on systems with qubit loss may generally different than other systems. For example, a non-qubit loss error may be a gate error. Similarly, loss of coherence may be expressed as a gate error. In a gate error or an error that is similar to a gate error, there is a comparatively smaller set of possible error values because the result of a gate error is like a measurement of the system. In some cases, the atom loss rate may be similar or larger than the gate error rate, thus it may be helpful to provide improved methods of correcting for atom loss. [0208] FIG.7 is a flowchart of an example method 700 for error corrected quantum computation. In some cases, an error correcting code which accounts for qubit loss may comprise identifying that a qubit has been lost (710); replacing the qubit (720); reimplementing the qubit into the circuit which may be in the wrong state when it is replaced 730; and flagging measurements taken while the qubit was missing as untrustworthy (740). [0209] Referring to FIG.7, at an operation 710 of a method 700 of error correction with atom loss, atom loss can be detected. For example, atom loss may be detected at the end of each syndrome extraction cycle. Methods of detecting atom loss are described herein. At an operation 720 of a method 700 of error correction with atom loss, once a qubit is identified as lost it may be replaced with a new qubit. The new qubit may be, at least initially, in a random state.
WSGR Docket No.55436-729.601 [0210] At an operation 730 of a method 700 of error correction with atom loss, the qubit may be reimplemented into the circuit. In some cases, operation 730 comprises use of a decoder algorithm. The decoder algorithm may take in a graph and determine a set of edges. In some cases, operation 1030 comprises prior implementing the decoder algorithm, updating a matching graph passed to the decoder algorithm based on a predicted probability distribution of a lost qubit replaced in operation 1020. Methods of updating the decoder algorithm are described herein. [0211] In systems not subject to atom loss, the errors may be discrete Pauli errors on physical qubits. But when qubits are stored on atoms, the atoms—and therefore the qubits they contain— can be lost. The effect on syndrome extraction in the presence of loss depends on hardware details. For neutral atoms using Rydberg gates, the effect of atom loss manifests as a non- interaction instead of a two-qubit gate. Practically, two-qubit interactions between a lost and present atom may be treated as affecting an Identity gate on the present atom. [0212] In an example, systems and methods described herein may comprise a case in which a two-qubit interaction between a qubit and a lost qubit has an effect of a Pauli operation or identity operation on the qubit, as described in International Application PCT/US2024/018180, which is incorporated by reference in its entirety herein for all purposes. In such a case, a first atom A and second atom B may be neighboring qubits. The qubits may be trapped ion qubits. The qubits may be trapped atom qubits. In some cases, one qubit acquires a phase conditioned on state-selective excitation of the other. In some cases, the state selective excitation is from a state |1> to a state |r>. [0213] In some cases, a state |r> is a Rydberg state. In some cases, a state |r> is a Rydberg state of a neutral atom qubit. If at A is excited to a Rydberg state, then Atom B (if present) experiences a shift due to the Rydberg interaction. In some cases, an optical excitation may be tuned to a frequency difference between the |1> state and the Rydberg state. If Atom A is in state |1>, then Atom A is at least transiently driven to the Rydberg state and Atom B (if present) experiences a shift due to the Rydberg interaction. If Atom B is not present and Atom A is in state |1>, there is no Shift to Atom B. If Atom A is in state |0> and Atom B is present, then the energy gap is too large, and nothing happens to Atom A or Atom B. If Atom A is in state |0> and Atom B is not present, then the energy gap is still too large, and nothing happens to Atom A or Atom B (which isn’t present). Accordingly, two-qubit interactions between a lost and present atom may be treated as affecting an Identity gate on the present atom. While this example describes a case where the two-qubit interaction with a lost atom affects the Identity operation, methods and system of the present disclosure also work when the two-qubit interaction with a lost atom affects a Pauli operation. A Pauli operation may comprise a Pauli-X gate, a Pauli-Y
WSGR Docket No.55436-729.601 gate, or a Pauli-Z gate. For example, the Pauli-X gate is a single-qubit rotation the pi radians around the X-axis. For example, the Pauli-Y gate is a single-qubit rotation the pi radians around the Y-axis. For example, the Pauli-Z gate is a single-qubit rotation the pi radians around the Z- axis. A rotation about an axis of two pi radians is an Identity operation. [0214] The above works similarly if Atom A is also a lost qubit. In some cases, the qubit is a non-lost qubit. In some cases, the qubit is a lost qubit. For example, when the qubit is a lost qubit, a two-qubit gate between two lost qubits similarly affects the identity. Because the two- qubit operation between an atom a lost atom affects the identity. The protocol does not propagate errors (to first order) forward in time. For example, if the two-qubit operation is imperfect. The operation may propagate forward higher order errors in time. Examples of Identifying Qubit Loss [0215] In an aspect, the present disclosure provides at least two methods of identifying qubit loss; however, various methods of identifying qubit loss may be integrated into methods and systems of the present disclosure. An example method of implementing an error correcting code which accounts for atom loss may comprise implementing a plurality of SWAP gates. Another example method of implementing an error correcting code which accounts for atom loss may comprise a modified knock-knock protocol. Systems and methods for the implementation of a plurality of SWAP gates and modified knock-knock protocols are described in International Application PCT/US2024/018180, incorporated in its entirety herein for all purposes. Examples of Qubit Replacement [0216] Methods and systems of the present disclosure may replace qubits into a quantum circuit after a vacancy has been identified. In some cases, the qubit is an atomic qubit. In some cases, the qubit is atom trapped in a spatially distinct optical trapping site. Examples presented herein may be recite qubits comprising neutral atoms; however, the methods and systems herein may be combined with various types of qubits. [0217] In some examples, present techniques may be combined with methods for probabilistic, deterministic, or near-deterministic loading of optical or other traps, such as those disclosed herein. In some cases, atoms within the science region may or may not be rearranged as the science array is replenished. In some cases, atoms can be transferred between sites by optical tweezers. In some cases, atoms can be transferred between sites by optical lattices. In some cases, atoms can be transferred between sites by tunneling/hopping between sites. In some cases, atoms can be transferred between sites by autonomous stabilization techniques. [0218] In some cases, atom replacement is performed using one or both of a moving optical trap (e.g., moving optical lattice) or one or more optical tweezers. In some cases, an optical tweezer
WSGR Docket No.55436-729.601 may be used to move a single atom (e.g., pick and place) or a subset of atoms between arrays or within an array. In some cases, a moving optical trap can be used to translate or compress an array. A moving optical trap (e.g., moving optical lattice) may implement a tone to sweep atoms from one location to another. The atom movement units may be configured to move the one or more replacement atoms from the one or more atoms reservoirs to the one or more optical trapping sites. For instance, the one or more atom movement units may comprise one or more electrically tunable lenses, acousto-optic deflectors (AODs), or spatial light modulators (SLMs). [0219] The atom rearrangement unit may be configured to alter the spatial arrangement in order to obtain an increase in a filling factor of the plurality of optical trapping sites. A filling factor may be defined as a ratio of the number of computationally active optical trapping sites occupied by one or more atoms to the total number of computationally active optical trapping sites available in the optical trapping unit or in a portion of the optical trapping unit. For instance, initial loading of atoms within the computationally active optical trapping sites may give rise to a filling factor of less than 100%, 90%, 80%, 70%, 60%, 50%, or less, such that atoms occupy fewer than 100%, 90%, 70%, 60%, 50%, or less of the available computationally active optical trapping sites, respectively. It may be desirable to rearrange the atoms to achieve a filling factor of at least about 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing the imaging information obtained by the imaging unit, the atom rearrangement unit may attain a filling factor of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The atom rearrangement unit may attain a filling factor of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The atom rearrangement unit may attain a filling factor that is within a range defined by any two of the preceding values. Examples of Modifying Decoding Algorithms [0220] Systems and methods of the present disclosure may be used in connection with error correction methodologies for quantum computing systems. An error correcting scheme (e.g., an implementation of an error correcting code) of the present disclosure may comprise a decoder and an error correcting code. A decoder may decode which errors occurred on which qubits. Once identified, these errors can be tracked and the information used to correct any subsequent measurement outcomes using the classical control software. The methods of updating the decoder described herein may not depend on the type of atom, the type of qubit, the type of error
WSGR Docket No.55436-729.601 correction code, or the specific decoder used in the error correcting code. In some cases, an error correcting code may be of the class of stabilizer codes. If the two qubit-gate operation affects the Identity or a Pauli operation, then the matching graph passed to the decoder may be updated as described herein. [0221] Systems and methods of the present disclosure may be used with various error correcting codes. An error correcting code may be a Shor style code. For example, in a Shor style code, repeated rounds of syndrome extraction may be implemented to overcome readout error and build confidence about the state of the system. The extracted syndrome information is then passed to a decoder to determine which errors have occurred and which corrections need to be applied. [0222] Systems and methods of the present disclosure may be used with various stabilizer codes. A stabilizer code may be an error correcting code which uses stabilizers. A stabilizer code may be a class of error correcting code. The class of stabilizer codes may include toric codes, surface codes, etc. By repeatedly measuring a quantum system using a complete set of commuting stabilizers, the system may be forced into a simultaneous and unique eigenstate of all the stabilizers. One can measure the stabilizers without perturbing the system; when the measurement outcomes change, this corresponds to one or more qubit errors, and the quantum state is projected by the measurements onto a different stabilizer eigenstate. [0223] An error correcting code may comprise a topological code. The class of topological codes may overlap with the class of stabilizer codes. A topological code may comprise a surface code, a color code, a toric code, etc. A topological code may also be referred to as a homological code. A topological code may comprise an array or lattice of qubits arranged on a surface (or higher dimensional structure). Systems and methods of the present disclosure may not generally change the underlying topology of a topological code. [0224] Systems and methods disclosed herein may be used with various surface codes. A surface code may be implemented as a stabilizer code. For example, in the surface code literature, surface codes may comprise two types of qubits data qubits and measurement qubits (e.g., ancilla qubits). The data qubits may contain the information carried by the quantum circuit, whose error is to be corrected. The measurement qubits may be used to stabilize and manipulate the quantum state of the data qubit. In a surface code, the measurement qubits may comprise two types: measure-Z qubits and measure-X qubits. These two types of qubits may be called Z syndrome qubits and X-syndrome qubits respectively. The measure Z-qubits may measure the Z stabilizer. The measure X-qubits may measure the X stabilizer. In some cases, a surface code may be implemented with a decoder. In some cases, a surface code can address errors that occur
WSGR Docket No.55436-729.601 during a surface code cycle as long as the errors that occur during each surface code cycle can be identified. [0225] Systems and methods disclosed herein may employ surface codes. Surface codes disclosed herein may include, for example, variations upon the minimum-weight perfect matching algorithm to decode the surface code. However, many surface codes may be applicable to the systems and methods disclosed herein. A general description of surface codes is provided for example at Fowler, A. G., et al., Surface codes: Towards Practical Large-scale Quantum Computation, arXiv: 1208.0928 [quant-ph] 4 Aug 2012, available at https://arxiv.org/pdf/1208.0928.pdf, which is incorporated by reference herein in its entirety. [0226] Systems and methods disclosed herein may be used with various color codes. A color code may be implemented as a stabilizer code. For example, a color code may comprise a Steane code, etc. Systems and methods disclosed herein may be used with various Shor style codes, for example, a Bacon-shor code. A Shor style code may be implemented as a stabilizer code. Systems and methods disclosed herein may be used with various qLDPC codes, for example, hypergraph product codes. A qLDPC code may be implemented as a stabilizer code. [0227] Systems and methods disclosed herein may be used with various decoders. An error correcting scheme (e.g., an implementation of an error correcting code) of the present disclosure may comprise a decoder and an error correction code. A decoder may decode which errors occurred on which qubits. Once identified, these errors can be tracked and the information used to correct subsequent measurement outcomes using the classical control software. Decoder algorithms may include, for example, minimum-weight perfect matching, union find, tensor network decoder, belief propagation with ordered statistics decoder, maximum likelihood decoder, and look up table decoders. Methods and systems of the present disclosure may be integrated with variations on the minimum-weight perfect matching such as sparse bloom and fusion blossom. A decoder may take in a matching graph. Systems and method of the present disclosure may update the matching graph passed to the decoder to account for a lost qubit. [0228] In some cases, qubit loss may involve modification to surface code techniques that do not experience qubit loss errors. For example, error correcting code which does not account for qubit loss errors may keep track of a particular qubit changing from 1 to 0 or 0 to 1 unexpectedly. If a qubit has been lost, there is no change in state; instead, there is no value to measure. [0229] Modifying the decoding algorithm may be a sub-operation of an operation for reimplementing the qubit into the circuit. A qubit reimplementing operation may comprise an embodiment, variation, or example of operation 730 of the method 700. In some cases,
WSGR Docket No.55436-729.601 modifying the decoding algorithm may be performed subsequent to or during a reimplementing operation such as operation 730 of a method 700. [0230] To augment the decoding algorithm, systems and methods disclosed herein may update existing decoders to incorporate the change in error type. In some cases, systems and methods disclosed herein may update the matching graph passed to a decoder. In some cases, systems and methods disclosed herein may update the matching graph passed to a minimum-weight perfect matching decoder algorithm or any other decoder algorithm which takes in a matching graph. [0231] In some examples, to update the matching graph, each node in the graph corresponds to a change-of-value of a particular stabilizer. Certain nodes are connected by edges corresponding to possible physical errors. These edges are weighted based on the likelihood of that particular error occurring. When an atom is lost and then replaced, the loss may be treated like a gate error that occurs with a probability of 50%. The procedure may change slightly if data vs. ancilla qubits are lost. [0232] Systems and methods described herein may be combined with systems and methods for reimplementing a qubit, as described in International Application PCT/US2024/018180, incorporated herein in its entirety for all purposes. Examples of Measurement Operation [0233] At an operation 740 of a method 700 of error correction with atom loss, measurements taken while the qubit was missing may be flagged as untrustworthy. In some cases, operation 740 comprises flagging measurement taken during a window of time that includes a time when the qubit was missing as untrustworthy. For example, a window of time may comprise a round of syndrome measurements. It may not be necessary to know exactly which measurements were taken while the atom was lost, only what set of measurements were taken during a window of time that includes a lost atom. A flagging operation may comprise, prior to flagging a measurement, performing a measurement of one or more qubits. The measurement may result in an emission of a photon. In some cases, the measurement may be state selective. For example, a measurement may selectively probe either a |0> state or a |1> state. After a round of measurement, it may be possible to know whether a measured atom is lost. In some cases, measurement of ancilla atoms during an error correction protocol may indicate whether an ancilla atom is lost. An identification operation in 710 (e.g., swap gates, a knock-knock protocol, etc.) may be performed in order to determine if a qubit is missing within a set of qubits including a data qubit without measurement of the data qubit. [0234] In systems and methods of the present disclosure, the fact of a missing qubit may not be immediately heralded. For example, it may become apparent that qubit is lost after completing a
WSGR Docket No.55436-729.601 round of syndrome extraction, rather than immediately upon taking a measurement implicating a lost qubit. Once the round of syndrome extraction is complete, it may become apparent that there was a qubit loss, in order to proceed with the calculation in may be beneficial to flag a series of measurements taken during a window of time that includes a time when the qubit was missing. Each of these measurements may be flagged as untrustworthy. In some cases, the series of time which includes the flagged qubit may not be limited to the time in which the qubit is definitively lost. The series of time which includes the flagged qubit may include at least the time with the qubit was lost. [0235] Advantageously, a qubit loss may be identified before measurement of the data qubits (and after a round of syndrome extraction). Since the data qubits have not yet been measured, it may be possible to continue on with a quantum circuit after replacing a lost qubit. The circuit may be adapted to retake or restart portions of the calculation implicating the lost qubit. As a consequence, systems and methods disclosed herein may allow for loss detection between cycles (or possibly less frequently) rather than after every gate. [0236] For example, the error correcting code may be directed to updating the calculation to address for error. In some cases, knowing about the error may be needed in order to implement error correcting code. However, in other cases, the error correcting code may be modified to account for missing data without explicit knowledge that qubit is lost. Examples of Systems for Error Corrected Quantum Computing [0237] FIG.8 shows a system for error corrected quantum computing that is programmed or otherwise configured to implement methods provided herein. A system for error corrected computing may comprise an error correcting code. The present disclosure provides systems for error corrected quantum computing. The system may comprise an error correction code. An implementation of the error correcting code may comprise a decoder. The decoder may be configured to receive a matching graph and to determine a set of edges, and the matching graph received by the decoder may be updated based on a predicted probability distribution of a lost qubit. In some cases, the error correction code comprises an operation in which a two-qubit interaction between a qubit and a lost qubit has an effect of a Pauli operation or an identity operation on the qubit. In some cases, the qubit is a non-lost qubit. In some cases, the qubit is a lost qubit. [0238] In some cases, a system for error corrected quantum computing may comprise a non- classical computing system 850. The non-classical computing system may be quantum computing system. The non-classical computing system may be a trapped atom quantum computing system. The trapped atom quantum computing system may comprise: an atom
WSGR Docket No.55436-729.601 movement unit, an atom rearrangement unit, an optical trapping unit, an imaging unit, an optical pumping unit, an entanglement unit, a Rydberg unit, a non-classical computation unit, an electromagnetic deliver unit, or any combination thereof. [0239] In some cases, the non-classical computing system may comprise a plurality of qubits. [0240] In some cases, the non-classical computing system may comprise one or more electromagnetic delivery units. The electromagnetic delivery units may be configured to produce electromagnetic excitations to perform various operations, such as for example, atom movement, atom rearrangement, optical trapping, imaging, and various operations on atoms that may comprises portions of a nonclassical computation. Portions of a non-classical computation on trapped atoms may comprise optical pumping, entanglement operations, Rydberg operations, gate operations (e.g., one qubit operations, two qubit operations, etc.). In some cases, an atom movement unit may comprise an atom rearrangement unit. The components of a non-classical computing system are discussed herein above with respect to the operations they implement. [0241] In some cases, the system further comprises a non-classical computing system, wherein the non-classical computing system comprises trapped atom qubits. In some cases, the trapped atom qubits comprise neutral atom qubits. In some cases, the neutral atom qubits comprise a Group II element or a Group II-like element. In some cases, the Group II element or a Group II- like element comprises Ytterbium, Rubidium, Cesium, or Strontium. In some cases, the plurality of qubits comprises qubit states comprising nuclear spin states on the 1S0 manifold. In some cases, the two-qubit interaction comprises an excitation of a nuclear spin state of a neutral atom to a Rydberg state of the neutral atom. [0242] In some cases, the system further comprises a non-classical computing system, wherein the non-classical computing system comprises a plurality of qubits, wherein the plurality of qubits comprises atomic qubits, and wherein an atom replacement operation is implemented using optical tweezers. [0243] In some cases, the non-classical computing system may be configured to interact with a processor 801. The processor may be classical processing system. The processor may be digital processing system. [0244] In some cases, the system further comprises a processor configured to implement an error correcting code. In some cases, the processor is further configured to provide instructions to a non-classical computing system, wherein the non-classical computing system is configured to implement the instructions to: (i) identify that a qubit has been lost; (ii) replace the qubit; and (iii) reimplement the qubit into the circuit. In some cases, the processor is further configured to (iv) flag measurements taken while the qubit was missing as untrustworthy. In some cases, (i) comprises using a plurality of swap gates. In some cases, a swap gate within the plurality of
WSGR Docket No.55436-729.601 swap gates is implemented as a plurality of CNOT gates. In some cases, the processor is further configured to provide instructions to the non-classical computing system to measure alternating atoms in a lattice; perform the plurality of swap gates to transfer data stored on data qubits to ancilla qubits; and measure the swapped data qubits to identify one or more lost atoms. [0245] In some cases, (i) comprises using a modified knock-knock protocol, wherein the modified knock-knock protocol comprises: providing a first atom to be probed using a second atom, wherein the second atom is an ancilla qubit; preparing the second atom in a |+> state; applying a modified control-Z gate between the first atom and the second atom based on a Rydberg interaction; and rotating the second qubit back to a computational basis and performing a measurement. In some cases, (iii) comprises (A) use of a decoder algorithm, wherein the decoder algorithm takes in a graph and determines a set of edges. In some cases, prior to (A) the processor is further configured to update a matching graph passed to the decoder algorithm based on a predicted probability distribution of a lost qubit replaced in (ii). In some cases, (iii) comprises use of a minimum-weight perfect matching decoder algorithm. In some cases, the processor is further configured to update a matching graph passed to the minimum-weight perfect matching decoder algorithm based on a predicted probability distribution of a lost qubit replaced in (ii). In some cases, the processor is further configured to: if an ancilla qubit is lost, update the matching graph so that a node involving the ancilla qubit is connected by edges corresponding to the predicted probability distribution; and if a data qubit is lost, update the matching graph by assigning the predicted probability distribution to each node involving the data qubit. In some cases, each node involving the ancilla qubit is updated. [0246] In some cases, the error correcting code is configured to be implemented during a quantum computation circuit. In some cases, the error correcting code is configured to be implemented without measurement of each or a plurality of data qubits. In some cases, the error correcting code is configured to be implemented substantially without loss of coherence of each or a plurality of data qubits. In some cases, processor is further configured to flag measurements taken during a window of time that includes a time when the lost qubit was missing as untrustworthy. [0247] In some cases, the decoder comprises union find, tensor network decoder, belief propagation with ordered statistics decoder, maximum likelihood decoder, or a look up table decoder. In some cases, the decoder comprises minimum weight perfect matching. In some cases, the decoder comprises sparse blossom or fusion blossom. In some cases, the error correcting code comprises a topological code. In some cases, the topological code is a stabilizer code. In some cases, the error correcting code is a surface code, a color code, a toric code, a shor style code, or a qLDPC code. In some cases, the color code is a Steane code. In some cases, the
WSGR Docket No.55436-729.601 shor style code is a Bacon-shor code. In some cases, the qLDPC code is a hypergraph product code. In some cases, each node in the matching graph corresponds to a change-of-value of a particular stabilizer and wherein pairs of nodes are connected by edges corresponding to possible physical errors. In some cases, the edges are weighted based on the likelihood of a particular error occurring. In some cases, atom loss is treated as a gate error that occurs with a probability of 50%. [0248] In some cases, the processor is further configured to provide instructions to the non- classical computing system to perform a measurement operation, wherein the measurement operation is state selective. In some cases, the measurement operation comprises applying electromagnetic energy to a qubit to be measured, wherein the electromagnetic energy is configured to selectively drive the qubit to be measured from an initial state to an excited state in a presence of an applied magnetic field, wherein a selectivity of a transition to the excited state is based at least in part on a strength of the applied magnetic field. In some cases, the processor is further configured to determine that the qubit to be measured was in the initial state based at least in part on the qubit returning to the initial state by emission of a photon in response to the electromagnetic energy. Examples of Processes for Performing Continuous, Non-Classical Computations [0249] FIG.9 illustrates an example process 900 for performing continuous, non-classical computation. In some examples, the process 900 may be implemented with the science region and the reservoir region being sufficiently distinct to enable loading of the reservoir region with sufficiently low disturbance of atoms within the science region. One way to accomplish this is for the reservoir and science regions to not be sub-regions of the same array, but rather to be distinct arrays in the sense that they may be formed using different lasers or different optical elements. Further, in some cases, the reservoir may be loaded from a third “transport” array, rather than from a MOT. This last technique may eliminate a use for near-resonant light, and so may allow for continuous coherent operations to be performed during the reloading process. [0250] At a high level, the process 900 may comprise: loading a reservoir array, transferring atoms to a science array from the reservoir array, performing a computation/simulation using the science array, atomic loss occurring in the science array, refilling the science array from the reservoir array, and reloading the reservoir array. A science array may comprise atoms that are actively being used for the application (e.g., quantum computing, optical clocks, sensing, or any other application disclosed herein). A science array in a quantum computer may comprise data qubits and ancilla qubits. A reservoir array may comprise atoms that are not actively in use but
WSGR Docket No.55436-729.601 which may be used at a later time to replace atoms lost from the science array, e.g., a lost ancilla qubit from a quantum computer. [0251] In some cases, the process 900 may begin with both the reservoir array and the science array being empty. Once atoms are loaded into the reservoir array, the atoms in the reservoir may then be imaged. In some examples, at least some of the atoms in the reservoir array may then transferred into the science array (for example using optical tweezers). The reservoir array can be reloaded to achieve a full or fuller reservoir array and enable further transfer of more atoms from the reservoir array into the science array. Once the science array is fully occupied (or occupied to a desired/predetermined amount), the computation/simulation may begin. During the computation/simulation, the science array may be periodically imaged to determine if and where atom loss has occurred. If an atom has been lost from a site in the science array, an atom from the reservoir array may be transferred to fill the site. This may continue provided there are sufficient atoms in the reservoir array. When there are not, new atoms may be loaded into the reservoir array, and the process continues. As illustrated in FIG.9, the process 900 may be iterative or repetitive in some examples, with potential repetition of one or more operations of the process 900. [0252] FIG.9 shows an example method and system for performing continuous, non-classical computations. The method 900 may comprise an operation 910. Operation 910 may comprise loading a plurality of atoms into a reservoir array that includes a first plurality of spatially distinct optical trapping sites. The first plurality of optical trapping sites may be configured to trap the plurality of atoms. In some cases, the plurality of atoms are qubits in a non-classical computational system, such as a quantum computer, a quantum annealer, etc. In some cases, the plurality of atoms comprises atoms in an atomic clock. [0253] The method 900 may comprise an operation 920. Operation 920 may comprise transferring a first subset of the plurality of atoms from the reservoir array into a science array. The science array may include a second plurality of spatially distinct optical trapping sites. The second plurality of optical trapping sites may be configured to trap a plurality of atoms. [0254] In some cases, at operation 915, operations 910 and 920 may be repeated a number of times. The operations may be repeated until a science array comprises a sufficient fill factor for quantum computation, quantum simulation, clock operations, metrology operations, sensing operations, etc. [0255] The method 900 may comprise an operation 930. Operation 930 may comprise performing a first application using at least some of the first subset of the plurality of atoms in the science array. The application may be quantum computation, quantum simulation, clock operations, metrology operations, sensing operations, etc.
WSGR Docket No.55436-729.601 [0256] The method 900 may comprise an operation 940. Operation 940 may comprise determining an atomic loss in one or more of the arrays. The operation may comprise determining an atomic loss number representing a difference between (i) a number of atoms in the first subset of the plurality of atoms and (ii) a number of atoms in a remaining subset of the first subset of the plurality of atoms that remain in the science array following the performing of the first non-classical computation. Atom loss may occur due to collisions with residual background gas, to leakage into un-trapped or otherwise undesirable internal states, due to heating associated with laser interactions, or other processes. [0257] The method 900 may comprise an operation 950. Operation 950 may comprise transferring a second subset of the plurality of atoms from the reservoir array into the science array. Operation 950 may comprise a reloading operation. In some cases, the second subset of the plurality of atoms includes at least a number of atoms equal to the atomic loss number. In some cases, the second subset of the plurality of atoms includes a number of atoms less than the atomic loss number. The second subset may be transferred substantially without loss of a coherence of the plurality of atoms in the science array. The second subset may be transferred substantially without stopping an application in the science array. [0258] In some cases, at operation 955, operations 930, 940, and 950 may be repeated a number of times. The operations may be repeated until a quantum computation, quantum simulation, clock operation, metrology operation, etc. is complete. The operations may be repeated while there are atoms in a reservoir to be filed into the science array. [0259] The method 900 may comprise an operation 960. Operation 960 may comprise reloading the reservoir array with additional atoms. The reservoir may be reloaded from an atom source. The atom source may be cooled atom source. In some examples, reservoir regions may be filled from a magneto-optical trap (MOT), from an atomic beam, from a thermal atomic gas, from another optical or other form of electromagnetic trap, or from any other source of atoms. In some examples, the initial loading of the science region may be direct (from any atomic source other than the reservoir array), from the reservoir array, or from a separate reservoir array than the one used for replenishing. In some examples, the reservoir region may be smaller, larger, or the same size/number of sites as the science region and similar techniques may be used to maintain an arbitrary number of atoms within each site of the science array. [0260] In some cases, at operation 965, operation 960 may be repeated a number of times. Operation 960 may be refilled a number of times to fill a reservoir array. The operation may be repeated such that an application in operation 930 may be performed continuously. [0261] In some cases, at operation 975, operations 955 and 965 may be both be repeated in order to maintain a fill factor in the science array. In some cases, the method may comprise
WSGR Docket No.55436-729.601 performing a second non-classical computation using at least some of one or both of (i) the remaining subset of the first subset of the plurality of atoms and (ii) the second subset of the plurality of atoms. [0262] The present disclosure comprises various sub-operations of the method 900. For example, one or more of the operations of the method 900 may be removed. For example, one or more of the operations of the method 900 may be repeated. Examples of Continuous Loading [0263] A useful error-corrected quantum computer should remove entropy faster than it can enter. One source of entropy in a trapped atom quantum computer may be atom loss. Accordingly, it may be useful to conditionally refill sites in a trapped atom quantum computer continuously with the calculation. Continuous operation during a non-classical computation may comprise refilling a lost atom during operations of computation. For example, continuation operation in a gate-model quantum computer may comprise refilling lost atoms “mid-circuit” or between gate operations in a quantum computation. Continuous operation in a quantum simulator may comprise refilling an atom during the simulation. Continuous operation in a clock operation may comprise refilling an atom during operation of the clock. In general, continuous operation may comprise refilling an atom during the time in which the application is being implemented. [0264] Continuous operation may comprise refilling an atom substantially without stopping the application. Substantially without stopping may comprise not performing recovery operations, such as repeating previous steps, to account for the atom loss. Such recovery operations may comprise repeating a calculation or a portion of a calculation to replace a “lost” portion. [0265] Similarly, since each of quantum computation, quantum simulation, clock operations, metrology, and quantum sensing may utilize phenomena such as quantum coherence, it may be useful to maintain coherence while refilling atoms in an atom-based implementation of these applications (e.g., an atomic clock, a neutral atom quantum computer, etc.). For example, atoms may be refilled substantially without loss of coherence of atoms in the array. Substantially without loss of coherence may comprise contrast loss on the order of 10% or better on seconds time scale. E.g., less than 10% loss of contrast over 2 seconds, about 5% contrast loss over two seconds or better. Substantially without loss of coherence may comprise a contrast of better than 0.8 (maximum of 1) over 1 second. [0266] The present disclosure provides systems and methods for continuous atom reloading. Systems and methods of the present disclosure may distinguish the science array and the reservoir array during atom transfer.
WSGR Docket No.55436-729.601 [0267] For example, the arrays may be physically distinguished. Systems and methods of the present disclosure may employ distinct sets or subsets of atoms within an array. For example, FIG.9 and FIG.11 show a science array and a reservoir array. In some cases, the science array is distinct from the reservoir array. In some cases, the science array is spatially distinct from the reservoir array. For example, the science array may be physically separated from the reservoir array. For example, the science array may be energetically separated from the reservoir array during atom movement. In some cases, both physical and energetic separation methods may be used to facilitate atom movement without disruption of the science array. [0268] Physical separation of the science array and the reservoir array may be useful in at least some respects for example. If the science array and the reservoir array are physically distinct, the reservoir array can be more easily spatially separated from the science region. This can allow loading into the reservoir array without disturbing atoms in the science region while the reservoir region is being loaded. For example, a disturbance may occur from unwanted scattering, unwanted light shifts, etc. during transfer. In some cases, a separate optical system from the trap excitation may be used to move atoms from a first array to a second array disclosed herein. For example, the reservoir array may be loaded from a separate optical potential or array, which may disturb atoms in the science region if the reservoir and science arrays were too close. Using separate optical systems to generate the two arrays may be helpful for separating the science array and the reservoir array arrays. Using a separate (e.g., a third) optical system, for atom movement may further insulate the arrays. [0269] In some cases, the reservoir and science regions may be separated either parallel or transverse to the axis along which imaging is performed. If the separation is parallel to the imaging axis, atoms may be transferred from reservoir to science region by means of translating the focus of focused trapping lasers, or by shifting the phase of a trapping optical lattice. In some cases, the transfer of the at least one atom that is from the reservoir array into the science array is a long-range transfer. Methods and systems described herein may be combined with methods and systems for long range transport, such as those described in International Applications PCT/US2023/026595 and PCT/US2023/075948, which are incorporated herein in their entirety. [0270] Electronic separation of the science array and the reservoir array may be useful in at least some respects for example. In some examples, the reservoir and science regions are distinguished by the internal or motional state occupied by the atoms (perhaps instead of being spatially separated). In some examples, the traps may be formed with spatially or temporally incoherent or coherent light, or by non-optical electromagnetic fields. [0271] In some cases, coherence may be protected by applying a “hiding” excitations during or partially during atom reloading into the science array. A hiding excitation may comprise placing
WSGR Docket No.55436-729.601 an atom being transferred or an atom already in an array into a dark state, a clock state, or another state forbidden by selection rules from an optical excitation used for the transferring. In some examples, hiding excitations may be applied to atoms in the science array during imaging or excitation of atoms to be moved into the science portion of the array. In some cases, the at least one atom that is transferred from the reservoir array into the science array is in a dark state. In some cases, an atom that is transferred from the reservoir array into the science array is in a dark state, a clock state, or another state forbidden by selection rules from an optical excitation used for the transferring. Methods and systems described herein may be combined with methods and systems for state selective movement, as described in International Application PCT/US2023/075948 which is incorporated herein in its entirety. Examples of Methods and Systems for Alignment of Modules [0272] Trapping arrays of atoms using laser beams may have various applications in the field of atomic physics. However, increasing the size of these arrays may be accompanied by more laser power or multiplying effective laser power with an optical cavity. Some optical cavities comprise two mirrors facing one another, e.g., a linear cavity, a folded cavity, a bowtie cavity, etc., while others may comprise a plurality of mirrors arranged to direct a beam in closed loop, e.g., a ring cavity. Optical cavities may generally allow a power to build up over multiple reflection within the cavity. [0273] Optical cavities may be sensitive to alignment. For example, if a mirror is mispositioned, subsequent passes of the beam may not traverse the same optical path leading to leakage, divergence, and loss of amplification. In some cases, the beam directing elements may be fixed during manufacturing. While fixing beam directing elements decreases likelihood that an aligned optical cavity will fall out of alignment, it may be important to align the cavity optics precisely before fixing the optical elements in place. [0274] In some cases, a spacer may hold several optical elements, e.g., mirrors, prisms, etc., in about the correct position and orientation. During manufacturing, they may be aligned before being affixed to the spacer. In the case of an optical cavity where the optical axis has a point of intersection, the optical alignment may create an overlap of multiple laser beams or optical passes of the same laser beam (either from multiple cavities or inside a single cavity). In the case of trapped array of atoms, micrometer precision or better may be advantageous. It may be advantageous to 1) measure the relative positions between beams with a high degree of precision, and to 2) adjust mirror positions to compensate for imperfect overlap. [0275] Methods and systems disclosed herein may be directed to measuring the relative positions between beams with a high degree of precision. This may allow for adjustment of
WSGR Docket No.55436-729.601 mirror positions to compensate for imperfect overlap. Methods and systems of the present disclosure may improve upon existing methods for cavity alignment in at least some aspects. Methods of alignment that involve moving an obstruction may lack adequate precision or may not be scalable to multiple beams. [0276] For example, moving slit methods can be used to align two cavity beams. In a moving slit alignment, a slit may be placed around the beams, and transmission of each cavity through the slit as the slit is moved up and down may be monitored. When the slit moves too far to either side it clips and finally completely blocks the cavity beam. When the transmission is maximized, the slit is exactly centered on the cavity beam, and if this maximum occurs at the same slit position for both cavity beams, then the beams are aligned in the direction of slit movement. [0277] This measurement may be relatively simple for two beams, because as long as they lie in a single plane, two beams are guaranteed to intersect. However, it is difficult to extend this method to more than two beams (which do not necessarily have a single intersection point), multiple beams inside a single cavity (where the obstruction can unintentionally clip multiple beams at once), or beam arrangements that do not lie in a single plane. Also, it is slow to mechanically move a slit while monitoring transmission, and additionally clipping the modes can change their shape, altering the measurement. Some of these problems can be addressed by replacing the moving slit with a fine tip, see for example Cai (incorporated by reference above), but it remains difficult to infer the exact 3D geometry of the beams from the response of the cavity transmission to a moving obstruction. [0278] Systems and methods describe herein may use direct imaging of Rayleigh scattering from the beams onto one or more cameras. In some cases, the position of intracavity laser beams can be directly measured by imaging Rayleigh scattering from the beams onto one or more cameras. A beam propagating through air scatters a small amount of light outside the beam path, which can be collected by a standard imaging system. The amount of scattered light is relatively small, but it may be multiplied by the power buildup factor of an optical cavity, accordingly the method may be suitable for cavities. [0279] In some cases, a cavity of the present disclosure may be used in combination with a frequency stabilized (alternatively, a wavelength stabilized) laser. A frequency stabilized laser may in some cases also be phase stabilized; however, some frequency stabilized lasers may not be phase stabilized. In some cases, a frequency stabilized laser may be an optical clock. In an optical clock, the frequency stability may be less than sub-Hertz. In some cases, a laser may be actively frequency stabilize. In some cases, a laser may be passively frequency stabilized. [0280] In some cases, a frequency-stabilized laser may be resonant with a cavity of the present disclosure.
WSGR Docket No.55436-729.601 [0281] In some cases, a camera is used in combination with the cavity to monitor the Rayleigh scattered light. In some cases, a single camera may measure the projection of all beams onto a 2D plane. In some examples, imaging from at least one other angle allows extraction of the beam paths in 3D space. The combination of the two cameras may provide complete information about the cavity beams. This method may be nonintrusive. The method may not involve blocking any of the beams. In some cases, all beams can be measured simultaneously, in 3D space, within a single image exposure time. Cavity Spacers [0282] FIG.10A shows different views of a plurality of mirrors to provide a plurality of optical cavities, according to some embodiments. The plurality of mirrors may be contained or held in place within a cavity spacer 1001. The cavity spacer 1001 may be constructed such that the plurality of mirrors may be oriented in one operable configuration. The cavity spacer 1001 may be constructed such that the plurality of mirrors may be oriented in at least two or more operable configurations. In some cases, the plurality of mirrors may comprise fold end mirrors 1002.1 and 1002.2 and two end mirrors 1002.3 and 1002.4 of a standing wave cavity. A plurality of mirrors 1003.1 and 1003.2 may be in an operable configuration to enable the generation of a three optical trap interaction propagating within the standing wave cavity and light propagating within the running wave cavity at interaction that may lie at the center of the cavity or cavity spacer. [0283] In some examples, FIG.10A shows different views of a plurality of mirrors configured to provide a plurality of optical cavities, according to some embodiments. A cavity spacer 1301 may be configured to hold a plurality of mirrors 1002.1-1002.4 and 1003.1-1003.3. The cavity spacer can be a low thermal expansion glass, thereby maintaining the configuration of the mirrors. The mirrors 1002.1-1002.4 can be configured to provide a first standing wave pattern (e.g., the mirrors can be configured to form a cavity configured to form a first standing wave pattern). The mirrors 1003.1-1003.3 can be configured to provide a second standing wave pattern. While not shown, there may be a fourth mirror of the mirrors 1003.1-1003.3 that is opposite the mirror 1003.3. The combination of the mirrors 1002.1-1002.4 and 1003.1-1003.3 can provide a plurality of optical traps as described elsewhere herein. [0284] FIGs.10A show different views of a plurality of mirrors to provide a plurality of optical cavities, according to some embodiments. Any mirror among a plurality of mirrors may independently comprise an optical substrate. Any mirror among a plurality of mirrors may independently comprise an optical substrate and a coating. Optical substrates may include but are not limited to α-BBO, barium fluoride, calcite, calcium fluoride, F2, germanium, magnesium fluoride, N-BK7, N-F2, N-SF11, potassium bromide, PTFE, rutile, sapphire, silicon, UV fused silica, YVO4, ZERODUR®, zinc selenide, or any combination thereof. Optical substrates may
WSGR Docket No.55436-729.601 comprise a crystalline component. Optical substrates may comprise an amorphous component. Optical coatings may include but are not limited to glass (e.g., glass with a dielectric coating), silver, aluminum, gold, nickel, anti-reflective coatings, dielectric coatings, highly reflective coatings and any combination thereof. Optical coatings may comprise a crystalline component. Optical coatings may comprise an amorphous component. The selection of optical substrate and optical coating may consider the wavelength of light to be utilized. The selection of optical substrate and optical coating may consider the operating temperature or operating pressure of a mirror or optical cavity that they may comprise. An optical substrate may be selected for having a low coefficient of thermal expansion. An optical coating may be selected for having a low coefficient of thermal expansion. [0285] In some cases, FIG.10A show different views of a plurality of mirrors to provide a plurality of optical cavities, according to some embodiments. The plurality of mirrors may be contained or held in place within a cavity spacer 1001. The cavity spacer 1001 may comprise a single piece of material. The cavity spacer 1001 may comprise two or more pieces of material. The material used to construct the cavity spacer 1001 may include but is not limited to glass, borosilicate glass, ultra low expansion glass, silicon, germanium, ZERODUR®, silicon carbide, silicon nitride, diamond, or any combination thereof. The selection of a material to construct a cavity spacer 1001 may consider the operating temperature or operating pressure of a device that the cavity spacer may comprise. A material to construct a cavity spacer 1001 may be selected for having a low coefficient of thermal expansion. [0286] The physical size of the cavity spacer 1001 may be on the order of millimeters, in some examples. For example, the footprint of the cavity spacer 1001 may be approximated as about a 10 mm cube, about a 20 mm cube, about a 30 mm cube, about a 40 mm cube, about a 50 mm cube, about a 60 mm cube, about a 70 mm cube, about an 80 mm cube, about a 90 mm cube, about a 100 mm cube, etc. For example, the cavity spacer 1001 may be about 65 x 58 x 30 mm (thickness). The mirrors 1002.1-1002.4 and 1003.1-1003.3 may also be on the order of millimeters, in some examples. For example, the mirrors 1002.1-1002.4 and 1003.1-1003.3 may be about 4 mm thick with about 10 mm diameter. The mirrors 1002.1-1002.4 and 1003.1-1003.3 may be optically bonded on the outer surface of the cavity spacer 1001, such that the overall cavity profile becomes larger in the transverse plane. In some examples, the exact dimensions of the cavity spacer 1001 or the mirrors 1002.1-1002.4 and 1003.1-1003.3 may be dictated by other apparatuses, such as the size of the vacuum chamber in which the cavity spacer 1001 is installed. In some examples, once the cavity lengths have been designed (e.g., based on the available space), the cavity spacer 1001 dimensions may be designed and implemented to be quite accurate, e.g., including both the surface-normal and the angular tolerances of the surfaces, since
WSGR Docket No.55436-729.601 light bounces off these mirrors and the slightest offset can deflect the light elsewhere. In some cases, machining tolerance for the surface-normals may be a few microns and 20 arcseconds for more stringent angular tolerances. In some examples, flatness for the mirrors 1002.1-1002.4 and 1003.1-1003.3 may be specified to λ/10. In some examples, surface quality of the mirrors may be specified to a couple angstroms. Cavity Alignment [0287] FIG.10B shows an example of a complicated set of cavities to be aligned. In the illustrated example, there are two cavities, each with multiple beams, where 4 beams lie in two planes and intersect at a single point. As shown, where two cavities, each containing two beams intersecting in an X, each in a different plane, are to be aligned at a single point. [0288] FIG.10B additionally shows modeled views from three different camera angles, as well as actual images of Rayleigh scattered light taken from each view. The combination of views may provide full information about the alignment. As shown in the top images, three modeled views of the modes of the cavity are shown. The bottom three images show measured Rayleigh scattering from these three views, clearly showing all laser beams, allowing full knowledge of the beam positions in 3D space. [0289] In some cases, a single intersection point is of interest. If a single intersection point is of interest, the camera depth may generally not be limiting. However, if multiple points of intersection are of interest, the camera depth of focused can be varied by changing the aperture of the camera. If there is a blur from limited resolution of the camera the systematic blur pattern can be fit to extract the beam center position. [0290] The angles into which light is Rayleigh scattered may be generally affected by polarization of the light. Accordingly, some camera angles may see little light for certain polarizations. In some cases, both polarization states of the laser are coupled into the cavity sequentially. [0291] Often optical cavities are placed into vacuum, where there is no Rayleigh scattering. In some cases, the alignment of the mirrors can be fixed during construction in air by epoxy, optical contacting, hydroxide catalysis bonding, or similar methods using alignment in air, and the alignment may be maintained after the air is evacuated. [0292] The procedure disclosed herein may be used in combination with line-fitting algorithms to extract beam positions. For example, images of the beams in the camera may be fit to a line and a line may be superimposed on the image. In some cases, the array of lines may be used to direct the alignment of the cavity optics. Similarly, a line-fitting algorithm may be used to measure an overlap between free-space beams. This may include folded beams that are reflected back to intersect with themselves.
WSGR Docket No.55436-729.601 [0293] In some cases, cavity mirrors may be manufactured to reflect strongly at a short “alignment wavelength,” in addition to the wavelengths of interest for other applications. Because short wavelengths have stronger Rayleigh scattering, an alignment wavelength may be useful. [0294] In some cases, the magnifications and apertures of the cameras may be chosen to improve signal-to-noise, depth of field, and resolution. [0295] In some cases, higher-order spatial modes of the cavity may be imaged to optimize focus, by examining the fine structure of these modes. [0296] In some cases, a depth of field of a camera may be intentionally limited to extract 3D information from a single image, using the blur from out-of-focus points. [0297] In some cases, a telecentric lens may be used to eliminate perspective error in images. [0298] In some cases, the images may be used to extract other properties of the beam such as beam size and intensity, which reflect properties of the cavity mode and build-up factor. [0299] These techniques may be used to align optical cavities for purposes besides trapping atoms, such as laser interferometers, ring-laser gyroscopes, and enhancement cavities for nonlinear optics. In some cases, trapping atoms may be useful for generating a neutral atom quantum computer or other non-classical computation. Continuous Non-Classical Computation [0300] In some cases, the methods and systems described herein may be used in combination with methods and systems for continuous non-classical computation, as described in International Application PCT/US20258/012599 which is incorporated herein in its entirety. In some cases, methods and systems for continuous non-classical computation comprise incremental filling of a target array from a repetitively filled reservoir, to maintain an equilibrium number of atoms for on which multi-photon gates are performed. In this protocol, the tweezers provide microscopic rearrangement of atoms, while the cavity-enhanced lattices enable the creation of large numbers of deep optical potentials that allow for rapid low-loss imaging of atoms. In a non-limiting example, deterministic filling (99% per-site occupancy) of 1225-site arrays is demonstrated. Because the reservoir may be repeatedly filled with fresh atoms, the array can be maintained in a filled state indefinitely. Methods and systems disclosed herein are compatible with mid-circuit reloading, which may be useful for running large-scale error-corrected quantum computations whose durations exceed the lifetime of a single atom in the system. The methods and systems for continuous non-classical computation, combined with multi-qubit gates, provide a promising means for fast, high-fidelity, and scalable quantum computation.
WSGR Docket No.55436-729.601 [0301] FIG.11 shows a diagram of a repeated loading sequence for continuous non-classical computation. One or more first “reservoir” optical tweezer arrays 1110 are repeatedly filled with one or more atoms transported from one or more spatially separated magneto-optical traps (MOTs), and ultimately transferred into one or more second “target” tweezer arrays 1120 using one or more third “rearrangement” tweezer arrays. In some cases, said one or more atoms are a plurality of atoms, as described herein. The rearrangement moves required for the loading sequence are determined from low-loss images obtained by transferring atoms from the one or more first reservoir tweezers and one or more second target tweezers into one or more cavity- enhanced optical lattices and performing site-resolved fluorescence detection. The one or more cavity-enhanced optical lattices allow for the scalable generation of large numbers of deep traps. In some cases, images are shown prior to the 20th cycle of rearrangement 1030, and after the 70th cycle 1040, with the final reservoir reloading step omitted. In a non-limiting example, this repeated loading sequence allows the loading of over 1200 atoms into 1225 target sites. [0302] In some cases, methods and systems disclosed herein combine the capabilities of one or more tweezers and one or more cavity-enhanced optical lattices to demonstrate an iterative approach to creating large arrays of individually controlled atoms. Systems and method disclosed herein may be combined with system and methods for continuous operation using a separate reservoir array, such as in FIG.9, and disclosed in International Application No. PCT/US2023/075948, which is incorporated by reference herein for all purposes. [0303] In some cases, tweezer rearrangement may be performed by stochastically loading up to a single atom into each trap within an array, imaging the atoms to determine trap occupancy, and then rearranging atoms within the array to create a deterministically occupied sub-array. In some cases, the number of atoms contained in the final array with this approach is no greater than the number initially loaded. Further, because the initial loading is stochastic, the number of sites in the array must generally be substantially larger than the desired final sub-array (though under certain conditions, near-deterministic loading may be achieved). In some cases, repeated loading of a “buffer” array from an optical dipole trap “reservoir” demonstrates that one can decouple the filling of a multi-site target array from a single loading of a cold reservoir. Disclosed herein are methods and systems which extend this example to repeated loading of one or more first reservoir tweezer arrays, from which one or more deterministically filled second target tweezer arrays may be formed (e.g., with a filling factor, described herein). [0304] Systems and methods for a repeated loading sequence described herein may be, in some cases, configured to alter the spatial arrangement in order to obtain an increase in a filling factor of the plurality of optical trapping sites. A filling factor may be defined as a ratio of the number of computationally active optical trapping sites occupied by one or more atoms to the total
WSGR Docket No.55436-729.601 number of computationally active optical trapping sites available in the one or more second target tweezer arrays it or in a portion of the one or more second target tweezer arrays. For instance, initial loading of atoms within the computationally active one or more second target tweezer arrays may give rise to a filling factor of less than about: 100%, 90%, 80%, 70%, 60%, 50%, or less, such that atoms occupy fewer than 100%, 90%, 70%, 60%, 50%, or less of the available computationally active optical trapping sites, respectively. It may be desirable to rearrange the atoms to achieve a filling factor of at least about: 50%, 60%, 70%, 80%, 90%, or 100%. By analyzing the imaging information obtained by systems and methods described herein, the one or more second target tweezer arrays may attain a filling factor of at least about: 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The one or more second target tweezer arrays may attain a filling factor of at most about: 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The one or more second target tweezer arrays may attain a filling factor that is within a range defined by any two of the preceding values. Computer Systems [0305] FIG.12 shows a computer system 1201 that is programmed or otherwise configured to operate the systems, the methods, the computer-readable media, or the techniques described herein (such as the systems, the methods, the computer-readable media, or the techniques implementing polarized driving fields for non-classical computing). The computer system 1201 can regulate various aspects of the present disclosure. The computer system 1201 can be an electronic device of a user or a computer system that is remotely located with respect to the electronic device. The electronic device can be a mobile electronic device. [0306] The computer system 1201 includes a central processing unit (CPU, also “processor” and “computer processor” herein) 1205, which can be a single core or multi core processor, or a plurality of processors for parallel processing. The computer system 1201 also includes memory or memory location 1210 (e.g., random-access memory, read-only memory, flash memory), electronic storage unit 1215 (e.g., hard disk), communication interface 1220 (e.g., network adapter) for communicating with one or more other systems, and peripheral devices 1225, such as cache, other memory, data storage or electronic display adapters. The memory 1210, storage unit 1215, interface 1220 and peripheral devices 1225 are in communication with the CPU 1205 through a communication bus (solid lines), such as a motherboard. The storage unit 1215 can be
WSGR Docket No.55436-729.601 a data storage unit (or data repository) for storing data. The computer system 1201 can be operatively coupled to a computer network (“network”) 1230 with the aid of the communication interface 1220. The network 1230 can be the Internet, an internet or extranet, or an intranet or extranet that is in communication with the Internet. The network 1230 in some cases is a telecommunication or data network. The network 1230 can include one or more computer servers, which can enable distributed computing, such as cloud computing. The network 1230, in some cases with the aid of the computer system 1201, can implement a peer-to-peer network, which may enable devices coupled to the computer system 1201 to behave as a client or a server. [0307] The CPU 1205 can execute a sequence of machine-readable instructions, which can be embodied in a program or software. The instructions may be stored in a memory location, such as the memory 1210. The instructions can be directed to the CPU 1205, which can subsequently program or otherwise configure the CPU 1205 to implement methods of the present disclosure. Examples of operations performed by the CPU 1205 can include fetch, decode, execute, and writeback. [0308] The CPU 1205 can be part of a circuit, such as an integrated circuit. One or more other components of the system 1201 can be included in the circuit. In some cases, the circuit is an application specific integrated circuit (ASIC). [0309] The storage unit 1215 can store files, such as drivers, libraries, and saved programs. The storage unit 1215 can store user data, e.g., user preferences and user programs. The computer system 1201 in some cases can include one or more additional data storage units that are external to the computer system 1201, such as located on a remote server that is in communication with the computer system 1201 through an intranet or the Internet. [0310] The computer system 1201 can communicate with one or more remote computer systems through the network 1230. For instance, the computer system 1201 can communicate with a remote computer system of a user. Examples of remote computer systems include personal computers (e.g., portable PC), slate or tablet PC’s (e.g., Apple® iPad, Samsung® Galaxy Tab), telephones, Smart phones (e.g., Apple® iPhone, Android-enabled device, Blackberry®), or personal digital assistants. The user can access the computer system 101 via the network 1230. [0311] Methods as described herein can be implemented by way of machine (e.g., computer processor) executable code stored on an electronic storage location of the computer system 1201, such as, for example, on the memory 1210 or electronic storage unit 1215. The machine executable or machine-readable code can be provided in the form of software. During use, the code can be executed by the processor 1205. In some cases, the code can be retrieved from the storage unit 1215 and stored on the memory 1210 for ready access by the processor 1205. In
WSGR Docket No.55436-729.601 some situations, the electronic storage unit 1215 can be precluded, and machine-executable instructions are stored on memory 1210. [0312] The code can be pre-compiled and configured for use with a machine having a processer adapted to execute the code or can be compiled during runtime. The code can be supplied in a programming language that can be selected to enable the code to execute in a pre-compiled or as-compiled fashion. [0313] Aspects of the systems and methods provided herein, such as the computer system 1201, can be embodied in programming. Various aspects of the technology may be thought of as “products” or “articles of manufacture” typically in the form of machine (or processor) executable code or associated data that is carried on or embodied in a type of machine readable medium. Machine-executable code can be stored on an electronic storage unit, such as memory (e.g., read-only memory, random-access memory, flash memory) or a hard disk. “Storage” type media can include any or all of the tangible memory of the computers, processors or the like, or associated modules thereof, such as various semiconductor memories, tape drives, disk drives and the like, which may provide non-transitory storage at any time for the software programming. All or portions of the software may at times be communicated through the Internet or various other telecommunication networks. Such communications, for example, may enable loading of the software from one computer or processor into another, for example, from a management server or host computer into the computer platform of an application server. Thus, another type of media that may bear the software elements includes optical, electrical, and electromagnetic waves, such as used across physical interfaces between local devices, through wired and optical landline networks and over various air-links. The physical elements that carry such waves, such as wired or wireless links, optical links or the like, also may be considered as media bearing the software. As used herein, unless restricted to non-transitory, tangible “storage” media, terms such as computer or machine “readable medium” refer to any medium that participates in providing instructions to a processor for execution. [0314] Hence, a machine readable medium, such as computer-executable code (e.g., computer- readable media), may take many forms, including but not limited to, a tangible storage medium, a carrier wave medium or physical transmission medium. Non-volatile storage media include, for example, optical or magnetic disks, such as any of the storage devices in any computer(s) or the like, such as may be used to implement the databases, etc. shown in the drawings. Volatile storage media include dynamic memory, such as main memory of such a computer platform. Tangible transmission media include coaxial cables; copper wire and fiber optics, including the wires that comprise a bus within a computer system. Carrier-wave transmission media may take the form of electric or electromagnetic signals, or acoustic or light waves such as those
WSGR Docket No.55436-729.601 generated during radio frequency (RF) and infrared (IR) data communications. Common forms of computer-readable media therefore include for example: a floppy disk, a flexible disk, hard disk, magnetic tape, any other magnetic medium, a CD-ROM, DVD or DVD-ROM, any other optical medium, punch cards paper tape, any other physical storage medium with patterns of holes, a RAM, a ROM, a PROM and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave transporting data or instructions, cables or links transporting such a carrier wave, or any other medium from which a computer may read programming code or data. Many of these forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to a processor for execution. [0315] The computer system 1201 can include or be in communication with an electronic display 1235 that comprises a user interface (UI) 1240. Examples of UI’s include, without limitation, a graphical user interface (GUI) and web-based user interface. [0316] Methods and systems of the present disclosure can be implemented by way of one or more algorithms. An algorithm can be implemented by way of software upon execution by the central processing unit 1205. Certain Examples [0317] As described, the systems, the methods, the computer-readable media, and the techniques disclosed herein may be applied in a neutral atom quantum computer with one or more spin ½ atoms. For example, the one or more atoms may comprise alkali atoms. The one or more atoms may comprise lithium (Li) atoms, sodium (Na) atoms, potassium (K) atoms, rubidium (Rb) atoms, or cesium (Cs) atoms. The one or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, or caesium-133 atoms. The one or more atoms may comprise alkaline earth atoms. The one or more atoms may comprise beryllium (Be) atoms, magnesium (Mg) atoms, calcium (Ca) atoms, strontium (Sr) atoms, or barium (Ba) atoms. The one or more atoms may comprise beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium- 44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, or barium-138 atoms. The one or more atoms may comprise rare earth atoms. The one or more atoms may comprise scandium (Sc) atoms, yttrium (Y) atoms, lanthanum (La) atoms, cerium (Ce) atoms, praseodymium (Pr) atoms, neodymium (Nd) atoms, samarium (Sm) atoms, europium (Eu) atoms, gadolinium (Gd) atoms, terbium (Tb) atoms, dysprosium (Dy) atoms, holmium (Ho) atoms, erbium (Er) atoms, thulium (Tm) atoms, ytterbium (Yb) atoms, or lutetium (Lu) atoms. The one or more atoms may
WSGR Docket No.55436-729.601 comprise scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium-136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium- 166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms. [0318] The one or more atoms may comprise a single element selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The one or more atoms may comprise a mixture of elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The one or more atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The one or more atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Li, Na, K, Rb, Cs, Be, Mg, Ca, Sr, and Ba. The one or more atoms may comprise a natural isotopic mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. The one or more atoms may comprise an isotopically enriched mixture of one or more elements selected from the group consisting of Sc, Y, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu. atoms may comprise rare earth atoms. For instance, the one or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium- 136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms,
WSGR Docket No.55436-729.601 samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium- 166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at least about 50%, 60%, 70%, 80%, 90%, 91%, 92%, 93%, 94%, 95%, 96%, 97%, 98%, 99%, 99.1%, 99.2%, 99.3%, 99.4%, 99.5%, 99.6%, 99.7%, 99.8%, 99.9%, 99.91%, 99.92%, 99.93%, 99.94%, 99.95%, 99.96%, 99.97%, 99.98%, 99.99%, or more. The one or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium-42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium- 136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium- 166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance of at most about 99.99%, 99.98%, 99.97%, 99.96%, 99.95%, 99.94%, 99.93%, 99.92%, 99.91%, 99.9%, 99.8%, 99.7%, 99.6%, 99.5%, 99.4%, 99.3%, 99.2%, 99.1%, 99%, 98%, 97%, 96%, 95%, 94%, 93%, 92%, 91%, 90%, 80%, 70%, 60%, 50%, or less. The one or more atoms may comprise lithium-6 atoms, lithium-7 atoms, sodium-23 atoms, potassium-39 atoms, potassium-40 atoms, potassium-
WSGR Docket No.55436-729.601 41 atoms, rubidium-85 atoms, rubidium-87 atoms, caesium-133 atoms, beryllium-9 atoms, magnesium-24 atoms, magnesium-25 atoms, magnesium-26 atoms, calcium-40 atoms, calcium- 42 atoms, calcium-43 atoms, calcium-44 atoms, calcium-46 atoms, calcium-48 atoms, strontium-84 atoms, strontium-86 atoms, strontium-87 atoms, strontium-88 atoms, barium-130 atoms, barium-132 atoms, barium-134 atoms, barium-135 atoms, barium-136 atoms, barium-137 atoms, barium-138 atoms, scandium-45 atoms, yttrium-89 atoms, lanthanum-139 atoms, cerium- 136 atoms, cerium-138 atoms, cerium-140 atoms, cerium-142 atoms, praseodymium-141 atoms, neodymium-142 atoms, neodymium-143 atoms, neodymium-145 atoms, neodymium-146 atoms, neodymium-148 atoms, samarium-144 atoms, samarium-149 atoms, samarium-150 atoms, samarium-152 atoms, samarium-154 atoms, europium-151 atoms, europium-153 atoms, gadolinium-154 atoms, gadolinium-155 atoms, gadolinium-156 atoms, gadolinium-157 atoms, gadolinium-158 atoms, gadolinium-160 atoms, terbium-159 atoms, dysprosium-156 atoms, dysprosium-158 atoms, dysprosium-160 atoms, dysprosium-161 atoms, dysprosium-162 atoms, dysprosium-163 atoms, dysprosium-164 atoms, erbium-162 atoms, erbium-164 atoms, erbium- 166 atoms, erbium-167 atoms, erbium-168 atoms, erbium-170 atoms, holmium-165 atoms, thulium-169 atoms, ytterbium-168 atoms, ytterbium-170 atoms, ytterbium-171 atoms, ytterbium-172 atoms, ytterbium-173 atoms, ytterbium-174 atoms, ytterbium-176 atoms, lutetium-175 atoms, or lutetium-176 atoms enriched to an isotopic abundance that is within a range defined by any two of the preceding values. [0319] The one or more atoms may comprise at least about 1, 2, 3, 4, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 300, 400, 500, or more atoms. The one or more atoms may comprise at most about 500, 400, 300, 200, 190, 180, 170, 160, 150, 140, 130, 120, 110, 100, 95, 90, 85, 80, 75, 70, 65, 60, 55, 50, 45, 40, 35, 30, 25, 20, 15, 10, 5, 4, 3, 2, or fewer atoms. The one or more atoms may comprise a number of atoms as defined by any two of the proceeding values. For example, the one or more atoms may comprise from about 75 to about 150 atoms. The one or more atoms may comprise neutral atoms. For example, the one or more atoms may comprise atoms that are not ionized (e.g., are in a neutral state). Each atom of the one or more atoms may be a neutral atom. For example, each atom of an array of atoms can be not ionized. The one or more atoms may comprise rare earth atoms (e.g., lanthanide series atoms (e.g., ytterbium, neodymium, lanthanum, erbium, etc.), scandium, yttrium, etc.), alkali atoms (e.g., sodium, potassium, etc.), alkali earth atoms (e.g., calcium, strontium (e.g., strontium-87 atoms), etc.), or the like, or any combination thereof. [0320] As described, the driving fields of the systems, the methods, the computer-readable media, and the techniques disclosed herein may be created by one or more lasers. The lasers
WSGR Docket No.55436-729.601 may comprise one or more continuous wave lasers. The lasers may comprise one or more pulsed lasers. The lasers may comprise one or more gas lasers, such as one or more helium-neon (HeNe) lasers, argon (Ar) lasers, krypton (Kr) lasers, xenon (Xe) ion lasers, nitrogen (N2) lasers, carbon dioxide (CO2) lasers, carbon monoxide (CO) lasers, transversely excited atmospheric (TEA) lasers, or excimer lasers. For instance, the lasers may comprise one or more argon dimer (Ar2) excimer lasers, krypton dimer (Kr2) excimer lasers, fluorine dimer (F2) excimer lasers, xenon dimer (Xe2) excimer lasers, argon fluoride (ArF) excimer lasers, krypton chloride (KrCl) excimer lasers, krypton fluoride (KrF) excimer lasers, xenon bromide (XeBr) excimer lasers, xenon chloride (XeCl) excimer lasers, or xenon fluoride (XeF) excimer lasers. The laser may comprise one or more dye lasers. [0321] The lasers may comprise one or more metal-vapor lasers, such as one or more helium- cadmium (HeCd) metal-vapor lasers, helium-mercury (HeHg) metal-vapor lasers, helium- selenium (HeSe) metal-vapor lasers, helium-silver (HeAg) metal-vapor lasers, strontium (Sr) metal-vapor lasers, neon-copper (NeCu) metal-vapor lasers, copper (Cu) metal-vapor lasers, gold (Au) metal-vapor lasers, manganese (Mn) metal-vapor laser, or manganese chloride (MnCl2) metal-vapor lasers. [0322] The lasers may comprise one or more solid-state lasers, such as one or more ruby lasers, metal-doped crystal lasers, or metal-doped fiber lasers. For instance, the lasers may comprise one or more neodymium-doped yttrium aluminum garnet (Nd:YAG) lasers, neodymium/chromium doped yttrium aluminum garnet (Nd/Cr:YAG) lasers, erbium-doped yttrium aluminum garnet (Er:YAG) lasers, neodymium-doped yttrium lithium fluoride (Nd:YLF) lasers, neodymium-doped yttrium orthovanadate (ND:YVO4) lasers, neodymium- doped yttrium calcium oxoborate (Nd:YCOB) lasers, neodymium glass (Nd:glass) lasers, titanium sapphire (Ti:sapphire) lasers, thulium-doped ytrium aluminum garnet (Tm:YAG) lasers, ytterbium-doped ytrrium aluminum garnet (Yb:YAG) lasers, ytterbium-doped glass (Yt:glass) lasers, holmium ytrrium aluminum garnet (Ho:YAG) lasers, chromium-doped zinc selenide (Cr:ZnSe) lasers, cerium-doped lithium strontium aluminum fluoride (Ce:LiSAF) lasers, cerium-doped lithium calcium aluminum fluoride (Ce:LiCAF) lasers, erbium-doped glass (Er:glass) lasers, erbium-ytterbium-codoped glass (Er/Yt:glass) lasers, uranium-doped calcium fluoride (U:CaF2) lasers, or samarium-doped calcium fluoride (Sm:CaF2) lasers. [0323] The lasers may comprise one or more semiconductor lasers or diode lasers, such as one or more gallium nitride (GaN) lasers, indium gallium nitride (InGaN) lasers, aluminum gallium indium phosphide (AlGaInP) lasers, aluminum gallium arsenide (AlGaAs) lasers, indium gallium arsenic phosphide (InGaAsP) lasers, vertical cavity surface emitting lasers (VCSELs), or quantum cascade lasers.
WSGR Docket No.55436-729.601 [0324] The lasers may emit continuous wave laser light. The lasers may emit pulsed laser light. The lasers may have a pulse length of at least about 1 femtoseconds (fs), 2 fs, 3 fs, 4 fs, 5 fs, 6 fs, 7 fs, 8 fs, 9 fs, 10 fs, 20 fs, 30 fs, 40 fs, 50 fs, 60 fs, 70 fs, 80 fs, 90 fs, 100 fs, 200 fs, 300 fs, 400 fs, 500 fs, 600 fs, 700 fs, 800 fs, 900 fs, 1 picosecond (ps), 2 ps, 3 ps, 4 ps, 5 ps, 6 ps, 7 ps, 8 ps, 9 ps, 10 ps, 20 ps, 30 ps, 40 ps, 50 ps, 60 ps, 70 ps, 80 ps, 90 ps, 100 ps, 200 ps, 300 ps, 400 ps, 500 ps, 600 ps, 700 ps, 800 ps, 900 ps, 1 nanosecond (ns), 2 ns, 3 ns, 4 ns, 5 ns, 6 ns, 7 ns, 8 ns, 9 ns, 10 ns, 20 ns, 30 ns, 40 ns, 50 ns, 60 ns, 70 ns, 80 ns, 90 ns, 100 ns, 200 ns, 300 ns, 400 ns, 500 ns, 600 ns, 700 ns, 800 ns, 900 ns, 1,000 ns, or more. The lasers may have a pulse length of at most about 1,000 ns, 900 ns, 800 ns, 700 ns, 600 ns, 500 ns, 400 ns, 300 ns, 200 ns, 100 ns, 90 ns, 80 ns, 70 ns, 60 ns, 50 ns, 40 ns, 30 ns, 20 ns, 10 ns, 9 ns, 8 ns, 7 ns, 6 ns, 5 ns, 4 ns, 3 ns, 2 ns, 1 ns, 900 ps, 800 ps, 700 ps, 600 ps, 500 ps, 400 ps, 300 ps, 200 ps, 100 ps, 90 ps, 80 ps, 70 ps, 60 ps, 50 ps, 40 ps, 30 ps, 20 ps, 10 ps, 9 ps, 8 ps, 7 ps, 6 ps, 5 ps, 4 ps, 3 ps, 2 ps, 1 ps, 900 fs, 800 fs, 700 fs, 600 fs, 500 fs, 400 fs, 300 fs, 200 fs, 100 fs, 90 fs, 80 fs, 70 fs, 60 fs, 50 fs, 40 fs, 30 fs, 20 fs, 10 fs, 9 fs, 8 fs, 7 fs, 6 fs, 5 fs, 4 fs, 3 fs, 2 fs, 1 fs, or less. The lasers may have a pulse length that is within a range defined by any two of the preceding values. [0325] The lasers may have a repetition rate of at least about 1 hertz (Hz), 2 Hz, 3 Hz, 4 Hz, 5 Hz, 6 Hz, 7 Hz, 8 Hz, 9 Hz, 10 Hz, 20 Hz, 30 Hz, 40 Hz, 50 Hz, 60 Hz, 70 Hz, 80 Hz, 90 Hz, 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz, 700 Hz, 800 Hz, 900 Hz, 1 kilohertz (kHz), 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, 9 kHz, 10 kHz, 20 kHz, 30 kHz, 40 kHz, 50 kHz, 60 kHz, 70 kHz, 80 kHz, 90 kHz, 100 kHz, 200 kHz, 300 kHz, 400 kHz, 500 kHz, 600 kHz, 700 kHz, 800 kHz, 900 kHz, 1 megahertz (MHz), 2 MHz, 3 MHz, 4 MHz, 5 MHz, 6 MHz, 7 MHz, 8 MHz, 9 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, 50 MHz, 60 MHz, 70 MHz, 80 MHz, 90 MHz, 100 MHz, 200 MHz, 300 MHz, 400 MHz, 500 MHz, 600 MHz, 700 MHz, 800 MHz, 900 MHz, 1,000 MHz, or more. The lasers may have a repetition rate of at most about 1,000 MHz, 900 MHz, 800 MHz, 700 MHz, 600 MHz, 500 MHz, 400 MHz, 300 MHz, 200 MHz, 100 MHz, 90 MHz, 80 MHz, 70 MHz, 60 MHz, 50 MHz, 40 MHz, 30 MHz, 20 MHz, 10 MHz, 9 MHz, 8 MHz, 7 MHz, 6 MHz, 5 MHz, 4 MHz, 3 MHz, 2 MHz, 1 MHz, 900 kHz, 800 kHz, 700 kHz, 600 kHz, 500 kHz, 400 kHz, 300 kHz, 200 kHz, 100 kHz, 90 kHz, 80 kHz, 70 kHz, 60 kHz, 50 kHz, 40 kHz, 30 kHz, 20 kHz, 10 kHz, 9 kHz, 8 kHz, 7 kHz, 6 kHz, 5 kHz, 4 kHz, 3 kHz, 2 kHz, 1 kHz, 900 Hz, 800 Hz, 700 Hz, 600 Hz, 500 Hz, 400 Hz, 300 Hz, 200 Hz, 100 Hz, 90 Hz, 80 Hz, 70 Hz, 60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, 10 Hz, 9 Hz, 8 Hz, 7 Hz, 6 Hz, 5 Hz, 4 Hz, 3 Hz, 2 Hz, 1 Hz, or less. The lasers may have a repetition rate that is within a range defined by any two of the preceding values. [0326] The lasers may emit light having a pulse energy of at least about 1 nanojoule (nJ), 2 nJ, 3 nJ, 4 nJ, 5 nJ, 6 nJ, 7 nJ, 8 nJ, 9 nJ, 10 nJ, 20 nJ, 30 nJ, 40 nJ, 50 nJ, 60 nJ, 70 nJ, 80 nJ, 90 nJ,
WSGR Docket No.55436-729.601 100 nJ, 200 nJ, 300 nJ, 400 nJ, 500 nJ, 600 nJ, 700 nJ, 800 nJ, 900 nJ, 1 microjoule (µJ), 2 µJ, 3 µJ, 4 µJ, 5 µJ, 6 µJ, 7 µJ, 8 µJ, 9 µJ, 10 µJ, 20 µJ, 30 µJ, 40 µJ, 50 µJ, 60 µJ, 70 µJ, 80 µJ, 90 µJ, 100 µJ, 200 µJ, 300 µJ, 400 µJ, 500 µJ, 600 µJ, 700 µJ, 800 µJ, 900 µJ, a least 1 millijoule (mJ), 2 mJ, 3 mJ, 4 mJ, 5 mJ, 6 mJ, 7 mJ, 8 mJ, 9 mJ, 10 mJ, 20 mJ, 30 mJ, 40 mJ, 50 mJ, 60 mJ, 70 mJ, 80 mJ, 90 mJ, 100 mJ, 200 mJ, 300 mJ, 400 mJ, 500 mJ, 600 mJ, 700 mJ, 800 mJ, 900 mJ, a least 1 Joule (J), or more. The lasers may emit light having a pulse energy of at most about 1 J, 900 mJ, 800 mJ, 700 mJ, 600 mJ, 500 mJ, 400 mJ, 300 mJ, 200 mJ, 100 mJ, 90 mJ, 80 mJ, 70 mJ, 60 mJ, 50 mJ, 40 mJ, 30 mJ, 20 mJ, 10 mJ, 9 mJ, 8 mJ, 7 mJ, 6 mJ, 5 mJ, 4 mJ, 3 mJ, 2 mJ, 1 mJ, 900 µJ, 800 µJ, 700 µJ, 600 µJ, 500 µJ, 400 µJ, 300 µJ, 200 µJ, 100 µJ, 90 µJ, 80 µJ, 70 µJ, 60 µJ, 50 µJ, 40 µJ, 30 µJ, 20 µJ, 10 µJ, 9 µJ, 8 µJ, 7 µJ, 6 µJ, 5 µJ, 4 µJ, 3 µJ, 2 µJ, 1 µJ, 900 nJ, 800 nJ, 700 nJ, 600 nJ, 500 nJ, 400 nJ, 300 nJ, 200 nJ, 100 nJ, 90 nJ, 80 nJ, 70 nJ, 60 nJ, 50 nJ, 40 nJ, 30 nJ, 20 nJ, 10 nJ, 9 nJ, 8 nJ, 7 nJ, 6 nJ, 5 nJ, 4 nJ, 3 nJ, 2 nJ, 1 nJ, or less. The lasers may emit light having a pulse energy that is within a range defined by any two of the preceding values. [0327] The lasers may emit light having an average power of at least about 1 microwatt (µW), 2 µW, 3 µW, 4 µW, 5 µW, 6 µW, 7 µW, 8 µW, 9 µW, 10 µW, 20 µW, 30 µW, 40 µW, 50 µW, 60 µW, 70 µW, 80 µW, 90 µW, 100 µW, 200 µW, 300 µW, 400 µW, 500 µW, 600 µW, 700 µW, 800 µW, 900 µW, 1 milliwatt (mW), 2 mW, 3 mW, 4 mW, 5 mW, 6 mW, 7 mW, 8 mW, 9 mW, 10 mW, 20 mW, 30 mW, 40 mW, 50 mW, 60 mW, 70 mW, 80 mW, 90 mW, 100 mW, 200 mW, 300 mW, 400 mW, 500 mW, 600 mW, 700 mW, 800 mW, 900 mW, 1 watt (W), 2 W, 3 W, 4 W, 5 W, 6 W, 7 W, 8 W, 9 W, 10 W, 20 W, 30 W, 40 W, 50 W, 60 W, 70 W, 80 W, 90 W, 100 W, 200 W, 300 W, 400 W, 500 W, 600 W, 700 W, 800W, 900 W, 1,000 W, or more. The lasers may emit light having an average power of at most about 1,000 W, 900 W, 800 W, 700 W, 600 W, 500 W, 400 W, 300 W, 200 W, 100 W, 90 W, 80 W, 70 W, 60 W, 50 W, 40 W, 30 W, 20 W, 10 W, 9 W, 8 W, 7 W, 6 W, 5 W, 4 W, 3 W, 2 W, 1 W, 900 mW, 800 mW, 700 mW, 600 mW, 500 mW, 400 mW, 300 mW, 200 mW, 100 mW, 90 mW, 80 mW, 70 mW, 60 mW, 50 mW, 40 mW, 30 mW, 20 mW, 10 mW, 9 mW, 8 mW, 7 mW, 6 mW, 5 mW, 4 mW, 3 mW, 2 mW, 1 mW, 900 µW, 800 µW, 700 µW, 600 µW, 500 µW, 400 µW, 300 µW, 200 µW, 100 µW, 90 µW, 80 µW, 70 µW, 60 µW, 50 µW, 40 µW, 30 µW, 20 µW, 10 µW, 9 µW, 8 µW, 7 µW, 6 µW, 5 µW, 4 µW, 3 µW, 2 µW, 1 µW, or more. The lasers may emit light having a power that is within a range defined by any two of the preceding values. [0328] The lasers may emit light comprising one or more wavelengths in the ultraviolet (UV), visible, or infrared (IR) portions of the electromagnetic spectrum. The lasers may emit light comprising one or more wavelengths of at least about 200 nm, 210 nm, 220 nm, 230 nm, 240 nm, 250 nm, 260 nm, 270 nm, 280 nm, 290 nm, 300 nm, 310 nm, 320 nm, 330 nm, 340 nm, 350
WSGR Docket No.55436-729.601 nm, 360 nm, 370 nm, 380 nm, 390 nm, 400 nm, 410 nm, 420 nm, 430 nm, 440 nm, 450 nm, 460 nm, 470 nm, 480 nm, 490 nm, 500 nm, 510 nm, 520 nm, 530 nm, 540 nm, 550 nm, 560 nm, 570 nm, 580 nm, 590 nm, 600 nm, 610 nm, 620 nm, 630 nm, 640 nm, 650 nm, 660 nm, 670 nm, 680 nm, 690 nm, 700 nm, 710 nm, 720 nm, 730 nm, 740 nm, 750 nm, 760 nm, 770 nm, 780 nm, 790 nm, 800 nm, 810 nm, 820 nm, 830 nm, 840 nm, 850 nm, 860 nm, 870 nm, 880 nm, 890 nm, 900 nm, 910 nm, 920 nm, 930 nm, 940 nm, 950 nm, 960 nm, 970 nm, 980 nm, 990 nm, 1,000 nm, 1,010 nm, 1,020 nm, 1,030 nm, 1,040 nm, 1,050 nm, 1,060 nm, 1,070 nm, 1,080 nm, 1,090 nm, 1,100 nm, 1,110 nm, 1,120 nm, 1,130 nm, 1,140 nm, 1,150 nm, 1,160 nm, 1,170 nm, 1,180 nm, 1,190 nm, 1,200 nm, 1,210 nm, 1,220 nm, 1,230 nm, 1,240 nm, 1,250 nm, 1,260 nm, 1,270 nm, 1,280 nm, 1,290 nm, 1,300 nm, 1,310 nm, 1,320 nm, 1,330 nm, 1,340 nm, 1,350 nm, 1,360 nm, 1,370 nm, 1,380 nm, 1,390 nm, 1,400 nm, or more. The lasers may emit light comprising one or more wavelengths of at most about 1,400 nm, 1,390 nm, 1,380 nm, 1,370 n, 1,360 nm, 1,350 nm, 1,340 nm, 1,330 nm, 1,320 nm, 1,310 nm, 1,300 nm, 1,290 nm, 1,280 nm, 1,270 n, 1,260 nm, 1,250 nm, 1,240 nm, 1,230 nm, 1,220 nm, 1,210 nm, 1,200 nm, 1,190 nm, 1,180 nm, 1,170 n, 1,160 nm, 1,150 nm, 1,140 nm, 1,130 nm, 1,120 nm, 1,110 nm, 1,100 nm, 1,090 nm, 1,080 nm, 1,070 n, 1,060 nm, 1,050 nm, 1,040 nm, 1,030 nm, 1,020 nm, 1,010 nm, 1,000 nm, 990 nm, 980 nm, 970 nm, 960 nm, 950 nm, 940 nm, 930 nm, 920 nm, 910 nm, 900 nm, 890 nm, 880 nm, 870 nm, 860 nm, 850 nm, 840 nm, 830 nm, 820 nm, 810 nm, 800 nm, 790 nm, 780 nm, 770 nm, 760 nm, 750 nm, 740 nm, 730 nm, 720 nm, 710 nm, 700 nm, 690 nm, 680 nm, 670 nm, 660 nm, 650 nm, 640 nm, 630 nm, 620 nm, 610 nm, 600 nm, 590 nm, 580 nm, 570 nm, 560 nm, 550 nm, 540 nm, 530 nm, 520 nm, 510 nm, 500 nm, 490 nm, 480 nm, 470 nm, 460 nm, 450 nm, 440 nm, 430 nm, 420 nm, 410 nm, 400 nm, 390 nm, 380 nm, 370 nm, 360 nm, 350 nm, 340 nm, 330 nm, 320 nm, 310 nm, 300 nm, 290 nm, 280 nm, 270 nm, 260 nm, 250 nm, 240 nm, 230 nm, 220 nm, 210 nm, 200 nm. The lasers may emit light comprising one or more wavelengths that are within a range defined by any two of the preceding values. [0329] The lasers may emit light having a bandwidth of at least about 1 x 10-15 nm, 2 x 10-15 nm, 3 x 10-15 nm, 4 x 10-15 nm, 5 x 10-15 nm, 6 x 10-15 nm, 7 x 10-15 nm, 8 x 10-15 nm, 9 x 10-15 nm, 1 x 10-14 nm, 2 x 10-14 nm, 3 x 10-14 nm, 4 x 10-14 nm, 5 x 10-14 nm, 6 x 10-14 nm, 7 x 10-14 nm, 8 x 10-14 nm, 9 x 10-14 nm, 1 x 10-13 nm, 2 x 10-13 nm, 3 x 10-13 nm, 4 x 10-13 nm, 5 x 10-13 nm, 6 x 10-13 nm, 7 x 10-13 nm, 8 x 10-13 nm, 9 x 10-13 nm, 1 x 10-12 nm, 2 x 10-12 nm, 3 x 10-12 nm, 4 x 10-12 nm, 5 x 10-12 nm, 6 x 10-12 nm, 7 x 10-12 nm, 8 x 10-12 nm, 9 x 10-12 nm, 1 x 10-11 nm, 2 x 10-11 nm, 3 x 10-11 nm, 4 x 10-11 nm, 5 x 10-11 nm, 6 x 10-11 nm, 7 x 10-11 nm, 8 x 10-11 nm, 9 x 10-11 nm, 1 x 10-10 nm, 2 x 10-10 nm, 3 x 10-10 nm, 4 x 10-10 nm, 5 x 10-10 nm, 6 x 10-10 nm, 7 x 10-10 nm, 8 x 10-10 nm, 9 x 10-10 nm, 1 x 10-9 nm, 2 x 10-9 nm, 3 x 10-9 nm, 4 x 10-9 nm, 5 x 10-9 nm, 6 x 10-9 nm, 7 x 10-9 nm, 8 x 10-9 nm, 9 x 10-9 nm, 1 x 10-8 nm, 2 x 10-8 nm, 3 x 10-8 nm, 4 x
WSGR Docket No.55436-729.601 10-8 nm, 5 x 10-8 nm, 6 x 10-8 nm, 7 x 10-8 nm, 8 x 10-8 nm, 9 x 10-8 nm, 1 x 10-7 nm, 2 x 10-7 nm, 3 x 10-7 nm, 4 x 10-7 nm, 5 x 10-7 nm, 6 x 10-7 nm, 7 x 10-7 nm, 8 x 10-7 nm, 9 x 10-7 nm, 1 x 10-6 nm, 2 x 10-6 nm, 3 x 10-6 nm, 4 x 10-6 nm, 5 x 10-6 nm, 6 x 10-6 nm, 7 x 10-6 nm, 8 x 10-6 nm, 9 x 10-6 nm, 1 x 10-5 nm, 2 x 10-5 nm, 3 x 10-5 nm, 4 x 10-5 nm, 5 x 10-5 nm, 6 x 10-5 nm, 7 x 10-5 nm, 8 x 10-5 nm, 9 x 10-5 nm, 1 x 10-4 nm, 2 x 10-4 nm, 3 x 10-4 nm, 4 x 10-4 nm, 5 x 10-4 nm, 6 x 10-4 nm, 7 x 10-4 nm, 8 x 10-4 nm, 9 x 10-4 nm, 1 x 10-3 nm, or more. The lasers may emit light having a bandwidth of at most about1 x 10-3 nm, 9 x 10-4 nm, 8 x 10-4 nm, 7 x 10-4 nm, 6 x 10-4 nm, 5 x 10-4 nm, 4 x 10-4 nm, 3 x 10-4 nm, 2 x 10-4 nm, 1 x 10-4 nm, 9 x 10-5 nm, 8 x 10-5 nm, 7 x 10-5 nm, 6 x 10-5 nm, 5 x 10-5 nm, 4 x 10-5 nm, 3 x 10-5 nm, 2 x 10-5 nm, 1 x 10-5 nm, 9 x 10-6 nm, 8 x 10-6 nm, 7 x 10-6 nm, 6 x 10-6 nm, 5 x 10-6 nm, 4 x 10-6 nm, 3 x 10-6 nm, 2 x 10-6 nm, 1 x 10-6 nm, 9 x 10-7 nm, 8 x 10-7 nm, 7 x 10-7 nm, 6 x 10-7 nm, 5 x 10-7 nm, 4 x 10-7 nm, 3 x 10-7 nm, 2 x 10-7 nm, 1 x 10-7 nm, 9 x 10-8 nm, 8 x 10-8 nm, 7 x 10-8 nm, 6 x 10-8 nm, 5 x 10-8 nm, 4 x 10-8 nm, 3 x 10-8 nm, 2 x 10-8 nm, 1 x 10-8 nm, 9 x 10-9 nm, 8 x 10-9 nm, 7 x 10-9 nm, 6 x 10-9 nm, 5 x 10-9 nm, 4 x 10-9 nm, 3 x 10-9 nm, 2 x 10-9 nm, 1 x 10-9 nm, 9 x 10-10 nm, 8 x 10-10 nm, 7 x 10-10 nm, 6 x 10-10 nm, 5 x 10-10 nm, 4 x 10-10 nm, 3 x 10-10 nm, 2 x 10-10 nm, 1 x 10-10 nm, 9 x 10-11 nm, 8 x 10-11 nm, 7 x 10-11 nm, 6 x 10-11 nm, 5 x 10-11 nm, 4 x 10-11 nm, 3 x 10-11 nm, 2 x 10-11 nm, 1 x 10-11 nm, 9 x 10-12 nm, 8 x 10-12 nm, 7 x 10-12 nm, 6 x 10-12 nm, 5 x 10-12 nm, 4 x 10-12 nm, 3 x 10-12 nm, 2 x 10-12 nm, 1 x 10-12 nm, 9 x 10-13 nm, 8 x 10-13 nm, 7 x 10-13 nm, 6 x 10-13 nm, 5 x 10-13 nm, 4 x 10-13 nm, 3 x 10-13 nm, 2 x 10-13 nm, 1 x 10-13 nm, 9 x 10-14 nm, 8 x 10-14 nm, 7 x 10-14 nm, 6 x 10-14 nm, 5 x 10-14 nm, 4 x 10-14 nm, 3 x 10-14 nm, 2 x 10-14 nm, 1 x 10-14 nm, 9 x 10-15 nm, 8 x 10-15 nm, 7 x 10-15 nm, 6 x 10-15 nm, 5 x 10-15 nm, 4 x 10-15 nm, 3 x 10-15 nm, 2 x 10-15 nm, 1 x 10-15 nm, or less. The lasers may emit light having a bandwidth that is within a range defined by any two of the preceding values. Certain Definitions and Additional Considerations [0330] Unless otherwise defined, all technical terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise. Any reference to “or” herein is intended to encompass “and/or” unless otherwise stated. [0331] Whenever the term “at least,” “greater than,” or “greater than or equal to” precedes the first numerical value in a series of two or more numerical values, the term “at least,” “greater than” or “greater than or equal to” applies to each of the numerical values in that series of numerical values. For example, greater than or equal to 1, 2, or 3 is equivalent to greater than or equal to 1, greater than or equal to 2, or greater than or equal to 3.
WSGR Docket No.55436-729.601 [0332] Whenever the term “no more than,” “less than,” or “less than or equal to” precedes the first numerical value in a series of two or more numerical values, the term “no more than,” “less than,” or “less than or equal to” applies to each of the numerical values in that series of numerical values. For example, less than or equal to 3, 2, or 1 is equivalent to less than or equal to 3, less than or equal to 2, or less than or equal to 1. [0333] As used herein, like characters refer to like elements. [0334] The term “about” or “approximately” may mean within an acceptable error range for the particular value, which will depend in part on how the value is measured or determined, e.g., the limitations of the measurement system. For example, “about” may mean within 1 or more than 1 standard deviation, per the practice in the art. Alternatively, “about” may mean a range of up to 20%, up to 10%, up to 5%, or up to 1% of a given value. Where particular values are described in the application and claims, unless otherwise stated the term “about” meaning within an acceptable error range for the particular value may be assumed. [0335] As used herein, the terms “non-classical computation,” “non-classical procedure,” “non- classical operation,” any “non-classical computer” generally refer to any method, system, or computer-readable media for performing computational procedures outside of the paradigm of classical computing. A non-classical computation, non-classical procedure, non-classical operation, or non-classical computer may comprise a quantum computation, quantum procedure, quantum operation, or quantum computer. [0336] As used herein, the terms “quantum computation,” “quantum procedure,” “quantum operation,” and “quantum computer” generally refer to any method, system, or computer- readable media for performing computations using quantum mechanical operations (such as unitary transformations or completely positive trace-preserving (CPTP) maps on quantum channels) on a Hilbert space represented by a quantum device. As such, quantum and classical (or digital) computation may be similar in the following aspect: both computations may comprise sequences of instructions performed on input information to then provide an output. Various paradigms of quantum computation may break the quantum operations down into sequences of basic quantum operations that affect a subset of qubits of the quantum device simultaneously. The quantum operations may be selected based on, for instance, their locality or their ease of physical implementation. A quantum procedure or computation may then consist of a sequence of such instructions that in various applications may represent different quantum evolutions on the quantum device. For example, procedures to compute or simulate quantum chemistry may represent the quantum states and the annihilation and creation operators of electron spin-orbitals by using qubits (such as two-level quantum systems) and a universal
WSGR Docket No.55436-729.601 quantum gate set (such as the Hadamard, controlled-not (CNOT), and ^^/8 rotations) through the so-called Jordan-Wigner transformation or Bravyi-Kitaev transformation. [0337] Additional examples of quantum procedures or computations may include procedures for optimization such as quantum approximate optimization algorithm (QAOA) or quantum minimum finding. QAOA may comprise performing rotations of single qubits and entangling gates of multiple qubits. In quantum adiabatic computation, the instructions may carry stochastic or non-stochastic paths of evolution of an initial quantum system to a final one. [0338] Quantum-inspired procedures may include simulated annealing, parallel tempering, master equation solver, Monte Carlo procedures and the like. Quantum-classical or hybrid algorithms or procedures may comprise such procedures as variational quantum eigensolver (VQE) and the variational and adiabatically navigated quantum eigensolver (VanQver). [0339] A quantum computer may comprise one or more adiabatic quantum computers, quantum gate arrays, one-way quantum computers, topological quantum computers, quantum Turing machines, quantum annealers, Ising solvers, or gate models of quantum computing. [0340] As used herein, the term “adiabatic” refers to any process performed on a quantum mechanical system in which the parameters of the Hamiltonian are changed slowly in comparison to the natural timescale of evolution of the system. [0341] As used herein, the term “non-adiabatic” refers to any process performed quantum mechanical system in which the parameters of the Hamiltonian are changed quickly in comparison to the natural timescale of evolution of the system or on a similar timescale as the natural timescale of evolution of the system. [0342] While preferred embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. It is not intended that the invention be limited by the specific examples provided within the specification. While the invention has been described with reference to the aforementioned specification, the descriptions and illustrations of the embodiments herein are not meant to be construed in a limiting sense. Numerous variations, changes, and substitutions will now occur to those skilled in the art without departing from the invention. Furthermore, it shall be understood that all aspects of the invention are not limited to the specific depictions, configurations or relative proportions set forth herein which depend upon a variety of conditions and variables. It should be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. It is therefore contemplated that the invention shall also cover any such alternatives, modifications, variations, or equivalents. It is intended that the following claims define the scope of the invention and that
WSGR Docket No.55436-729.601 methods and structures within the scope of these claims and their equivalents be covered thereby. [0343] It should be noted that various illustrative or suggested ranges set forth herein are specific to their example embodiments and are not intended to limit the scope or range of disclosed technologies, but, again, merely provide example ranges for frequency, amplitudes, etc. associated with their respective embodiments or use cases. Where values are described as ranges, it will be understood that such disclosure includes the disclosure of all possible sub- ranges within such ranges, as well as specific numerical values that fall within such ranges irrespective of whether a specific numerical value or specific sub-range is expressly stated. [0344] It should be understood that, unless a term is expressly defined in this patent, there is no intent to limit the meaning of that term, either expressly or by implication, beyond its plain or ordinary meaning, and such term should not be interpreted to be limited in scope based on any statement made in any section of this patent (other than the language of the claims). To the extent that any term recited in the claims at the end of this patent is referred to in this patent in a manner consistent with a single meaning, that is done for sake of clarity only so as to not confuse the reader, and it is not intended that such claim term be limited, by implication or otherwise, to that single meaning. Finally, unless a claim element is defined by reciting the word “means” and a function without the recital of any structure, it is not intended that the scope of any claim element be interpreted based on the application of 35 U.S.C. §112, sixth paragraph. [0345] Throughout this specification, plural instances may implement components, operations, or structures described as a single instance. Although individual operations of one or more methods are illustrated and described as separate operations, one or more of the individual operations may be performed concurrently, and nothing requires that the operations be performed in the order illustrated. Structures and functionality presented as separate components in example configurations may be implemented as a combined structure or component. Similarly, structures and functionality presented as a single component may be implemented as separate components. These and other variations, modifications, additions, and improvements fall within the scope of the subject matter herein. [0346] Additionally, certain embodiments are described herein as including logic or a number of routines, subroutines, applications, or instructions. These may constitute either software (e.g., code embodied on a machine-readable medium) or hardware. In hardware, the routines, etc., are tangible units capable of performing certain operations and may be configured or arranged in a certain manner. In example embodiments, one or more computer systems (e.g., a standalone, client or server computer system) or one or more hardware modules of a computer system (e.g., a processor or a group of processors) may be configured by software (e.g., an application or
WSGR Docket No.55436-729.601 application portion) as a hardware module that operates to perform certain operations as described herein. [0347] In various embodiments, a hardware module may be implemented mechanically or electronically. For example, a hardware module may comprise dedicated circuitry or logic that is permanently configured (e.g., as a special-purpose processor, such as a field programmable gate array (FPGA) or an application-specific integrated circuit (ASIC) to perform certain operations. A hardware module may also comprise programmable logic or circuitry (e.g., as encompassed within a general-purpose processor or other programmable processor) that is temporarily configured by software to perform certain operations. It will be appreciated that the decision to implement a hardware module mechanically, in dedicated and permanently configured circuitry, or in temporarily configured circuitry (e.g., configured by software) may be driven by cost and time considerations. [0348] Accordingly, hardware modules may encompass a tangible entity, be that an entity that is physically constructed, permanently configured (e.g., hardwired), or temporarily configured (e.g., programmed) to operate in a certain manner or to perform certain operations described herein. Considering embodiments in which hardware modules are temporarily configured (e.g., programmed), each of the hardware modules need not be configured or instantiated at any one instance in time. For example, where the hardware modules comprise a general-purpose processor configured using software, the general-purpose processor may be configured as respective different hardware modules at different times. Software may accordingly configure a processor, for example, to constitute a particular hardware module at one instance of time and to constitute a different hardware module at a different instance of time. [0349] Hardware modules can provide information to, and receive information from, other hardware modules. Accordingly, the described hardware modules may be regarded as being communicatively coupled. Where multiple of such hardware modules exist contemporaneously, communications may be achieved through signal transmission (e.g., over appropriate circuits and buses) that connect the hardware modules. In embodiments in which multiple hardware modules are configured or instantiated at different times, communications between such hardware modules may be achieved, for example, through the storage and retrieval of information in memory structures to which the multiple hardware modules have access. For example, one hardware module may perform an operation and store the output of that operation in a memory device to which it is communicatively coupled. A further hardware module may then, at a later time, access the memory device to retrieve and process the stored output. Hardware modules may also initiate communications with input or output devices, and can operate on a resource (e.g., a collection of information). Elements that are described as being
WSGR Docket No.55436-729.601 coupled and or connected may refer to two or more elements that may be (e.g., direct physical contact) or may not be (e.g., electrically connected, communicatively coupled, etc.) in direct contact with each other, but yet still cooperate or interact with each other. [0350] The various operations of example methods described herein may be performed, at least partially, by one or more processors that are temporarily configured (e.g., by software) or permanently configured to perform the relevant operations. Whether temporarily or permanently configured, such processors may constitute processor-implemented modules that operate to perform one or more operations or functions. The modules referred to herein may, in some example embodiments, comprise processor-implemented modules. [0351] Similarly, the methods or routines described herein may be at least partially processor- implemented. For example, at least some of the operations of a method may be performed by one or more processors or processor-implemented hardware modules. The performance of certain operations may be distributed among the one or more processors, not only residing within a single machine, but deployed across a number of machines. In some example embodiments, the processor or processors may be located in a single location (e.g., within a home environment, an office environment or as a server farm), while in other embodiments the processors may be distributed across a number of locations. [0352] The performance of certain operations may be distributed among the one or more processors, not only residing within a single machine, but deployed across a number of machines. In some example embodiments, the one or more processors or processor-implemented modules may be located in a single geographic location (e.g., within a home environment, an office environment, or a server farm). In other example embodiments, the one or more processors or processor-implemented modules may be distributed across a number of geographic locations.