US20210390444A9 - Quantum information processing with an asymmetric error channel - Google Patents

Quantum information processing with an asymmetric error channel Download PDF

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US20210390444A9
US20210390444A9 US17/253,460 US201917253460A US2021390444A9 US 20210390444 A9 US20210390444 A9 US 20210390444A9 US 201917253460 A US201917253460 A US 201917253460A US 2021390444 A9 US2021390444 A9 US 2021390444A9
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qubit
microwave field
ancilla
state
data
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US20210125096A1 (en
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Shruti PURI
Alexander Grimm
Philippe Campagne-lbarcq
Steven M. Girvin
Michel Devoret
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Yale University
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/40—Physical realisations or architectures of quantum processors or components for manipulating qubits, e.g. qubit coupling or qubit control
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N10/00—Quantum computing, i.e. information processing based on quantum-mechanical phenomena
    • G06N10/70—Quantum error correction, detection or prevention, e.g. surface codes or magic state distillation
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/02—Arrangements for detecting or preventing errors in the information received by diversity reception
    • H04L1/04—Arrangements for detecting or preventing errors in the information received by diversity reception using frequency diversity
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B82—NANOTECHNOLOGY
    • B82Y—SPECIFIC USES OR APPLICATIONS OF NANOSTRUCTURES; MEASUREMENT OR ANALYSIS OF NANOSTRUCTURES; MANUFACTURE OR TREATMENT OF NANOSTRUCTURES
    • B82Y10/00—Nanotechnology for information processing, storage or transmission, e.g. quantum computing or single electron logic

Definitions

  • the technology described herein relates generally to quantum information systems. Specifically, the present application is directed to systems and methods for performing quantum information processing (QIP) using at least one qubit with an asymmetric error channel.
  • QIP quantum information processing
  • QIP uses quantum mechanical phenomena, such as energy quantization, superposition, and entanglement, to encode and process information in a way not utilized by conventional information processing. For example, it is known that certain computational problems may be solved more efficiently using quantum computation rather than conventional classical computation.
  • quantum computation requires the ability to precisely control a large number of quantum bits, known as “qubits,” and the interactions between these qubits.
  • qubits should have long coherence times, be able to be individually manipulated, be able to interact with one or more other qubits to implement multi-qubit gates, be able to be initialized and measured efficiently, and be scalable to large numbers of qubits.
  • a qubit may be formed from any physical quantum mechanical system with at least two orthogonal states.
  • the two states of the system used to encode information are referred to as the “computational basis.”
  • photon polarization, electron spin, and nuclear spin are two-level systems that may encode information and may therefore be used as a qubit for QIP.
  • Different physical implementations of qubits have different advantages and disadvantages. For example, photon polarization benefits from long coherence times and simple single qubit manipulation, but suffers from the inability to create simple multi-qubit gates.
  • qubits are physical two-level systems.
  • quantum information may also be stored in logical qubits, which are formed from multiple physical two-level systems or quantum systems with more than two states.
  • states of a quantum mechanical oscillator of which there are an infinite number of energy eigenstates, may also be used to form the computational basis for QIP.
  • coherent states of a quantum mechanical oscillator that are sufficiently displaced from one another in phase space are quasi-orthogonal states and may be used as a computational basis.
  • states that are superpositions of coherent states known as “cat states” may be exactly orthogonal to one another and used to form a computational basis.
  • phase qubits where the computational basis is the quantized energy states of Cooper pairs in a Josephson Junction
  • flux qubits where the computational basis is the direction of circulating current flow in a superconducting loop
  • charge qubits where the computational basis is the presence or absence of a Cooper pair on a superconducting island.
  • superconducting qubits are an advantageous choice of qubit because the coupling between two qubits is strong making two-qubit gates relatively simple to implement, and superconducting qubits are scalable because they are mesoscopic components that may be formed using conventional electronic circuitry techniques. Additionally, superconducting qubits exhibit excellent quantum coherence and a strong non-linearity associated with the Josephson effect. All superconducting qubit designs use at least one Josephson junction as a non-linear non-dissipative element.
  • a quantum information processing (QIP) system includes a data qubit and an ancilla qubit, the ancilla qubit having an asymmetric error channel.
  • the data qubit is coupled to the ancilla qubit.
  • a method of performing QIP in a system comprising a data qubit coupled to an ancilla qubit.
  • the method includes driving the ancilla qubit with a stabilizing microwave field to create an asymmetric error channel.
  • FIG. 1 is block diagram of a quantum information processing system, according to some embodiments.
  • FIG. 2 is a diagram of the joint system of the data qubit and the ancilla qubit of FIG. 1 , according to some embodiments.
  • FIG. 3 is a diagram of a superconducting circuit element of FIG. 2 that includes a transmon, according to some embodiments.
  • FIG. 4 is a diagram of a superconducting circuit element of FIG. 2 that includes a superconducting nonlinear asymmetric inductor element (SNAIL), according to some embodiments.
  • SNAIL superconducting nonlinear asymmetric inductor element
  • FIG. 5 is a block diagram of a quantum information system based on cavity quantum electrodynamics, according to some embodiments.
  • FIG. 6 depicts a Bloch sphere based on cat states, according to some embodiments.
  • FIG. 7A depicts an eigenspectrum of a pumped cat oscillator, according to some embodiments.
  • FIG. 7B depicts the potential of a pumped cat oscillator (PCO) in the limit of large parametric drive, according to some embodiments.
  • PCO pumped cat oscillator
  • FIG. 8 is a quantum circuit diagram of error syndrome detection, according to some embodiments.
  • FIG. 9A is a plot of the dynamics of a PCO and associated qubits during a stabilizer measurement for a four-qubit toric code, according to some embodiments.
  • FIG. 9B is a plot of the dynamics of a PCO and associated qubits during a stabilizer measurement for a four-qubit toric code, according to some embodiments.
  • FIG. 10A is a plot of the dynamics of a PCO and associated qubits during a stabilizer measurement for a cat code, according to some embodiments.
  • FIG. 10B is a plot of the dynamics of a PCO and associated qubits during a stabilizer measurement for a cat code, according to some embodiments.
  • FIG. 11A is a quantum circuit diagram for performing adaptive phase estimation, according to some embodiments.
  • FIG. 11B is a quantum circuit diagram for performing adaptive phase estimation, according to some embodiments.
  • FIG. 12 is a diagram illustrating a readout process in terms of a Bloch sphere, according to some embodiments.
  • FIG. 13 is a schematic of a quantum information processing system, according to some embodiments.
  • FIG. 14A is a plot of the amplitude as a function of time of multiple driving fields used to implement a bias-preserving CNOT gate, according to some embodiments.
  • FIG. 14B is a plot of the phases as a function of time of multiple driving fields used to implement a bias-preserving CNOT gate, according to some embodiments.
  • FIG. 15 is a quantum circuit diagram of a technique for detecting errors, according to some embodiments.
  • FIG. 16 is a flowchart of a quantum information processing method, according to some embodiments.
  • FIG. 17 is a flowchart of a readout method, according to some embodiments.
  • Quantums The state of a single qubit may be represented by the quantum state
  • ⁇ a
  • 1 is the computational basis, which may be implemented physically using any physical system with two orthogonal states.
  • a first type of conventional quantum gate is a single-qubit gate, which transforms the quantum state of a single qubit from a first quantum state to a second quantum state.
  • single-qubit quantum gates include the set of rotations of the qubit on a Bloch sphere.
  • a second type of conventional quantum gate is a two-qubit gate, which transforms the quantum state of a first qubit based on the quantum state of a second qubit. Examples of two-qubit gates include the controlled NOT (CNOT) gate and the controlled phase gate.
  • CNOT controlled NOT
  • Conventional single-qubit gates and two-qubit gates unitarily evolve the quantum state of the qubits from a first quantum state to a second quantum state.
  • quantum information is protected from errors, which result from inevitable and uncontrolled interactions with the environment.
  • Techniques for mitigating such errors include quantum error correction (QEC) schemes.
  • QEC quantum error correction
  • quantum information is protected by linking errors and undesirable interactions with low-weight quantum operators.
  • quantum information may be encoded in a logical qubit using the non-local degrees of freedom of a high-dimensional system rather than simply encoding the information in a the two quantum states of a physical qubit.
  • high-weight operators imply many-body operators arising, for example, in a system of several qubits or operators involving many quantum states of a single high-dimensional physical system (e.g., a quantum mechanical oscillator).
  • the high-weight operators characterizing a codespace of quantum information are referred to as “stabilizers” and are designed to commute with the logical qubit operators but anti-commute with the errors in the system. In the absence of errors, the system lies in the +1 eigenspace of the stabilizer and after an error occurs the system moves to the ⁇ 1 eigenspace.
  • the inventors have recognized and appreciated that the above types of QEC techniques are undesirable for practical implementations of QIP. Instead, the inventors have recognized and appreciated that it is desirable to synthesize stabilizer measurements via naturally available couplings between the data qubits of the system and an ancillary system. Coupling the data qubits of the QIP system exposes the data qubits to a different set of errors that may be just as challenging to mitigate. For example, if the measurement of the ancillary system is not designed intelligently, errors from the ancillary system may propagate to the data qubits, damaging the encoded quantum information beyond repair. Recognizing this, the inventors have developed techniques for reducing and/or, in some instances, eliminating such catastrophic backaction from the ancillary system.
  • a stabilizer measurement technique is described here.
  • a system representing the logical data qubit, encodes the quantum information in N subsystems implemented using physical qubits.
  • a code is defined by multiple stabilizers but, for simplicity, a single stabilizer, ⁇ , is considered here.
  • a set of low-weight operators, ⁇ circumflex over (M) ⁇ i , where i 1, 2, . . . N, commute with the stabilizer ⁇ and can be used to synthesize ⁇ through coupling with an ancilla.
  • the ancillary system may be, for example, an ancilla qubit which is coupled to the data qubit via an interaction Hamiltonian
  • ⁇ z is the z-Pauli operator of the ancilla qubit and g i are controllable interaction strengths between the ancilla and each of the physical qubits used to form the logical data qubit.
  • g i are controllable interaction strengths between the ancilla and each of the physical qubits used to form the logical data qubit.
  • the coupling strength and duration of the interaction, T, between the data qubit and the ancilla qubit may be chosen such that the unitary operator acting on the joint system (up to local rotations) becomes:
  • the result of the interaction with an interaction time T is therefore a phase-flip of the ancilla qubit conditioned on whether the stabilizer is +1 or ⁇ 1.
  • This phase-flip in the ancilla qubit is the error syndrome.
  • the inventors have recognized and appreciated that, during the interaction time, the data qubit and the ancilla qubit are entangled and, to be a successful QEC scheme, it is desirable to engineer the joint system such that errors in the ancilla qubit do not propagate as uncorrectable errors to the data qubit, which is known as “fault-tolerance.”
  • all possible errors in the ancilla qubit should commute with the unitary operator ⁇ (t) at all times.
  • the phase flip error ⁇ circumflex over ( ⁇ ) ⁇ Z satisfies this condition. Therefore, if a phase-flip error occurs at any time ⁇ during the interaction time duration, then at time T, the state of the system is described by the unitary operator:
  • phase flip in the ancilla qubit only introduces an error in the measurement of the syndrome, but does not cause any backaction on the data qubit.
  • bit flip errors represented by the Pauli matrix ⁇ circumflex over ( ⁇ ) ⁇ x
  • amplitude damping errors represented by the Pauli matrix ⁇ circumflex over ( ⁇ ) ⁇ ⁇
  • a bit flip error, ⁇ circumflex over ( ⁇ ) ⁇ x on the ancilla qubit propagates as a high-weight error to the data qubit.
  • the unitary operator ⁇ (t) would result in no backaction on the data qubit if the ancilla qubit did not have bit flip ⁇ circumflex over ( ⁇ ) ⁇ x errors.
  • some aspects of the present application are directed to using an ancilla qubit with an asymmetric error channel where bit flip errors are suppressed relative to phase flip errors.
  • aspects of the present disclosure include a method for making fault-tolerant measurements in quantum systems.
  • the techniques described herein may be used in at least three possible applications.
  • the techniques may be used in quantum error correction schemes by allowing fault-tolerant extraction of error syndromes.
  • the techniques may be used for new, more efficient error correcting codes and procedures.
  • the techniques may be used to create bias-preserving gates, such as a controller-NOT (CNOT) gate.
  • CNOT controller-NOT
  • Some embodiments improve upon the overhead requirements of relative to conventional schemes fault-tolerant syndrome measurements.
  • Some embodiments include a hardware efficient realization of such a syndrome extraction scheme using a two-component cat state in a parametrically driven nonlinear oscillator that exhibits a highly-biased noise channel.
  • the inventors have further recognized and appreciated the flexibility of the above approach.
  • different codes may be used.
  • the syndrome extraction process is used for a variety of codes such as qubit-based toric codes, bosonic cat-codes (and in extension, binomial and pair-cat code) and Gottesman-Kitaev-Preskill (GKP) codes.
  • GKP Geckman-Kitaev-Preskill
  • a challenge for error correction with biased noise is to be able to maintain the bias while performing elementary gate operations such as a CNOT gate, which is an important ingredient for many error correction codes and for universal computation.
  • a native bias-preserving CNOT is not possible even if the underlying noise is biased.
  • the inventors have recognized and appreciated that the aforementioned techniques developed for fault-tolerant syndrome extraction can be utilized and extended to realize a bias-preserving CNOT gate between two stabilized cat states.
  • a CNOT gate is based on the structure of cat states in phase space. In this case, a stabilized cat state can be realized in a parametrically driven nonlinear cavity or via dissipation engineering.
  • Some embodiments that include a bias preserving CNOT gate may achieve gains in the threshold for topological error correcting codes (e.g. toric and surface codes).
  • the ZZ( ⁇ ) gate when combined with an ZZ( ⁇ ) gate, it may be possible to reduce the thresholds for what is known as “magic state preparation” (which is an important but expensive ingredient, in terms of overhead costs, to achieve universality).
  • the ZZ( ⁇ ) gate inherently preserves bias and may be implemented with stabilized cats.
  • the inventors have recognized and appreciated that combining bias preserving CNOT gates, ZZ( ⁇ ) gates and syndrome measurements provides the basis for a fault-tolerant architecture for large-scale quantum computation with ultrahigh thresholds and drastically reduced overhead requirements.
  • Such an architecture which exploits the bias in the noise channel of stabilized cat qubits, does not have any equivalent in conventional systems based on physical qubits.
  • FIG. 1 illustrates a QIP system according to some embodiments.
  • the QIP system 100 includes at least a data qubit 110 and an ancilla qubit 120 .
  • Some embodiments further include a microwave field source 150 and/or a measurement device 125 .
  • the measurement device 125 may include a read-out cavity 130 and a cavity state detector 140 .
  • the microwave field source 150 may be considered to be part of the measurement device 125 as microwave fields emitted by the microwave field source 150 play a role in the measurement.
  • the data qubit 110 may be any physical or logical qubit capable of being coupled to the ancilla qubit 120 .
  • the data qubit 110 may include a superconducting circuit component.
  • the data qubit 110 may include at least one Josephson junction.
  • the data qubit 110 may include a transmon.
  • the data qubit 110 may include a superconducting nonlinear asymmetric inductor element (SNAIL), which is an example of a superconducting circuit component that includes multiple Josephson Junctions.
  • SNAIL superconducting nonlinear asymmetric inductor element
  • the data qubit 110 may include an oscillator.
  • An example of a linear oscillator that may be used includes the electromagnetic field, e.g., microwave radiation, supported by a cavity.
  • a cavity may include a three-dimensional (3D) cavity or a planar transmission line cavity.
  • the cavity may be driven to include a specific type of quantum state.
  • the cavity may be driven to include a cat state or a GKP state.
  • a superconducting circuit component may be coupled to a cavity to form a Kerr-nonlinear cavity.
  • the ancilla qubit 120 may be any physical or logical qubit capable of being coupled to the data qubit 110 .
  • the ancilla qubit 120 may include a superconducting circuit component.
  • the ancilla qubit 120 may include at least one Josephson junction.
  • the ancilla qubit 120 may include a transmon.
  • the ancilla qubit 120 may include a SNAIL.
  • the ancilla qubit 120 may include an oscillator.
  • An example of a linear oscillator that may be used includes the electromagnetic field, e.g., microwave radiation, supported by a cavity.
  • a cavity may include a three-dimensional cavity or a planar transmission line cavity.
  • the cavity may be driven to include a specific type of quantum state.
  • the cavity may be driven to include a cat state or a GKP state.
  • a superconducting circuit component may be coupled to a cavity to form a Kerr-nonlinear cavity.
  • the ancilla qubit 120 may be used by the measurement device 125 to measure one or more properties of the data qubit 110 .
  • an interaction between the data qubit 110 and the ancilla qubit 120 may be engineered such that the state of the ancilla qubit 120 is based on a particular property of the data qubit 110 .
  • the measurement of the data qubit 110 is a quantum nondemolition measurement, meaning the state of the data qubit 110 is left unaffected by the measurement process.
  • the quantum nondemolition measurement may be performed by using the measurement device 125 to measure the state of the ancilla qubit 120 , after the data qubit 110 and the ancilla qubit 120 interact, to determine a property of the ancilla qubit 120 .
  • the interaction between the data qubit 110 and the ancilla qubit 120 may be turned on by driving the data qubit 110 and/or the ancilla qubit 120 with one or more microwave fields using the microwave field source 150 .
  • the read-out cavity 130 is a cavity coupled to the ancilla qubit 120 and configured to support multiple electromagnetic radiation, e.g., microwave radiation, states based on a property of the ancilla qubit 120 .
  • an interaction between the read-out cavity 130 and the ancilla qubit 120 is engineered such that the state of the read-out cavity 130 is dependent on a particular property of the ancilla qubit 120 , which itself may be based on a property of the data qubit 110 .
  • the interaction results in the read-out cavity 130 being in a first state; and if the property of the ancilla qubit 120 is a second value, then the interaction results in the read-out cavity being in a second state.
  • the two states of the read-out cavity 130 may be two different quasi-orthogonal coherent states. In other words, the read-out cavity 130 may be displaced in different ways depending on the value of the property of the ancilla qubit 120 .
  • this process may be performed using what is referred to herein as a “Q-switch,” which uses a frequency conversion technique to conditionally displace the read-out cavity 130 based on the property of the ancilla qubit 130 .
  • the interaction between the read-out cavity 130 and the ancilla qubit 120 may be turned on by driving the read out cavity 130 and/or the ancilla qubit 120 with one or more microwave fields using the microwave field source 150 .
  • the cavity state detector 140 may be, for example a microwave radiation detector capable of distinguishing between the possible states of the read-out cavity 130 that result from the interaction between the read-out cavity 130 and the ancilla qubit 120 .
  • the cavity state detector may be a phase-sensitive detector that is capable of measuring not only amplitude, but phase of the electromagnetic field of the read-out cavity 130 .
  • the cavity state detector 140 may be a homodyne detector or a heterodyne detector. The result of the detection, in some embodiments, is directly related to the error syndrome.
  • the ancilla qubit 130 includes a cavity, the state of which may be measured directly using a homodyne detector. However, if the cavity of the ancilla qubit 130 is a high-Q cavity, the homodyne detection would be slow. Accordingly, the read-out cavity may be a low-Q cavity that may be readout quickly.
  • FIG. 2 is a diagram of a particular embodiment of a joint system 200 that includes an example of the data qubit 110 and an example of the ancilla qubit 120 , according to some embodiments.
  • the data qubit 110 includes a data cavity 210 and a data superconducting circuit 212 .
  • the ancilla qubit 120 includes an ancilla cavity 220 and an ancilla superconducting circuit 222 .
  • the two cavities are coupled together with via an interface 230 , which may include, for example, a microwave waveguide and/or a pin connector.
  • the data cavity 210 may be a three-dimensional cavity and includes at least one microwave port 214 for receiving microwave fields 216 from the microwave field source 150 .
  • the ancilla cavity 220 may be a three-dimensional cavity that includes at least one microwave port 224 for receiving microwave fields 226 from the microwave field source 150 .
  • the microwave ports may include pin connectors and/or microwave waveguides. While FIG. 2 illustrates only a single port for each cavity, each cavity may include more than one port for receiving and/or transmitting microwave fields. For example, not shown in FIG. 2 is a port for coupling the ancilla cavity 220 to the read-out cavity 130 .
  • the data superconducting circuit element 212 and the ancilla superconducting circuit element 222 may include a nonlinear circuit element.
  • the superconducting circuit elements may be a transmon or a SNAIL.
  • FIG. 3 illustrates an example of a superconducting circuit element 300 that may be used as the data superconducting circuit element 212 and/or the ancilla superconducting circuit element 222 .
  • the superconducting circuit element 300 includes a transmon 301 that consists of a single Josephson junction and an antenna that includes a first antenna portion 303 and a second antenna portion 305 . The two antenna portions together form a dipole antenna through which the transmon 301 is coupled to the three-dimensional cavity in which the superconducting circuit element 300 is located.
  • FIG. 4 illustrates an example of a superconducting circuit element 400 that may be used as the data superconducting circuit element 212 and/or the ancilla superconducting circuit element 222 .
  • the superconducting circuit element 400 includes a SNAIL 401 that consists of a single Josephson junction and an antenna that includes a first antenna portion 403 and a second antenna portion 405 .
  • the two antenna portions together form a dipole antenna through which the SNAIL 401 is coupled to the three-dimensional cavity in which the superconducting circuit element 400 is located.
  • the SNAIL 401 is a nonlinear superconducting circuit element that has additional tenability relative to a transmon.
  • FIG. 5 is a schematic diagram of a SNAIL 500 , according to some embodiments.
  • the SNAIL 500 includes a superconducting ring 501 with two nodes 511 and 512 . There are two path along two different portions of the superconducting ring 501 that connect the first node 511 and the second node 512 .
  • the first ring portion includes multiple Josephson junctions 505 - 507 connected in series.
  • a Josephson junction is a dipole circuit element (i.e., it has two nodes).
  • a first node of a first Josephson junction 505 is directly connected to the first node 511 of the SNAIL, which may lead to some other external circuit element (not shown).
  • a second node of the first Josephson junction 505 is directly connected to a first node of a second Josephson junction 506 .
  • a second node of the second Josephson junction 506 is directly connected to a first node of a third Josephson junction 507 .
  • a second node of the third Josephson junction 507 is directly connected to a second node 512 of the SNAIL, which may lead to some other external circuit element (not shown), such as a portion of an antenna.
  • FIG. 5 illustrates the first ring portion including three Josephson junctions
  • any suitable number of Josephson junctions greater than one may be used.
  • three, four, five, six, or seven Josephson junctions may be used.
  • Three Josephson junctions are selected for the example shown because three Josephson junctions is the lowest number of Josephson junctions (other than zero or one) that can be formed using a Dolan bridge process of manufacturing, which may be used in some embodiments.
  • Josephson junctions 505 - 507 are formed to be identical.
  • the tunneling energies, the critical current, and the size of the Josephson junctions 505 - 507 are all the same.
  • the second ring portion of the SNAIL 500 includes a single Josephson junction 508 .
  • a first node of a single Josephson junction 508 is directly connected to the first node 511 of the SNAIL, which may lead to some other external circuit element (not shown), such as a portion of an antenna.
  • a second node of the single Josephson junction 508 is directly connected to the second node 512 of the SNAIL, which may lead to some other external circuit element (not shown), such as a portion of an antenna.
  • the single Josephson junction 508 has a smaller tunneling energy than each of Josephson junctions 505 - 507 .
  • the single Josephson junction 508 may be referred to as a “small” Josephson junction and Josephson junctions 505 - 507 may be referred to as “large” Josephson junctions.
  • the terms “large” and “small” are relative terms that are merely used to label the relative size of Josephson junction 508 as compared to Josephson junctions 505 - 507 .
  • the Josephson energy and the Josephson junction size are larger in the large Josephson junction than in the small Josephson junction.
  • the parameter a is introduced to represent the ratio of the small Josephson energy to the large Josephson energy.
  • the Josephson energy of the large Josephson junctions 505 - 507 is E j
  • the Josephson energy of the small Josephson junction 508 is ⁇ E j , where 0> ⁇ 1.
  • FIG. 5 illustrates the circuit element symbol for the SNAIL 500 , which is used in the superconducting circuit element 400 of FIG. 4 .
  • the parameters that characterize the SNAIL 500 are the Josephson energy E j and the superconducting phase difference, ⁇ , of the small Josephson junction 508 .
  • the SNAIL 500 has only two nodes 511 and 512 , which may be connected to respective portions of an antenna.
  • a the data superconducting circuit element 212 and the ancilla superconducting circuit element 222 may be located within a single, shared 3D cavity. In other embodiments, the data superconducting circuit element 212 and the ancilla superconducting circuit element 222 may be coupled to a respective two-dimensional (2D) transmission line cavity.
  • the superconducting circuit element coupled to a cavity forms a Kerr-nonlinear oscillator, which may be used as the data qubit and/or the ancilla qubit.
  • a two-photon pump, received from the microwave field generator 150 may be used to create a two-component cat state:
  • FIG. 6 illustrates the Bloch sphere 600 used in this application.
  • the basis of the logical qubit formed using these cat states is such that the +Z and ⁇ Z axis of the Bloch sphere 600 corresponds to the superposition states
  • FIG. 6 also illustrates a simplified phase space diagram of each of the states associated with the axes of the Bloch sphere 600 .
  • Cat states of the type described above have the property that natural couplings cause only rotations around the Z axis of the Bloch sphere 600 because the pump used to create the cat states creates a large energy barrier that prevents phase rotations (i.e., rotations from the coherent state
  • phase rotations i.e., rotations from the coherent state
  • a noise channel associated with photon loss corresponds to phase flip errors, which dominate the error channel for logical qubits in some embodiments, whereas bit flip errors are suppressed to create the asymmetric error channel, according to some embodiments.
  • phase-flip errors increase, e.g., linearly, with the size of the cat states, as determined by
  • bit-flip errors and the amplitude damping errors are exponentially suppressed based on the size of the cat state
  • fault-tolerant syndrome measurements may be performed when the pumped cat state of the Kerr-nonlinear cavity is used as the physical implementation of the ancilla qubit.
  • Some embodiments extract an error syndrome based on conditional rotation of a cat state around the Z axis. This may be accomplished, in some examples, using only low-weight local interactions.
  • this fault-tolerant technique may be used with a variety of error correcting codes, such as stabilizer codes. Examples of stabilizer codes include, but are not limited to toric codes, bosonic cat codes, and GKP codes.
  • Some embodiments may use non-stabilizer based error correcting codes, such as non-additive quantum codes. Additionally, some embodiments may use the asymmetric error channel of the ancilla qubit to perform fault-tolerant quantum gates.
  • the interactions between a data qubit and an ancilla qubit are realized using the inherent nonlinearity of the ancilla qubit implemented using a cat state in a Kerr-nonlinear cavity. As such, some embodiments require no additional coupling elements. Thus, by exploiting the techniques described herein, hardware-efficient quantum information processing schemes can be realized.
  • a Kerr-nonlinear oscillator implemented, for example, using the hardware described above, may be driven by a two-photon drive with a frequency equal to twice a resonance frequency of the oscillator.
  • the oscillator When driven by such a microwave field, the oscillator is referred to as a pumped-cat oscillator (PCO) and the Hamiltonian in the rotating wave approximation is:
  • ⁇ PCO ⁇ Kâ ⁇ 2 â 2 +P ( â ⁇ 2 +â 2 ),
  • ⁇ PCO ⁇ K ( â ⁇ 2 ⁇ 2 )( â 2 ⁇ 2 )+ K ⁇ 4 .
  • ⁇ are quasi-orthogonal ( ⁇
  • ⁇ exp( ⁇ 2 ⁇ 2 )) and the cat states
  • the eigenspace 700 of the PCO Hamiltonian can be divided into the even parity subspace 701 and odd parity subspace 702 denoted by the superscripts ⁇ , respectively.
  • the cat subspace of the eigenspace 700 is denoted by and is separated from the rest of the states of the eigenspace 700 by an energy gap ⁇ gap ⁇ 4K ⁇ 2 .
  • the first excited state of ⁇ ′ is the Fock state
  • n 1 , with an energy 4K ⁇ 2 below the vacuum state
  • n 1 are the two degenerate excited states in the original undisplaced frame. Since the eigenstates of the PCO Hamiltonian are also the eigenstates of the parity operator, it may be convenient to express the excited states as the two orthogonal states
  • ⁇ e,1 ⁇ N e,1 ⁇ [D ( ⁇ ) ⁇ D ( ⁇ )]
  • n 1 , which are even and odd parity states, respectively, where N e,1 ⁇ are normalization constants.
  • ⁇ e,1 ⁇ is therefore ⁇ gap ⁇ 4K ⁇ 2 .
  • FIG. 7B illustrates the potential of the PCO in the limit of large parametric drive, according to some embodiments.
  • the drive microwave field is large (e.g., large ⁇ , or equivalently, large P)
  • the PCO behaves like two harmonic oscillators displaced by ⁇ .
  • the tunneling between the two harmonic oscillators is suppressed exponentially as a function of ⁇ because the tunnel splitting can by approximated by the overlap n
  • n ⁇ ( ⁇ 2 )e ⁇ 2 ⁇ 2 , where ⁇ ( ⁇ 2 ) is a polynomial function of ⁇ 2 .
  • the eigenspectrum of the PCO Hamiltonian reduces to superpositions of pairs of degenerate displaced Fock states, [D ( ⁇ ) ⁇ D( ⁇ )]
  • this approximation becomes less valid for higher values of n and breaks down near n ⁇ 2 .
  • the Hamiltonian becomes that of an undriven nonlinear oscillator with Fock states
  • n 0 and
  • n 1 being degenerate and the next two excited states,
  • n 2 and
  • n 3 being non-degenerate.
  • ⁇ gap becomes equal to the gap between the Fock states
  • n 0 and
  • n 2 , which is equal to 2K.
  • the PCO interacts with the data qubit, represented by the system , in such a way that the interaction Hamiltonian in the rotating frame is:
  • ⁇ I ⁇ i ⁇ i ( t ) ⁇ circumflex over (M) ⁇ i ( â ⁇ +â ).
  • ⁇ I 2 ⁇ ⁇ i ⁇ i ′( t ) ⁇ circumflex over (M) ⁇ i ,
  • the interaction Hamiltonian is an entangling interaction that is identical to the interaction Hamiltonian, ⁇ circumflex over (V) ⁇ , described above in the example stabilizer measurement technique and therefore leads to the unitary evolution equivalent to the unitary operator ⁇ (t), above.
  • the error syndrome can be extracted, in some embodiments, by measuring the state of the PCO at time T.
  • such a coupling may be used to extract the error syndrome when using the GKP code.
  • FIG. 8 is a quantum circuit diagram of error syndrome detection 800 , according to some embodiments.
  • the three horizontal lines represents a read-out cavity 801 , an ancilla qubit 802 and a data qubit 803 .
  • Time increases from left to right such that operations that occur on the left of the drawing are performed before operations illustrated on the right of the drawing.
  • the read out cavity 801 is initialized in the vacuum state
  • the ancilla qubit 802 is initialized in the even number cat state,
  • the data qubit 803 is in whatever state
  • the ancilla qubit 802 includes a PCO, as described above.
  • the first act of the error syndrome detection 800 is to map the error syndrome on the state of the PCO.
  • This is referred to as the syndrome measurement 810 .
  • the ancilla qubit 802 may remain in the cat state
  • the syndrome measurement may be implemented using a control-Z rotation 811 , where the state of the PCO is conditionally rotated around the Z-axis of the block sphere based on the state of the data qubit 803 .
  • the syndrome measurement 810 does not change the state of the data qubit 803 .
  • the syndrome measurement 810 may be a quantum non-demolition measurement.
  • the error syndrome detection 800 includes a readout operation 820 .
  • the readout operation 820 determines the state of the ancilla qubit 802 , e.g., by determining the ancilla qubit 802 is in the cat state
  • the read-out of the ancilla qubit 802 may include mapping the state of the ancilla qubit 802 onto the read-out cavity 801 .
  • the read-out operation 820 may include two separate operations. The first operation may be a rotation operation 821 on the ancilla qubit 802 .
  • the rotation operation 821 may rotate the cat states
  • the second operation of the read-out operation 820 includes the “Q-Switch” operation 823 in which a single-photon exchange coupling between the PCO and the read-out cavity 801 is turned on by applying appropriate microwave fields from the microwave field generator 150 .
  • the result of the Q-Switch operation 823 is that the read-out cavity 801 is conditionally displaced based on the state of the PCO.
  • the read-out cavity 801 is measured using, for example a homodyne detection scheme, thereby yielding the error syndrome.
  • the error channel (sometimes referred to as the noise channel) of a PCO is dominated by single-photon loss in the oscillator, which arises from the single-photon exchange coupling with a bath.
  • the dynamics of the PCO is confined to the cat state subspace.
  • the single-photon exchange coupling with the bath results in phase-flip errors dominating over bit-flip errors, which are exponentially small with respect to the strength of the pump field, ⁇ .
  • the bath lifts the two-fold degeneracy of the cat state subspace by an amount exponentially small in the size of ⁇ 2 . This is because the number of photons in the odd cat state
  • ⁇ + ⁇ 2 p 2 , and the number of photons in the even cat state
  • the preservation of the degenerate cat subspace in some embodiments makes the PCO an good candidate for a meter for use in syndrome detection because coupling with the bath commutes with the interaction Hamiltonian and does not cause backaction on the data qubit, .
  • Single photon loss to the bath may, however, induce random flips between the two cat states
  • the measurement of the ancilla qubit is performed multiple times and a majority vote is used to determine the error syndrome.
  • the PCO has single-photon gain and pure dephasing spectral densities less than the energy gap. In such embodiments, irrespective of the underlying cause of noise, the PCO's error channel is dominated by phase-flip errors, while bit-flip errors are exponentially suppressed.
  • the action of â ⁇ on a cat state that is part of the cat state subspace causes both leakage out of the cat state subspace and a phase-flip error.
  • ⁇ e,1 ⁇ are approximately equal to the displaced single photon Fock states (see, e.g., FIG. 7B ).
  • ⁇ e,1 ⁇ and a single-photon gain excites the first excited subspace.
  • the excited states form another two-level ancilla with the same coupling to the data qubit as the ancilla in the cat state subspace.
  • the data qubit does not gain any information about whether the PCO was in cat subspace or not (e.g., in the first excited subspace).
  • the data qubit is transparent to leakage errors in the PCO. If the PCO is experiences n photon-gain events, then the PCO is excited to
  • an n-qubit ⁇ circumflex over ( ⁇ ) ⁇ z stabilizer is used in connection with a toric code, which is an example of a topological quantum error correcting code.
  • a toric code which is an example of a topological quantum error correcting code.
  • two-dimensional toric codes may be used.
  • a four-qubit stabilizer, ⁇ Z ⁇ circumflex over ( ⁇ ) ⁇ z,1 ⁇ circumflex over ( ⁇ ) ⁇ z,2 ⁇ circumflex over ( ⁇ ) ⁇ z,3 ⁇ circumflex over ( ⁇ ) ⁇ z,4 , may be measured using, e.g., a direct, eigenspace-preserving measurement.
  • the Hilbert space of the stabilizer ⁇ z may be classified into an even eigenspace, ⁇ , and an odd eigenspace, .
  • an eight-fold degenerate even (odd) subspace comprises the states which are +1 ( ⁇ 1) eigenstates of ⁇ z .
  • the even eigenspace, ⁇ , and an odd eigenspace, may be defined as the code and error subspace, respectively, such that a measurement of ⁇ z yields ⁇ 1 or +1 based on whether there was an error or not. Thus, the measurement indicates the error syndrome.
  • direct measurement of the stabilizer ⁇ z would require a five-body interaction between the data qubits and an ancilla qubit, which is challenging to realize experimentally. Instead, some embodiments perform a syndrome measurement using only two-body interactions. This may be accomplished by replacing ⁇ circumflex over (M) ⁇ i with ⁇ circumflex over ( ⁇ ) ⁇ z,i in the interaction Hamiltonian described above. The resulting interaction Hamiltonian is:
  • ⁇ I ⁇ ( t ) ⁇ ′ z ( â ⁇ +â ),
  • ⁇ z ′ ⁇ circumflex over ( ⁇ ) ⁇ z,1 + ⁇ circumflex over ( ⁇ ) ⁇ z,2 + ⁇ circumflex over ( ⁇ ) ⁇ z,3 + ⁇ circumflex over ( ⁇ ) ⁇ z,4 , which has the form of a longitudinal qubit-oscillator coupling.
  • all the interaction strengths are assumed to be equal, though it is not required for them to be equal.
  • the duration of interaction with each qubit can be adjusted to perform the syndrome measurement.
  • An alternate approach is to keep the duration of interaction fixed, but use a pair of bit-flip driving field pulses for each qubit appropriately separated in time.
  • ⁇ ( t ) i sin ⁇ 2 ⁇ ⁇ ′ z ⁇ 0 t ⁇ ( ⁇ ) d ⁇ +cos ⁇ 2 ⁇ ⁇ ′ z ⁇ 0 t ⁇ ( ⁇ ) d ⁇ .
  • the error syndrome may be extracted by first initializing the PCO to the cat state
  • the interaction time duration for the i-th qubit is T i,z , where
  • the exponential term at the beginning is a local phase rotation of the qubits.
  • the local phase rotation may be kept track of in classical software while performing subsequent operations on the qubits and accounted for later.
  • local ⁇ circumflex over ( ⁇ ) ⁇ z -gate may be applied to the qubit during or after syndrome measurement to compensate for these phase rotations.
  • the state of the PCO after time T z in some embodiments, is therefore
  • a time-dependent qubit-oscillator interaction is implemented by switching on and then turning off a coupling between the qubit and the oscillator.
  • the four qubits are initialized in a maximally entangled state
  • ⁇ o ⁇ 1 8 ⁇ ( ⁇ i ⁇ ⁇ ⁇ x , i + ⁇ i , j , k ⁇ ⁇ ⁇ x , i ⁇ ⁇ ⁇ x , j ⁇ ⁇ ⁇ x , k )
  • the PCO is initialized to the even number cat state
  • FIG. 9A is a plot 900 illustrating the probability 901 for the PCO to be in the state
  • the probability that the qubits, represented by the density matrix ⁇ circumflex over ( ⁇ ) ⁇ q are in the maximally entangled odd parity state is ⁇ o
  • ⁇ o 0.9999 ⁇ 1
  • the probability that the PCO, represented by the density matrix ⁇ circumflex over ( ⁇ ) ⁇ PCO is in the odd cat state is ⁇ ⁇
  • ⁇ ⁇ 0.9999 ⁇ 1.
  • FIG. 9B is a plot 910 illustrating the probability 911 for the PCO to be in the state
  • ⁇ e ⁇ 1 8 ⁇ ( ⁇ i , j ⁇ ⁇ ⁇ x , i ⁇ ⁇ ⁇ x , j + I ⁇ + ⁇ ⁇ x , 1 ⁇ ⁇ ⁇ x , 2 ⁇ ⁇ ⁇ x , 3 ⁇ ⁇ ⁇ x , 4 , )
  • a cat code stabilizer which is a type of bosonic error correcting code where the information is encoded in superpositions of coherent states.
  • the photon-number parity operator indicates whether a state of the data qubit has an even or an odd number of photons.
  • the two-fold degenerate code subspace is defined by the cat states with even photon numbers:
  • the error subspace is defined by the cat states with odd photon numbers:
  • a storage oscillator which encodes the cat codeword, is dispersively coupled to an ancilla qubit.
  • the dispersive coupling between the storage oscillator and the ancilla qubit may be used to map the parity of the storage cat, which is a property of the data qubit, onto the ancilla qubit.
  • a random relaxation of the ancilla during the measurement induces a random phase rotation of the cat codeword, making this scheme non-fault tolerant.
  • the interaction Hamiltonian of the storage oscillator and ancilla PCO is then given by:
  • ⁇ I ⁇ ( t ) â s ⁇ â s ( â ⁇ +â ).
  • This interaction is equivalent to a longitudinal interaction between the storage oscillator and the ancilla PCO. In some embodiments, this interaction can be created in a tunable manner.
  • ⁇ ( t ) i sin ⁇ 2 ⁇ â s ⁇ â s ⁇ 0 t ⁇ ( ⁇ ) d ⁇ +cos ⁇ 2 ⁇ â s ⁇ â s ⁇ 0 t ⁇ ( ⁇ ) d ⁇ .
  • the error syndrome may be extracted by first initializing the PCO to the cat state
  • T p the unitary operator reduces to:
  • the exponential term at the beginning is a deterministic rotation of the frame of reference of the storage cat.
  • the deterministic rotation may be kept track of in classical software while performing subsequent operations on the qubits and accounted for later. If the storage oscillator is in the code subspace x
  • the PCO if the storage oscillator is in the error subspace x
  • the PCO only measures the parity of the storage cat without revealing information about the actual photon statistics as long as ⁇ is small and the dynamics of the PCO can be restricted to the stabilized cat subspace.
  • a time-dependent qubit-oscillator interaction is implemented by switching on and then turning off a coupling between the storage cavity and the ancilla PCO.
  • the storage cavity is initialized in the odd-parity state
  • ⁇ o
  • the ancilla PCO is initialized in the cat state
  • FIG. 10A is a plot 1000 illustrating the probability 1001 for the PCO to be in the state
  • the probability that the storage cavity, represented by the density matrix ⁇ circumflex over ( ⁇ ) ⁇ s are in the maximally entangled odd-parity state is ⁇ o
  • ⁇ o 0.9999 ⁇ 1
  • the probability that the PCO, represented by the density matrix ⁇ circumflex over ( ⁇ ) ⁇ PCO is in the odd cat state is ⁇ ⁇
  • ⁇ ⁇ 0.9999 ⁇ 1.
  • FIG. 10B is a plot 1010 illustrating the probability 1011 for the PCO to be in the state
  • ⁇ e
  • the probability of the storage cavity at time T P being in the even-parity state is ⁇ o
  • ⁇ o 0.9999 ⁇ 1 and the probability that the PCO, is in the odd-parity state is ⁇ ⁇
  • ⁇ ⁇ 0.9999 ⁇ 1.
  • a GKP code which is a type of bosonic error correcting code designed to correct random displacement errors in phase space.
  • Two ideal GKP codewords are uniform superpositions of eigenstates of the position operator ⁇ circumflex over (q) ⁇ at even and odd integer multiples of ⁇ square root over ( ⁇ ) ⁇ , respectively. These GKP states are a sum of an infinite number of infinitely squeezed states and are unphysical (non-normalizable) because of their unbounded number of photons. More realistic codewords that may be used in some embodiments can be realized by replacing the infinitely squeezed state
  • ⁇ circumflex over (q) ⁇ 0 with a squeezed Gaussian state and replacing the uniform superposition over these states by an overall envelope function, such as a Gaussian, a binomial, etc.
  • the GKP code provides protection against low-rate errors which can be represented as small phase space displacements of the oscillator given by exp( ⁇ iu ⁇ circumflex over (q) ⁇ ) and exp( ⁇ iv ⁇ circumflex over (p) ⁇ ).
  • the displaced GKP states are also the eigenstates of the stabilizers ⁇ q and ⁇ p with eigenvalues exp(i2 ⁇ square root over ( ⁇ ) ⁇ u) and exp(i2 ⁇ square root over ( ⁇ ) ⁇ v), respectively.
  • a measurement of the stabilizers yields the eigenvalues and hence uniquely determines the displacement errors u and v. In some embodiments, this is possible when
  • a simple approach to measure the eigenvalues exp(i2 ⁇ square root over ( ⁇ ) ⁇ u) and exp(i2 ⁇ square root over ( ⁇ ) ⁇ v) of ⁇ q and ⁇ p , respectively, is based on an adaptive phase-estimation protocol (APE).
  • APE adaptive phase-estimation protocol
  • displacement operations are repetitively performed on the storage cavity, the displacement operations being conditioned on the state of the ancilla qubit.
  • some embodiments are directed to a fault-tolerant protocol for the APE of the stabilizers for a GKP code using a stabilized cat in a PCO.
  • the storage cavity is coupled to the PCO via a tunable single photon exchange interaction (also known as a beam splitter operation), defined by the Hamiltonian:
  • ⁇ BS ⁇ PCO +g ( t ) â ⁇ â s +g *( t ) ââ s ⁇ ,
  • this tunable beam splitter operation may be realized using the three- or four-wave mixing capability of the PCO and external microwave drives of appropriate frequencies received from the microwave field generator 150 .
  • the beam splitter Hamiltonian can be approximated as
  • H ⁇ BS ′ H ⁇ PCO + ⁇ ⁇ ( p + p - 1 2 ) ⁇ ( g ⁇ ( t ) ⁇ a ⁇ s + g * ⁇ ( t ) ⁇ a ⁇ s ⁇ ) ⁇ - i ⁇ ⁇ ⁇ ( p - p - 1 2 ) ⁇ ( g ⁇ ( t ) ⁇ a ⁇ s + g * ⁇ ( t ) ⁇ a ⁇ s ⁇ ) ⁇ .
  • the second term of the Hamiltonian ⁇ ′ BS becomes negligibly small and evolution under the Hamiltonian results in a controlled displacement along the position or momentum quadrature depending on the phase chosen for the coupling g(t).
  • dt ⁇ square root over ( ⁇ /2) ⁇
  • the unitary operator corresponding to the beam splitter interaction Hamiltonian above reduces to:
  • ⁇ 1 (T 1 ) is the conditional displacement of the storage cavity for APE of ⁇ q , according to some embodiments.
  • ⁇ 2 (T 2 ) is the conditional displacement of the storage cavity for APE of ⁇ p , according to some embodiments.
  • FIGS. 11A and 11B show a quantum circuit diagram for performing an APE protocol, according to some embodiments.
  • FIG. 11A illustrates the protocol 1100 for estimating ⁇ q
  • FIG. 11B illustrates the APE protocol 1150 for estimating ⁇ p .
  • the three horizontal lines represents a read-out cavity 1101 , an ancilla qubit 1102 and a data qubit 1103 . Time increases from left to right such that operations that occur on the left of the drawings are performed before operations illustrated on the right of the drawings.
  • the read-out cavity 1101 is initialized in the vacuum state
  • the ancilla qubit 1102 is initialized in the even number cat state,
  • the data qubit 1103 is in whatever state
  • the ancilla qubit 1102 includes a PCO, as described above.
  • the protocol 1100 for estimating ⁇ q includes performing a first joint unitary operation 1110 on the data qubit 1102 and the ancilla qubit 1103 such that ⁇ 1 (T 1 ) is implemented.
  • the first joint unitary operation 1110 includes two separate actions. First, a displacement operation 1111 that implements the displacement
  • conditional displacement operation 1113 that implements the displacement D( ⁇ square root over ( 2 ⁇ ) ⁇ ) on the data qubit 1103 based on the state of the ancilla qubit 1102 .
  • the protocol 1100 for estimating ⁇ q then includes a rotation operation 1120 performed on the ancilla qubit 1102 around the Z-axis by an angle ⁇ .
  • the rotation operation 1120 is performed by driving the ancilla qubit 1102 , which may be a PCO, with a microwave field.
  • the value of ⁇ may be determined by a previous iteration of the protocol 1100 for estimating ⁇ p .
  • the protocol 1100 for estimating ⁇ q then includes a read-out operation 1130 for determining the state of the ancilla qubit 1102 .
  • the readout operation 1130 determines the state of the ancilla qubit 1102 , e.g., by determining the ancilla qubit 1102 is in the cat state
  • the read-out of the ancilla qubit 1102 may include mapping the state of the ancilla qubit 1102 onto the read-out cavity 1101 .
  • the read-out operation 1130 may include two separate operations. The first operation may be a rotation operation 1131 on the ancilla qubit 1102 .
  • the rotation operation 1131 may rotate the cat states
  • the second operation of the read-out operation 1130 includes the “Q-Switch” operation 1133 in which a single-photon exchange coupling between the PCO 1102 and the read-out cavity 1101 is turned on by applying appropriate microwave fields from the microwave field generator 150 .
  • the result of the Q-Switch operation 1133 is that the read-out cavity 1101 is conditionally displaced based on the state of the PCO 1102 .
  • the read-out cavity 1101 is measured using, for example a homodyne detection scheme.
  • the read-out cavity 1101 and the ancilla qubit 1103 may be reset to their respective initialized states (
  • the protocol 1150 for estimating ⁇ p includes performing a second joint unitary operation 1160 on the data qubit 1102 and the ancilla qubit 1103 such that ⁇ 2 (T 2 ) is implemented.
  • the second joint unitary operation 1160 includes two separate actions. First, a displacement operation 1161 that implements the displacement
  • conditional displacement operation 1163 that implements the displacement D (i ⁇ square root over (2 ⁇ ) ⁇ ) on the data qubit 1103 based on the state of the ancilla qubit 1102 .
  • the protocol 1100 for estimating ⁇ p then includes a rotation operation 1170 performed on the ancilla qubit 1102 around the Z-axis by an angle ⁇ .
  • the rotation operation 1170 is performed by driving the ancilla qubit 1102 , which may be a PCO, with a microwave field.
  • the value of ⁇ may be determined by a previous iteration of the protocol 1100 for estimating ⁇ p .
  • the protocol 1100 for estimating ⁇ p then includes a read-out operation 1180 for determining the state of the ancilla qubit 1102 .
  • the readout operation 1180 determines the state of the ancilla qubit 1102 , e.g., by determining the ancilla qubit 1102 is in the cat state
  • the read-out of the ancilla qubit 1102 may include mapping the state of the ancilla qubit 1102 onto the read-out cavity 1101 .
  • the read-out operation 1180 may include two separate operations. The first operation may be a rotation operation 1181 on the ancilla qubit 1102 .
  • the rotation operation 1181 may rotate the cat states
  • the second operation of the read-out operation 1180 includes the “Q-Switch” operation 1183 in which a single-photon exchange coupling between the PCO 1102 and the read-out cavity 1101 is turned on by applying appropriate microwave fields from the microwave field generator 150 .
  • the result of the Q-Switch operation 1183 is that the read-out cavity 1101 is conditionally displaced based on the state of the PCO 1102 .
  • the read-out cavity 1101 is measured using, for example a homodyne detection scheme.
  • the read-out cavity 1101 and the ancilla qubit 1103 may be reset to their respective initialized states (
  • the amount of rotation performed in rotation operations 1120 and 1160 above are ⁇ and ⁇ , respectively, and may be determined based on a previous iteration of the respective estimation protocol. In this way, measurement results are fed back into subsequent iterations of the APE protocol.
  • the data qubit 1103 is in an eigenstate of the stabilizer ⁇ q with an eigenvalue exp(2i ⁇ square root over ( ⁇ ) ⁇ u).
  • the state of the ancilla qubit becomes i
  • the state of the ancilla qubit 1102 becomes
  • u) cos 2 ( ⁇ square root over ( ⁇ ) ⁇ u+ ⁇ /2). Consequently, to accurately predict the value of u, the sensitivity of the probability distribution ⁇ P ⁇ (+
  • this is achieved in APE by choosing the feedback phase ⁇ based on whether the ancilla qubit 1102 was measured to be in the
  • a similar analysis applies to performing the APE protocol 1150 for the eigenvalues of ⁇ p and the feedback phase ⁇ .
  • the APE protocols 1100 and 1150 may be iterated to estimate the stabilizer eigenvalues. As the number of iterations of phase estimation increases, the accuracy of the estimates of u, v also increases and, consequently, the uncertainty of the eigenvalues exp(2i ⁇ square root over ( ⁇ ) ⁇ u) and exp(2i ⁇ square root over ( ⁇ ) ⁇ v) decreases.
  • a readout operation is performed to measure the ancilla qubit. While the state of the ancilla qubit may be directly measured by, in the case the ancilla qubit is a PCO, by directly measuring the state of the cavity using homodyne detection, such a measurement would be slow due to the high Q of the PCO cavity. Thus, in some embodiments, a read-out cavity with a Q-value smaller than the Q-value of the ancilla cavity is measured using homodyne detection after mapping the state of the ancilla qubit onto the state of the read-out cavity.
  • the readout of the PCO may be a quantum nondemolition (QND) measurement, though it need not be (e.g., it may be that the readout introduces bit-flips or other errors in the state of the ancilla qubit).
  • QND quantum nondemolition
  • Such non-QND measurements are possible because the interaction between the ancilla PCO and the data qubit may be turned off while the PCO is being measured such that ancilla errors do not propagate to the data qubit.
  • Such direct measurements of the ancilla qubit may be performed using a superconducting transmon.
  • the readout of the ancilla PCO includes a measurement along the Z-axis of the Bloch sphere and does not introduce any additional nonlinearities into the system.
  • the states along the Z-axis of the Bloch sphere are approximately coherent states and may be measured using homodyne detection of the field at the output of the PCO.
  • a Q-switch operation is performed whereby the PCO stats is switched via frequency conversion into a low-Q read-out cavity.
  • the Q-switch operation conditionally displaces the readout cavity based on the state of the PCO along the Z-axis.
  • the read-out operation may include a first operation where the cat states of the PCO are rotated into coherent states. Then, the coherent state of the PCO is Q-switched into the readout cavity. Finally the readout cavity is measured.
  • FIG. 12 illustrates the read-out process 1200 in terms of the Bloch sphere.
  • ⁇ ⁇ of the PCO is performed using microwave drive fields from the microwave field generator 150 .
  • the Bloch sphere 1201 of the read-out process 1200 shows the cat states
  • the rotation around the Z-axis is performed using a single-photon drive with a Hamiltonian:
  • the free evolution under the Kerr-nonlinear Hamiltonian results in the states
  • the two-photon cat pump is reapplied so that the cat subspace is again stabilized against bit-flips.
  • the PCO remains in the coherent states, as shown in Bloch sphere 1205 of the read-out process 1200 .
  • the states of the PCO lies along the Z-axis of the Bloch sphere.
  • the PCO is then coupled to an off-resonance readout cavity. In the absence of an external microwave drive field, the coupling between the PCO and the readout cavity is negligible due to a large detuning between the two.
  • a single-photon exchange coupling (a beam splitter coupling) is turned on by applying at least one microwave drive field from the microwave field generator to compensate for the frequency difference between the PCO and the readout cavity.
  • a three- or four-wave mixing between the drives, the PCO and the readout cavity results in an interaction between the PCO and the readout cavity causing a resonant single photon exchange between the two.
  • This controllable coupling is referred to a Q-switch.
  • the result of the Q-switch operation is to displace the readout cavity conditions on the state of the PCO, as shown in phase space diagram 1207 of the read-out process 1200 .
  • ⁇ Q g(â ⁇ â r +ââ r ⁇ )
  • â r ⁇ and â r are the creation and annihilation operators of the readout cavity
  • g is the tunable coupling strength between the PCO and the readout cavity.
  • H ⁇ Q g ⁇ ⁇ ( p + p - 1 2 ) ⁇ ( a ⁇ r + a ⁇ r ⁇ ) ⁇ - i ⁇ g ⁇ ( p - p - 1 1 ) ⁇ ( a ⁇ r - a ⁇ r ⁇ ) ⁇ .
  • ⁇ a ⁇ r ⁇ ⁇ 2 ⁇ g ⁇ i ⁇ ⁇ ⁇ r ⁇ ( 1 - e - ⁇ r ⁇ t / 2 )
  • ⁇ r is the linewidth of the field.
  • a homodyne detector is used to determine the state of the readout cavity and, thereby, determine the state of the PCO, which is equivalent to extracting an error syndrome.
  • the inventors have recognized and appreciated that the above techniques of using an asymmetric error channel of an ancilla qubit to detect error syndromes may be extended to implement a bias-preserving quantum gate.
  • asymmetric error channels i.e., asymmetric error channels
  • operations that do not commute with the dominant error type can un-bias, or depolarize, the noise channel of the qubit, thereby reducing the benefits of the biased noise channel.
  • ⁇ circumflex over (Z) ⁇ i is the Z-Pauli operator for the i-th qubit and ⁇ is a tunable phase angle.
  • the ZZ( ⁇ ) gate becomes a controlled-phase gate, also referred to as a CZ gate, up to local Pauli rotations and an overall phase.
  • the erroneous gate operation is equivalent to an error-free gate followed by a phase flip. Accordingly, the ZZ( ⁇ ) gate preserves the error bias of the qubit.
  • controlled NOT (CNOT) gate also referred to as a CX gate
  • CX gate may be implemented using the following CX Hamiltonian:
  • H ⁇ cx V ⁇ [ ( I ⁇ 1 + Z ⁇ 1 2 ) ⁇ I ⁇ 2 + ( I ⁇ 1 + Z ⁇ 1 2 ) ⁇ X ⁇ 2 ]
  • the phase-flip error in the target qubit introduces a phase-flip error in the control qubit, depending on when the phase error in the target qubit occurs.
  • the phase-flip of the target qubit during the CNOT gate propagates as a combination of a phase-flip error and a bit-flip error in the same qubit. Consequently, the CNOT gate reduces the bias of the noise channel by introducing bit-flips in the target qubit.
  • coherent errors in the gate operation that arise from uncertainty in V and T also give rise to bit-flip errors in the target qubit. As a consequence, a native bias-preserving CNOT gate is not possible to implement.
  • the inventors have recognized and appreciated that in the absence of a bias-preserving CNOT gate, alternate circuits are required to extract an error syndrome. These alternate circuits add complexity and limit the gains in fault-tolerance thresholds for error correction that result from using qubits with biased noise. The inventors have therefore developed a novel solution to this problem by engineering a bias-preserving CNOT gate using the same two-component cat states realized in a parametrically driven nonlinear oscillator described above.
  • the Bloch sphere is oriented such that the superposition states are oriented along the Z-axis of the Bloch sphere.
  • the Z-axis is selected as the computational basis such that:
  • ⁇ 0 ⁇ ⁇ C ⁇ + ⁇ + ⁇ C ⁇ - ⁇ 2
  • ⁇ 1 ⁇ ⁇ C ⁇ + ⁇ - ⁇ C ⁇ - ⁇ 2
  • ⁇ is the complex amplitude of the coherent state associated with the cat states.
  • 1 are degenerate eigenstates of a parametrically driven Kerr-nonlinear oscillator.
  • the PCO exhibits strong noise bias such that bit-flips are exponentially suppressed.
  • Some embodiments use the PCO to implement a native CNOT gate while preserving the error bias, overcoming the problem with the example CNOT gate described above.
  • the CNOT gate is based on the topological phase that arises from the rotation of the cat states around the Bloch sphere generated by changing a phase of the parametric drive.
  • the topological nature of some embodiments allows the CNOT gate to preserve the error bias in the qubits.
  • the ability to preserve the noise bias demonstrates just one advantage of using continuous variable physical systems, such as the PCO, to implement a logical qubit rather than using two-level physical systems as the basis of a qubit.
  • the time-dependent unitary evolution of the qubits undergoing a CNOT gate does not contain an explicit ⁇ circumflex over (X) ⁇ operator (i.e., the X-Pauli operator) because, as described in the above example of a CNOT gate, the ⁇ circumflex over (X) ⁇ operator does not maintain the noise bias of the qubit.
  • evolution equivalent to the ⁇ circumflex over (X) ⁇ operator are engineered using alternative techniques that do preserve the noise bias. To see how this is accomplished, it is noted that the cat states are eigenstates of the ⁇ circumflex over (X) ⁇ operator such that ⁇ circumflex over (X) ⁇
  • ⁇ ⁇ ⁇
  • the orientation of the cat state on the Bloch sphere is defined by a phase ⁇ of the two-photon drive field that creates the cat state in the PCO, where the Hamiltonian of the PCO is given by:
  • This Hamiltonian is the same as the previously discussed PCO Hamiltonian, but the drive field is no longer considered to be real and positive, resulting in the inclusion of the phase ⁇ .
  • this phase of the two-photon pump is varied to implement the CNOT gate. For example, if the phase is adiabatically changes from 0 to rc, then the cat states transform from
  • ⁇ ⁇ ⁇
  • a two-qubit bias-preserving CNOT gate is based on a conditional phase-space rotation of a target qubit based on the state of the control qubit.
  • the first and second terms in the tensor product refer to the control and target qubits, respectively, and the terms c i and d i are simply the probability amplitudes for each of the components of the superposition and can be arbitrarily chosen to be any initial state. If the phase of the two-photon drive applied to the target PCO is conditioned on the state of the control PCO, then the state of the system evolves as follows such that any given time t, the state is:
  • ⁇ ⁇ ⁇ ( t ) ⁇ c 0 ⁇ ⁇ 0 ⁇ ⁇ [ ( d 0 + d 1 ) ⁇ ⁇ C ⁇ + ⁇ + ( d 0 - d 1 ) ⁇ ⁇ C ⁇ - ⁇ ] + c 1 ⁇ ⁇ 1 ⁇ ⁇ [ ( d 0 + d 1 ) ⁇ ⁇ C ⁇ ⁇ ⁇ e i ⁇ ⁇ ⁇ ⁇ ( t ) + ⁇ + ( d 0 - d 1 ) ⁇ ⁇ C ⁇ ⁇ ⁇ e i ⁇ ⁇ ⁇ ( t ) - ⁇ ] .
  • a CNOT gate is realized by rotating the phase of the cat in the target PCO by ⁇ conditioned on the state of the control PCO.
  • the CNOT operation is realized because, during this rotation, the
  • This acquired phased difference between the two cat states is a topological phase that results from the state
  • the topological phase does not depend on energy like a dynamic phase does. Nor is the topological phase dependent on the geometry of the path, as is the case with a geometric phase.
  • the CNOT gate based on topological phase described above preserves the bias in the noise channel of the qubits.
  • a phase-flip error occurs in the control PCO during the CNOT gate evolution is equivalent to a phase-flip occurring on the control qubit after an ideal CNOT gate is performed.
  • a phase-flip error on the target PCO during the CNOT gate evolution is equivalent to phase-flip errors on the control and target qubits occurring after an ideal CNOT gate. Therefore, the CNOT gate according to some embodiments, does not un-bias the noise channel. This result contrasts with the aforementioned CNOT gate implements between two strictly two-level qubits and shows one advantage of using a larger Hilbert space (e.g., an oscillator) to perform quantum information processing.
  • a particular Hamiltonian is used to implement the time evolution of the state
  • ⁇ (t) the amplitude of the cats in the control PCO, a, and the target PCO, ⁇ , are different.
  • the following is the time dependent interaction Hamiltoinian that implements a bias-preserving CNOT gate between two PCOs according to some embodiments:
  • the first line is the parametrically driven nonlinear oscillator stabilizing the control cat-qubit.
  • the second line of the above expression shows that the cat states
  • the phase ⁇ (t) is changed adiabatically, respecting the limitation ⁇ dot over ( ⁇ ) ⁇ (t) ⁇
  • the different in the geometric phases acquired by the two cat states ⁇ g ⁇ (t) reflects the fact that the mean photon numbers are different for the two cat states and the area of the path followed by
  • the different in the two geometric phases decreases exponentially in ⁇ 2 such that
  • ⁇ g - ⁇ ( t ) - ⁇ g + ⁇ ( t ) 4 ⁇ ⁇ ⁇ ( t ) ⁇ ⁇ 2 ⁇ e - 2 ⁇ ⁇ 2 1 - e - 4 ⁇ ⁇ 2 .
  • the physical realization of the bias-preserving CNOT gate using three-wave mixing between two oscillators is a beam splitter coupling.
  • the oscillators are themselves fourth-order, Kerr nonlinear.
  • the Kerr nonlinearity of the oscillators themselves is sufficient to realize the CNOT interaction Hamiltonian and no additional coupling elements are necessary.
  • the coupling is controllable.
  • FIG. 13 is a schematic of a quantum information processing device 1300 configured to implement a bias-preserving CNOT gate, according to some embodiments.
  • FIG. 13 provides additional detail about the driving fields than are provided in, e.g., the block diagrams of FIG. 1 and FIG. 2 .
  • the schematic of FIG. 13 is a circuit diagram equivalent of the quantum information processing device 1300 .
  • the physical system in some embodiments, is implemented as discussed in connection with FIGS. 1-5 above.
  • the quantum information processing device 1300 includes a control qubit 1301 and a target qubit 1303 .
  • the qubits 1301 and 1303 are a Kerr nonlinear cavity.
  • the nonlinearity of the cavity may be controlled using a superconducting circuit element, such as a transmon or a SNAIL, as described above.
  • both the control qubit 1301 and the target qubit 1303 include a SNAIL.
  • the SNAIL of the control qubit 1301 has a resonance frequency of co
  • the SNAIL of the target qubit 1303 has a resonance frequency of ⁇ t .
  • the SNAILs are biased with an external magnetic field to engineer three- and/or four-wave mixing interactions between the control qubit 1301 and the target qubit 1303 .
  • a two-photon driven Kerr nonlinear oscillator results and may be used to create a PCO with a biased noise channel.
  • the control qubit 1301 and the target qubit 1303 are capacitively coupled to one another, as illustrated by the capacitor 1309 .
  • Microwave fields may be coupled to the control qubit 1301 via an input port 1305 and microwave fields may be coupled to the target qubit 1303 via an input port 1307 .
  • Microwave fields may be received from the microwave field generator 150 , discussed in connection with FIG. 1 .
  • microwave fields of more than one frequency may be applied to a given input port at one time.
  • the CNOT Hamiltonian ⁇ CX described above is implemented. Expanding the terms of the CNOT Hamiltonian can help understand what driving fields are needed to implement this Hamiltonian.
  • the expanded CNOT Hamiltonian may be written as:
  • ⁇ CX ⁇ Kâ c ⁇ 2 â c 2 ⁇ Kâ t ⁇ 2 â t 2 + ⁇ 2 ( â c ⁇ 2 +h.c .)+ K ⁇ 2 cos( ⁇ ( t ))( e i ⁇ (t) â t ⁇ 2 +h.c .) ⁇ ( K ⁇ 2 sin( ⁇ ( t ))/( ⁇ )( iâ t ⁇ 2 â c +h.c .)+( K ⁇ 4 /2 ⁇ )sin(2 ⁇ ( t ))( iâ c ⁇ +h.c .) ⁇ ( K ⁇ 4 sin 2 ( ⁇ ( t ))/ ⁇ 2 ) â c ⁇ â c ⁇ dot over ( ⁇ ) ⁇ ( t ) â t ⁇ â t /2+( ⁇ dot over ( ⁇ ) ⁇ ( t )/4 ⁇ )) â t ⁇ â t ( â c ⁇ +
  • ⁇ ⁇ ( t ) ⁇ 0 t ⁇ ( K ⁇ ⁇ ⁇ 4 ⁇ 2 ) ⁇ sin 2 ⁇ ( ⁇ ⁇ ( s ) ) ⁇ ds .
  • the Hamiltonian can be parametrically engineered using for-wave mixing based on the Kerr-nonlinearity and driving fields.
  • ⁇ (t) is a phase shift that changes over time and adiabatically increases from 0 to ⁇ in the time T.
  • the driving microwave fields corresponding to the amplitudes A 1 , A 2 , A 3 , A 4 , and A 5 are applied at the frequencies 2 ⁇ c , 2 ⁇ t , 2 ⁇ t ⁇ c , ⁇ c , and ⁇ c , respectively.
  • a particular sequence of fields are applied to the control qubit 1301 and/or the target qubit 1303 during an interaction time duration T. It is during this interaction time that the execution of the CNOT gate is performed.
  • the phases ⁇ i (t) are time-varying and change from a value of 0 to ⁇ .
  • the value of the phases between 0 and T may change in any way, as long as the changes are adiabatic.
  • FIG. 14A is a plot of the amplitude as a function of time of the five driving fields used to implement the CNOT gate, according to some embodiments
  • FIG. 14B is a plot of the time-dependent phases as a function of time for the five driving fields used to implement the CNOT gate.
  • a first microwave field is applied to the control cavity at a frequency 2 ⁇ c with a fixed amplitude A 1 and a time-dependent phase ⁇ 1 (t).
  • This first microwave field provide the two-photon term to drive the control cavity via three-wave mixing.
  • the fixed amplitude A 1 is illustrated by line 1401 in FIG. 14A and the time-dependent phase ⁇ 1 (t) is illustrated by line 1411 in FIG. 14B .
  • the phase decreases linearly.
  • a second microwave field is applied to the target cavity at a frequency 2 ⁇ t with a time-dependent amplitude A 2 and a time dependent-phase ⁇ 2 (t).
  • This second microwave field provide the two-photon term to drive the target cavity via three-wave mixing.
  • the changing amplitude A 2 is illustrated by line 1402 in FIG. 14A and the time-dependent phase ⁇ 2 (t) is illustrated by line 1412 in FIG. 14B .
  • the amplitude changes sinusoidally over time.
  • the phase is constant at a first phase value during a first portion of the gate time duration and constant at a second phase value during a second portion of the gate time duration, wherein the first phase value is less than the second phase value. This is because the amplitudes are always taken to be positive.
  • the amplitude A 2 which is a sine function, crosses the zero amplitude point, rather than going negative, the amplitude begins to increase again, and the phase takes on different value instead.
  • a third microwave field at a frequency 2 ⁇ t ⁇ c is applied to the target cavity at with a time-dependent amplitude A 3 and a time-dependent phase ⁇ 3 (t).
  • This third microwave field realize the coupling terms proportional to â t ⁇ 2 â c inn the CNOT Hamiltonian.
  • the changing amplitude A 3 is illustrated by line 1403 in FIG. 14A and the time-dependent phase ⁇ 3 (t) is illustrated by line 1413 in FIG. 14B .
  • the amplitude changes as a cosine over time. In the example shown, the phase increases linearly as a function of time.
  • a fourth microwave field at a frequency ⁇ c is applied to the control cavity at a with a time-dependent amplitude A 4 and a time-dependent phase ⁇ 4 (t).
  • This fourth microwave field realizes the single-photon drive of the control cavity.
  • the changing amplitude A 4 is illustrated by line 1404 in FIG. 14A and the time-dependent phase ⁇ 4 (t) is illustrated by line 1414 in FIG. 14B .
  • the amplitude changes as a cosine over time.
  • the phase decreases linearly during the first portion of the gate time duration and decreases linearly during the second portion of the gate time duration.
  • the linear decrease has the same slope in both portions of the gate time duration, but there is a jump in the phase half way through the gate time duration.
  • a fifth microwave field is applied to the target cavity at a frequency co, with a fixed amplitude A 5 and a time-dependent phase ⁇ 5 (t).
  • This fifth microwave field provide realizes the final term in the CNOT Hamiltonian.
  • the fixed amplitude A 5 is illustrated by line 1405 in FIG. 14A and the time-dependent phase ⁇ 5 (t) is illustrated by line 1415 in FIG. 14B .
  • the phase decreases linearly.
  • the inventors have recognized and appreciated that aspects of the stabilizer measurement scheme described above may be used to efficiently implement an error-correction code tailored to the biased noise because the measurement scheme preserves the noise bias.
  • the preparation of cat states in data qubits and ancilla qubits is described.
  • Quantum gates such as Z-axis rotations and ZZ( ⁇ ) gates are also described above.
  • measurements along the Z-axis can be performed, for example, using homodyne detection using the techniques above.
  • Measurements along the X-axis can be performed using additional gates and ancilla.
  • the inventors have recognized and appreciated that these state preparation techniques, quantum gates, and detections can be combined with the bias preserving CNOT gate between the two cat-qubits to implement universal fault-tolerant quantum computation.
  • some embodiments use the bias-preserving set of operations ⁇ CNOT, Z( ⁇ ), ZZ( ⁇ ),
  • the biased-noise qubits are encoded in a repetition code 1 and corrections are made for the dominant error types (e.g., phase flip errors).
  • a repetition code with n qubits can correct for (n ⁇ 1)/2 phase flip errors.
  • the codewords are
  • 0 L (
  • 1 L (
  • + L
  • ⁇ L
  • the result of this first encoding is a more symmetric noise channel with reduced noise strength.
  • the repetition code with errors below a threshold may then be concatenated to a CSS code 2 to further reduce errors.
  • the n ⁇ 1 stabilizer generators for the repetition code are ⁇ circumflex over (X) ⁇ 1 ⁇ circumflex over (X) ⁇ 2 ⁇ Î 3 ⁇ Î 4 . . . , Î 1 ⁇ circumflex over (X) ⁇ 2 ⁇ circumflex over (X) ⁇ 3 ⁇ Î 4 . . . , etc.
  • the most naive way to detect errors is used, which is to measure each stabilizer generator using an ancilla.
  • Such a technique is shown by the quantum circuit diagram 1500 of FIG. 15 .
  • Each ancilla 1504 - 1505 is initialized in the state
  • This decoding scheme is equivalent to constructing an r-bit repetition code for each of the (n ⁇ 1) stabilizer generators of the repetition code.
  • each bit of syndrome from the inner code is itself encoded in an [r, 1, r] repetition code so that decoding can proceed by first decoding the syndrome bits and then decoding the resulting syndrome.
  • This naive way to decode the syndrome results in a simple analytic expressions for the logical error rates.
  • the two-stage decoder of FIG. 15 can be replaced by a decoder that directly infers the most likely error on the n-qubit repetition code given s measured syndrome bits.
  • the notion of a measurement code is introduced that exploits the above insights to improve on the naive scheme by constructing a block code that can directly correct the bit-flip errors on the n data qubits in a single decoding step.
  • a classical code with parameters [n+2, n, d].
  • the classical parity checks should be compatible with the stabilizers of the original quantum code, in this example the repetition code.
  • each parity check in the measurement code should have even weight when restricted to the data qubits so that it commutes with the logical ⁇ circumflex over (Z) ⁇ L operator of the quantum phase-flip code.
  • consistency with the stabilizer group of the base quantum code is the only constraint on a measurement code.
  • the general form of a measurement code can be specified by the parity check matrix H M .
  • This in turn is specified as a function of the (generally redundant) parity checks H Z of the quantum repetition code and an additional set of s ancilla bits that label the measurements.
  • the rows of the H Z have even weight because the rows come from the stabilizers of a quantum repetition code.
  • the rows are linearly independent, making the associated code have parameters [n+s, n, d] for some d ⁇ n.
  • the distance is never greater than n since a string of ⁇ circumflex over (Z) ⁇ operators on the data qubits, corresponding to 1's on exactly the first n bits, is always in the kernel of H M .
  • the measurement of the j-th parity check in the measurement code can be done by a standard choice of circuit.
  • a CNOT gate is applied to the i-th qubit if there is a 1 in the i-th column, and target the ancilla labeled in column n+j. Note that by construction there is a 1 in position (j; n+j) of H M .
  • the effective error rate of this bare-ancilla measurement gadget depends on the number of CNOT gates used, and hence on the weight of the stabilizer being measured. Therefore, all other things (such as code distance) being equal, lower weight rows are preferred when designing a measurement code.
  • the two examples considered here are generated from the following choices for H Z , displayed here in transpose to save space:
  • H Z T ( 1 1 0 1 0 1 )
  • ⁇ H Z T ( 1 0 0 1 1 1 0 0 1 1 )
  • ⁇ H Z T ( 1 0 0 0 1 1 0 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 1 0 ) .
  • H Z T ( 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 1 1 ) .
  • the 3 ⁇ 3 choice above corresponds to a [6, 3, 3] measurement code whereas the naive repeated generator method yields a [12, 3, 3] measurement code.
  • the naive scheme yields an [n (n ⁇ 1)r , n, d(n, r)] code, and for smaller r the distance will not yet saturate to n.
  • the naive scheme yields either an [13, 5, 3] code or a [21, 5, 5] code, which are inferior in either distance or rate respectively to the [14, 5, 5] code that results from the choice.
  • These examples also illustrate a counterintuitive feature of measurement codes, according to some embodiments.
  • a decoder that uses the structure of the associated measurement code can correct 1 or 2 arbitrary data errors with these respective parameters, which then reduces the leading order behavior of the code failure probability.
  • an optimal (maximum likelihood) decoder is infeasible to implement because it requires exponential resources in n and s to compute, so substantially larger codes will need decoding heuristics such as message passing algorithms to approach peak decoding performance.
  • the decoder declares failure whenever the data error is not guessed exactly right, even though this is not necessary. When repeated rounds of error correction occur, it is sufficient to define success as reducing the weight of any correctable error.
  • FIG. 16 is a flowchart of a method 1600 of performing QIP that applies generally to most of the embodiments described above that use a data qubit coupled to an ancilla qubit.
  • the physical realization of the data qubit and the ancilla qubit may be any of the physical systems described above.
  • the method 1600 includes driving an ancilla qubit with a stabilizing field.
  • the stabilizing field generates the asymmetry in the error channel of an ancilla qubit that is exploited to measure error syndromes and perform bias-preserving quantum gates.
  • the stabilizing field may be applied to the ancilla qubit using the microwave field generator 160 .
  • the method 1600 includes creating a Kerr-nonlinearity in the ancilla qubit using at least one Josephson junction of the ancilla.
  • coupling a superconducting circuit element to a cavity creates the Kerr-nonlinearity.
  • a transmon or a SNAIL may be located in a 3D cavity to create a Kerr-nonlinear cavity.
  • the method 1600 includes applying a plurality of microwave fields to the ancilla qubit and the data qubit.
  • these microwave fields may be applied to create pumped cat states in the Kerr-nonlinear cavity.
  • the microwave fields may be applied to perform rotation on the states of the data qubit or the ancilla qubit.
  • the microwave fields may be applied to perform conditional gates, such as conditional rotations, on one qubit based on the state of another qubit.
  • the microwave fields may be applied to couple the ancilla qubit to the data qubit or to couple the ancilla qubit to a readout cavity. Or, as discussed above, any number of operations may be performed by applying microwave fields to the data qubit and/or the ancilla qubit.
  • the method includes measuring the ancilla qubit.
  • the ancilla qubit may be measured directly by, e.g., performing homodyne detection of a cavity of the ancilla qubit.
  • the ancilla qubit may be measured by coupling the ancilla qubit to a readout cavity, conditionally displacing the state of the readout cavity based on the state of the ancilla qubit, and then measuring the state of the readout cavity.
  • the measurement of the ancilla is a QND measurement.
  • FIG. 17 is a flowchart of a method 1700 for performing readout of an ancilla qubit, according to some embodiments.
  • the method 1700 may be used to measure a property of a data qubit that is coupled to the ancilla qubit.
  • the method 1700 may implement a QND measurement.
  • the method 1700 includes applying at least one rotation microwave field to the ancilla cubit.
  • the rotation may be about the Z-axis of a Bloch sphere associated with the ancilla qubit.
  • the rotation may rotate cat states from
  • the method 1700 includes turning off a stabilizing microwave field for an amount of time. In some embodiments, this allows the ancilla qubit to freely evolve.
  • the ancilla qubit may include a Kerr-nonlinear cavity and the state of the ancilla qubit may freely evolve under the Kerr-nonlinear Hamiltonian.
  • the free evolution of the ancilla qubit results in a rotation of the state of the ancilla qubit that could not be performed if the stabilizing field was still applied to the ancilla qubit.
  • the method 1700 includes re-applying the stabilizing microwave field to the ancilla qubit.
  • re-applying the stabilizing microwave field stops the free evolution of the state of the ancilla.
  • re-applying the stabilizing microwave field keeps the state of the ancilla in one of two coherent states.
  • re-applying the stabilizing field suppresses a particular type of error such that the error channel of the ancilla qubit is asymmetric.
  • the stabilizing field may suppress bit-flip errors.
  • the method 1700 includes applying an exchange microwave field to the ancilla qubit.
  • the exchange microwave field creates an interaction between the ancilla qubit and a readout cavity.
  • applying the exchange microwave field creates a three- or four-wave mixing interaction.
  • applying the exchange microwave field causes a Q-switch operation.
  • the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements.
  • This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified.
  • the phrase “equal” or “the same” in reference to two values means that two values are the same within manufacturing tolerances. Thus, two values being equal, or the same, may mean that the two values are different from one another by ⁇ 5%.
  • a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

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