EP4634834A2

EP4634834A2 - Techniques for dual-rail encoding of qubits and related systems and methods

Info

Publication number: EP4634834A2
Application number: EP23847949.7A
Authority: EP
Inventors: Robert J. SCHOELKOPF III; Steven M. GIRVIN; Shruti Puri; James TEOH; Stijn DE GRAAF; Sophia XUE; Benjamin Chapman; John Garmon; Aniket MAITI; Yao LU; Harshvardhan BABLA; William KALFUS; Neel THAKUR; Takahiro Tsunoda; Patrick WINKEL
Original assignee: Yale University
Current assignee: Yale University
Priority date: 2022-12-16
Filing date: 2023-12-15
Publication date: 2025-10-22
Also published as: KR20250123889A; JP2025540827A; WO2024130146A2; WO2024130146A3; CN120530403A; WO2024130146A8

Abstract

Techniques are described for quantum error correction that utilize a dual-rail encoding of a single excitation in a pair of coupled quantum oscillators. The dual-rail qubit may be implemented in a cQED (circuit quantum electrodynamics) system comprising a pair of coupled resonators. In this dual-rail encoding, an excitation is stored in one of the two resonators. An excitation of the first resonator is treated as a logical 0, whereas an excitation of the other resonator is treated as a logical 1. Thus together the two resonators form a single logical dual-rail qubit in which the logical states are |0>_L = |01> and |1>_L = |10>.

Description

TECHNIQUES FOR DUAL-RAIL ENCODING OF QUBITS AND RELATED SYSTEMS AND METHODS GOVERNMENT FUNDING [0001] This invention was made with government support under W911NF-18-1- 0212 awarded by the Army Research Office (ARO), and under DE-SC0012704 awarded by the Department of Energy (DOE). The government has certain rights in the invention. BACKGROUND [0002] Quantum information processing techniques perform computation by manipulating one or more quantum objects. These techniques are sometimes referred to as “quantum computing.” In order to perform computations, a quantum information processor utilizes quantum objects to reliably store and retrieve information. According to some quantum information processing approaches, a quantum analogue to the classical computing “bit” (being equal to 1 or 0) has been developed, which is referred to as a quantum bit, or “qubit.” A qubit can be composed of any quantum system that has two distinct states (which may be thought of as 1 and 0 states), but also has the special property that the system can be placed into quantum superpositions and thereby potentially exist in both of those states at once. SUMMARY [0003] In some aspects, the techniques described herein relate to a dual-rail qubit including: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to operate the at least one energy source to initialize the dual-rail qubit in a 0 or 1 logical state by: when the dual-rail qubit is to be initialized in the 0 logical state, operating the at least one energy source to arrange the first quantum oscillator in a single photon state and the second quantum oscillator in its ground state; or when the dual-rail qubit is to be initialized in the 1 logical state, operating the at least one energy source to arrange the first quantum oscillator in its ground state and the second quantum oscillator in a single photon state. [0004] In some aspects, the techniques described herein relate to a dual-rail qubit including: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to operate the at least one energy source to perform a single-qubit rotation of a logical state of the dual-rail qubit by operating the at least one energy source to direct energy to the coupling element one or more times. [0005] In some aspects, the techniques described herein relate to a dual-rail qubit including: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator and having a ground state |^^, a first excited state |^^ and a second excited state |^^; at least one energy source; and at least one controller configured to operate the at least one energy source to: (a) direct energy to the ancilla qubit to perform a first rotation of the quantum state of the ancilla qubit; (b) subsequent to (a), direct energy to the coupling element to perform a beamsplitter operation on the first quantum oscillator and the second quantum oscillator; and (c) subsequent to (b), direct energy to the ancilla qubit to perform a second rotation of the quantum state of the ancilla qubit. [0006] In some aspects, the techniques described herein relate to a dual-rail qubit including: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to: operate the at least one energy source to measure a parity state of the first quantum oscillator; operate the at least one energy source to measure a parity state of the second quantum oscillator; and determine a logical state of the dual-rail qubit based on the measured parity states of the first and second quantum oscillators. [0007] In some aspects, the techniques described herein relate to a dual-rail qubit including: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator and having a ground state |^^, a first excited state |^^ and a second excited state |^^; at least one energy source; and at least one controller configured to: operate the at least one energy source to perform one or more gates and/or operations at least in part on the dual-rail qubit; detect an erasure error subsequent to performing the one or more gates and/or operations; and in response to detecting the erasure error, operate the at least one energy source to initialize the dual- rail qubit in a new logical state. [0008] In some aspects, the techniques described herein relate to a system including: a first dual-rail qubit including: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and to the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; a second dual-rail qubit including: a third quantum oscillator; a fourth quantum oscillator; a second coupling element coupled to the third quantum oscillator and to the fourth quantum oscillator; and a second ancilla qubit coupled to the third quantum oscillator; and a third coupling element coupled to the second quantum oscillator and to the third quantum oscillator. [0009] In some aspects, the techniques described herein relate to a system including: a plurality of dual-rail qubits, each dual-rail qubit of the plurality of dual-rail qubits including: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and to the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; a plurality of measurement qubits, each measurement qubit of the plurality of measurement qubits including: a first quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator of the measure qubit, wherein each of the plurality of measurement qubits is coupled to four of the plurality of dual-rail qubits, with each coupling provided via a respective coupling element. [0010] In some aspects, the techniques described herein relate to a module including: a plurality of dual-rail qubits, each dual-rail qubit of the plurality of dual-rail qubits including: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and to the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; and a plurality of cavities each coupled to: one of the plurality of dual-rail qubits via a respective coupling element; and an external port configured for coupling to a quantum bus. [0011] The foregoing apparatus and method embodiments may be implemented with any suitable combination of aspects, features, and acts described above or in further detail below. These and other aspects, embodiments, and features of the present teachings can be more fully understood from the following description in conjunction with the accompanying drawings. BRIEF DESCRIPTION OF DRAWINGS [0012] Various aspects and embodiments will be described with reference to the following figures. It should be appreciated that the figures are not necessarily drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures is represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing. [0013] FIG.1 depicts an illustrative system suitable for practicing the techniques described herein, according to some embodiments; [0014] FIG.2A depicts an illustrative implementation of the system of FIG.1 comprising microwave cavities, according to some embodiments; [0015] FIG.2B depicts couplings between the elements of FIG.2A, according to some embodiments; [0016] FIG.2C depicts logical states of the dual-rail qubit of FIG.2A, according to some embodiments; [0017] FIG.3A represents parity measurements of a dual-rail qubit, according to some embodiments; [0018] FIG.3B summarizes the different measurement outcomes of the parity measurement sequence of FIG.3A, according to some embodiments; [0019] FIG.4 depicts an illustrative process for implementing a dual-rail qubit with single qubit gates and erasure checks, according to some embodiments; [0020] FIGs.5A-5B depict an alternative scheme for erasure detection, according to some embodiments; [0021] FIG.6 depicts one aspect of the evolution of the dual-rail qubit over time during application of the drives shown in FIGs.5A-5B, according to some embodiments; [0022] FIG.7A depicts an illustrative implementation of the system of FIG.1 comprising two dual-rail logical qubits implemented with microwave cavities, according to some embodiments; [0023] FIG.7B depicts couplings between the elements of FIG.7A, according to some embodiments; [0024] FIG.8 depicts an illustrative implementation of a ^^^^^ gate between two dual-rail qubits, according to some embodiments; [0025] FIGs.9A-9B depict illustrative drives for performing a CPHASE^^^ gate between two dual-rail qubits, according to some embodiments; [0026] FIG.10 depicts an illustrative example of a network of dual-rail qubits, according to some embodiments; [0027] FIG.11 depicts an illustrative example of a physical arrangement of dual- rail qubits that may be grouped and operated as a logical qubit, according to some embodiments; [0028] FIG.12A depicts multiple modules each comprising multiple dual-rail qubits that may be grouped and operated as a logical qubit, according to some embodiments; and [0029] FIG.12B depicts an illustrative quantum bus that may act as an inter- module link in the example of FIG.12A, according to some embodiments. DETAILED DESCRIPTION [0030] Quantum multi-level systems such as superconducting qubits exhibit quantum states that, based on current experimental practices, decohere in around ~100µs. While experimental techniques will undoubtedly improve on this and produce qubits with longer decoherence times, it may nonetheless be beneficial to couple a multi-level system to another system that exhibits much longer decoherence times. A system configured with bosonic modes may be particularly desirable for coupling to a multi- level system. Through this coupling, the multi-level system’s state may be represented by the bosonic mode(s) instead, thereby maintaining the same information yet in a longer-lived state than would otherwise exist in the multi-level system alone. When used in this manner, the bosonic system is sometimes referred to as a “logical” qubit. [0031] Quantum information stored in bosonic modes may nonetheless still have a limited lifetime, such that errors will still occur within the bosonic system. It may therefore be desirable to manipulate a bosonic system when errors in its state occur to effectively correct those errors and thereby regain the prior state of the system. If a broad class of errors can be corrected for, it may be possible to maintain the state of the bosonic system indefinitely (or at least for long periods of time) by correcting for any type of error that might occur. [0032] Quantum information stored in bosonic modes may nonetheless still have a limited lifetime, such that errors will still occur within the bosonic system. It is desirable to manipulate a bosonic system when errors in its state occur to effectively correct those errors and thereby regain the prior state of the system. In some cases, it may be sufficient to simply detect that an error occurred during an operation. The desire to detect and/or correct errors in a quantum system requires that a number of requirements are simultaneously addressed, including the number of physical qubits used to encode logical information, a suitable scheme for performing gates and measurements on the quantum system, and the precision of the physical components of the system. [0033] The inventors have recognized and appreciated that not all types of errors in a given quantum error correction architecture are equally likely to occur, nor are they equally harmful to the logical information stored in a quantum system. Some types of errors, for instance, require that a particular error syndrome is measured to pinpoint an error and to gain the required information to properly fix the error. This does not always lead to successful error correction, however, as sometimes there is not enough information to unambiguously determine the error. On the other hand, erasure errors - errors in which the information in a qubit is erased but in a way that allows the qubit to be identified – can be much more easily addressed. [0034] The inventors have further recognized and appreciated techniques for error correction that utilize a dual-rail encoding of a single excitation in a pair of coupled quantum oscillators. In particular, the dual-rail qubit may be implemented in a cQED (circuit quantum electrodynamics) system comprising a pair of coupled resonators. In this dual-rail encoding, an excitation is stored in one of the two resonators. An excitation of the first resonator is treated as a logical 0, whereas an excitation of the other resonator is treated as a logical 1. Thus together the two resonators form a single logical dual-rail qubit in which the logical states are |0^_^ = |01^ and |1^_^ = |10^. This type of logical qubit has several benefits, which include the following. First, photon loss appears as an erasure error, which is among the easier types of errors to correct, as described above. Second, the single photon state is the lowest energy state of the cavity and thereby has the lowest error rate of any state of the cavity. As, such the dual-rail encoding minimizes the rate of loss errors. Third, photon gains or losses are readily detectable by measuring the joint parity of the cavities. [0035] According to some embodiments, a dual-rail qubit may be implemented as a cQED system comprising two quantum resonators (e.g., microwave cavity resonators) coupled together via a suitable non-linear coupling element such as a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL) or a superconducting quantum interference device (SQUID). One of the quantum oscillators may be coupled to an ancilla qubit (e.g., a transmon qubit). Although the ancilla qubit only couples to one of the bosonic modes of the system, due to a beamsplitter interaction provided by the coupling element (described below), both bosonic modes interact with the ancilla qubit, enabling various dual-rail operations. For instance, a dual-rail state may be prepared, and gates may be performed upon a dual-rail state, through application of energy (e.g., microwave pulses) to the ancilla qubit and/or to the coupling element, as described further below. [0036] According to some embodiments, the logical quantum state of a dual-rail qubit is prepared by preparing a single excitation of one of the two quantum oscillators of the dual-rail qubit. For instance, to prepare the |0^_^ state (the logical 0 state), both oscillators may be prepared in their respective ground state |0^, and the first oscillator excited to its |1^ state while the second oscillator remains in its |0^ state. Similarly, to prepare the |1^_^ state (the logical 1 state), both oscillators may be prepared in their respective ground state |0^, and the second oscillator excited to its |1^ state while the first oscillator remains in its |0^ state. In the case of a cQED dual-rail qubit comprising two quantum resonators, for example, one of the resonators may be prepared in a |1^ state by loading a single photon into the resonator. [0037] According to some embodiments, the logical state of a dual-rail qubit may be measured through one or more parity measurements of either and/or both of the two quantum oscillators (e.g., by performing one or more quantum non-destructive (QND) measurements). Since the dual-rail logical state has an odd joint parity, a change in ^{parity indicates an error, which is either an erasure error which produces the joint state} _{|00^, or an excitation into one of the states |11^, |20^ or |02^. When implementing the} dual-rail qubit as a cQED system comprising two quantum resonators, parity measurements may be performed by applying one or more drives to an ancilla qubit coupled to one of the resonators and/or to a non-linear coupling element that couples the resonators to one another, examples of which are described below. [0038] According to some embodiments, a plurality of dual-rail qubits may be arranged in a repeating arrangement, such as an array or lattice. Such an arrangement may provide for an improved response to errors by treating multiple dual-rail qubits as a single logical group. For instance, a plurality of dual-rail qubits may be coupled and operated in a surface code architecture in which nominally identical dual-rail qubits play a role of either a data qubit, or a measurement qubit on which stabilizer measurements are performed as an error correction process. [0039] Following below are more detailed descriptions of various concepts related to, and embodiments of, implementation and operation of dual-rail qubits. It should be appreciated that various aspects described herein may be implemented in any of numerous ways. Examples of specific implementations are provided herein for illustrative purposes only. In addition, the various aspects described in the embodiments below may be used alone or in any combination, and are not limited to the combinations explicitly described herein. [0040] An illustrative system suitable for practicing the techniques described herein is shown in FIG.1, according to some embodiments. In system 100, oscillators 101 and 102 are coupled to one another via coupling element 103. The oscillator 101 is also coupled to an ancilla qubit 104. Energy source 105 may be operated by controller 106 to direct energy to the ancilla qubit 104, the coupling element 103, and/or the readout resonator 107. [0041] According to some embodiments, the oscillator 101 and the oscillator 102 each includes a cavity that supports quantum states of microwave photons. For example, in some embodiments, the first oscillator 101 and the second oscillator 102 may each comprise a resonator, such as a transmission line resonator or three-dimensional cavity formed from a superconducting material, such as aluminum. [0042] Coupling element 103 may comprise a non-linear element that is coupled to the oscillators 101 and 102. The coupling element 103 mediates coupling between the quantum states of the two oscillators, allowing for interactions between the first oscillator 101 and the oscillator 102. In some embodiments, the coupling element 103 may be a superconducting nonlinear asymmetric inductive element (SNAIL), a superconducting quantum interference device (SQUID), a Josephson junction, or some other non-linear element. In some embodiments, the coupling element 103 may comprise a transmon qubit that is dispersively coupled to both the first oscillator 101 and the second oscillator 102. [0043] In some embodiments, the ancilla qubit 104 may be a transmon qubit, a SNAIL, a SQUID or some other non-linear element. [0044] An illustrative implementation of system 100 is shown as dual-rail qubit 200 in FIG.2A. In this implementation, the oscillators are implemented as bosonic modes stored in cavities 201 and 202 (e.g., microwave cavities). A microwave source (not shown in FIG.2A) may be configured as the energy source 105 in this system and configured to direct microwave pulses of desired amplitudes, frequencies and phases to the ancilla qubit 204 (e.g., a transmon qubit), to the coupling element 203, and/or to the readout resonator 207. Such a microwave source may be coupled to the ancilla qubit and to the coupling element. The coupling between the microwave source and these components provides a way for the microwave source to apply microwave radiation to the components. [0045] In some embodiments, the energy source 105 may be capacitively coupled to each of the ancilla qubit 204, the coupling element 203, and the readout resonator 207. In some embodiments, a plurality of instances of the dual-rail qubit shown in FIG.2A may be arranged coupled with one or more instances of the energy source 105, and with one or more controllers 106 configured to perform operations on the plurality of dual-rail qubits. [0046] In the example of FIG.2A, a microwave source (not shown) may be operated to readout the state of the ancilla qubit 204. For instance, the readout resonator 207 may be arranged such that its resonant frequency (e.g., ~GHz) is far from the transition frequency of the ancilla qubit 204 (e.g., dispersively coupled). The coupling between the ancilla qubit and readout resonator means that there is a shift in the resonator frequency that is dependent on the state of the qubit. This shift is small compared with the resonant frequency of the resonator (e.g., ~MHz). As a result, sending a tone to the readout resonator 207 near the resonant frequency will be reflected by the resonator and the form of the reflected tone (also referred to herein as the “readout signal”) can be analyzed to determine the state of the ancilla qubit. In this manner, the state of the ancilla qubit 204 can be probed non-destructively by sending probe tones to the readout resonator 207, which is dispersively coupled to the qubit. [0047] The dual-rail qubit 200 has logical code words that are encoded in the single-photon subspace of the qubit, with |0^_^ = |01^ and |1^_^ = |10^, for which the beamsplitter interaction is sufficient to facilitate arbitrary single-qubit rotations. These states are shown in FIG.2C as the “logical subspace.” By encoding the logical state of the qubit in the single-photon subspace of the joint cavity Hilbert space with odd joint photon-number parity, the dominant errors in the cavity system, namely single-photon loss to the common ground state |00^, can be converted into detectable erasure errors by means of joint-parity measurements. In addition, photon gain events in the cavities, which result in states |02^, |11^ or |20^ and are generally rare in thermal equilibrium, are also detectable by joint-parity measurements. This leaves cavity dephasing as the dominant Pauli error in the system. [0048] In the example of FIGs.2A-2B, the Hamiltonian of the dual-rail qubit 200 combines a beamsplitter interaction between the cavities 201 and 202, with a dispersive interaction between the ancilla qubit 204 and modes of the cavity 201. This Hamiltonian may be written as: where ^{and ^^^ ≡ |^^^^| − |^^^^| is the Pauli ^ operator in the two-level subspace defined by the} _{|^^ and |^^ levels of the ancilla. A three-level ancilla qubit is considered in this} example, which may have a benefit of allowing use of the |^^ level for detecting a single ancilla decay event. Although, in some implementations described below, a two-level ancilla qubit may be implemented. [0049] In the example of FIGs.2A-2B, the term ℋ^{^} _BS/ℏ represents the beamsplitter coupling between the cavities 201 and 202 as generated by the coupling element 203, and the term ℋ^{^} _^/ℏ represents the dispersive coupling between the cavity 201 and the ancilla qubit 204. These couplings are illustrated in FIG.2B. In the Hamiltonian ℋ^{^} _^, the annihilation operators ^^ and $^% act on the bosonic modes of cavities 201 and 202, respectively, ^_^^^^^ is the complex amplitude of the beamsplitter interaction between the cavities 201 and 202, (^^^ is an effective detuning between the modes of the two cavities, and ^ is the strength of the dispersive interaction between the ancilla qubit 204 (in the ^^-manifold) and mode ^^ of the cavity 201. This Hamiltonian ℋ^{^} _^BS is written in ^{a frame where the dispersive interaction is symmetric, shifting the frequency of ^^ by} d_{ependent on the ancilla state.} [0050] Parameters of the Hamiltonian ℋ^{^} _^BS may be controlled and varied through suitable selection of microwave drive signals applied to the coupling element 203 and to the ancilla qubit 204. For instance, the coupling strength its phase ,^^^, and detuning (^^^ can all be rapidly varied via microwave drive techniques. As described below, operations can be engineered in this system by actuating microwave drives while engineering these parameters, with different values of the parameters corresponding to different operations. [0051] According to some embodiments, the coupling element 203 may comprise a nonlinear asymmetric inductive element (SNAIL), which is a superconducting circuit ^{comprising a loop with multiple (e.g., three) Josephson junctions with tunnelling energy} _{E. on one arm of the loop and a single Josephson junction on the other arm of the loop} with tunnelling energy less than E_., and two large capacitor pads connected to the two arms of a loop. In this approach, the SNAIL is arranged between cavities 201 and 202 to generate a capacitive coupling between the SNAIL mode and each of the cavity modes. A magnetic flux is also thread through the loop of the SNAIL to bias to an operating point where the SNAIL has a 3rd order nonlinearity (e.g., using a superconducting flux transformer). The SNAIL is described in US Patent No.11,737,376, which is incorporated herein in its entirety. [0052] According to some embodiments, performing an operation on the dual-rail qubit 200 may comprise pumping the coupling element to initiate a so-called “beamsplitter” operation. In particular, the nonlinearity of the coupling element may enable three or four wave mixing, which is used to perform a frequency-converting bilinear coupling between the first cavity and the second cavity. In the case of a coupling element that is a SNAIL, for instance, the beamsplitter operation may comprise three wave mixing, whereas in the case of a coupling element that is a SQUID or transmon, the beamsplitter operation may comprise four wave mixing. [0053] According to some embodiments, a beamsplitter operation may be performed by applying a microwave drive to the coupling element that satisfies the frequency matching condition /₀ − /₁ = /₃₁ − /₃₀, where /₀ is the resonant frequency of the first cavity, /₁ is the resonant frequency of the second cavity, /₃₀ is the frequency of the first pump associated with a mode c, and /₃₁ is the frequency of the second pump associated with a mode d. This frequency matching condition relates to the four wave mixing case (e.g., where the coupling element is a SQUID or a transmon). In ^{the case of a coupling element that is a SNAIL, the frequency matching condition is} _{/0 − /1 = /3. That is, when the coupling element 203 is a SNAIL, a drive frequency} may be applied to the SNAIL that is equal to the difference between the resonant frequencies of the cavities 201 and 202. In general, the drive frequency /₃ may be detuned from this frequency /₃ = /₀ − /₁ − (. When ( = 0, the drive implements what is hereafter referred to as a “resonant beamsplitter operation,” which can be applied to swap photons between the two cavities. When ( ≠ 0, the drive implements what is hereafter referred to as a “detuned beamsplitter operation,” which can be applied to perform entangling gates between dual-rail qubits as described below. The amplitude and phase of the microwave drive may be selected to choose the strength and the phase, respectively, of the beamsplitter operation. [0054] The length of time during which the beamsplitter drive ^_^^ ^{^}^^{^} is applied to the coupling element 203 determines the particular operation that is applied to the two cavities 201 and 202. In particular, writing the integral of the beamsplitter drive during a time period ^1 as: ^{allows for parameterization of the type of beamsplitter operation. When ^ = 8 and ( =} _{0, the beamsplitter unitary operator implements a SWAP operation that exchanges the} ^{states between the cavity modes of the cavities 201 and 202. When ( = 0 and ^ =} _{±8/2, the operation produced by the resonant beamsplitter operation corresponds to a} 50/50 beamsplitter. Such operations may be referred to “beamsplitter” operations in that they can be viewed as playing the same role and implement the same unitary operation as beamsplitters do in linear optics quantum computation (LOQC) schemes, although here the beamsplitter operation is achieved by pumping a non-linear coupling element that couples together two cavities – a wholly distinct configuration and process from an optical beamsplitter. As described below, beamsplitter operations may be performed as part of various operations of a dual-rail qubit, such as photon number parity measurements, single-qubit gates, entangling gates between two dual-rail qubits, etc. [0055] According to some embodiments, state preparation of a dual-rail qubit may comprise loading a single photon into one of the two cavities, where the cavity selected depends on whether the |0^_^ or |1^_^ dual-rail qubit logical state is being prepared (e.g., a single photon may be loaded into cavity 201 to prepare the |0^_^ state, or a single photon may be loaded into cavity 202 to prepare the |1^_^ state). Loading of a single photon may, in some embodiments, be performed via optimal control pulses or via cavity-ancilla sideband drives. [0056] A logical state of a dual-rail qubit can be measured by determining whether the single photon is in cavity 201 (state |0^_^) or cavity 202 (state |1^_^). If it is instead determined that the photon is in neither cavity, this indicates a leakage event due to photon loss. [0057] According to some embodiments, the logical state of a dual-rail qubit may be determined through parity measurements of the two cavities, described below. The parity measurements may in some cases be performed subsequent to state preparation to verify that the state of the dual-rail qubit was prepared as intended. For example, subsequent to loading a single photon into one of the two cavities of a dual-rail qubit, the parities of each cavity in the dual-rail qubit are measured, and the results of the parity measurements examined to determine whether or not the state of the dual-rail qubit was prepared in the intended state. [0058] Parity measurements of a dual-rail qubit are represented in FIG.3A, according to some embodiments. One illustrative way to perform the parity measurement of cavity 201 is to arrange the ancilla qubit 204 in its ground state |^^ and ^{to drive the ancilla qubit with a 8/2 pulse to prepare the ancilla qubit in the state |^^ +} _{|^^ (normalization factor omitted). Subsequent to the 8/2 pulse, the ancilla qubit 204 is} not driven for a time period 8/^_9:, where ^_9: is the strength of the dispersive interaction between the ancilla qubit 204 and the cavity 201 in the ^^-manifold. During this time, the ancilla qubit 204 and the cavity 201 are interacting through their dispersive interaction ℋ^{^} _^/ℏ. Subsequent to this waiting period in which the ancilla qubit is not driven, the ancilla qubit is then driven with another 8/2 pulse. After this pulse, the ancilla qubit 204 will be in its excited state |^^ if there are an odd number of photons in the cavity 201, and in its ground state |^^ if there are an even number of photons in cavity 201. As such, measuring the state of the ancilla qubit at this stage produces a measurement result that is indicative of the photon number parity of cavity 201. [0059] The parity of one cavity may be measured in this way, then a beamsplitter operation may be performed to exchange the two cavity states, as described above. The parity of the other cavity may then be measured in the same way, and the cavity states exchanged again through another beamsplitter operation to return the states to their original configuration. For example, a sequence of parity measurements may comprise: a) measuring the parity in cavity 201 by driving the ancilla 204 with a 8/2 pulse, waiting for a time period 8/^_9:, driving the ancilla 204 with a 8/2 pulse, and measuring a state of the ancilla qubit 204; b) driving coupling element 203 to perform a beamsplitter operation that exchanges the states in cavities 201 and 202; c) a) measuring the parity in cavity 201 by driving the ancilla 204 with a 8/2 pulse, waiting for a time period 8/^_9:, driving the ancilla 204 with a 8/2 pulse, and measuring a state of the ancilla qubit 204; and d) driving coupling element 203 to perform a beamsplitter operation that exchanges the states in cavities 201 and 202. ^{[0060] If the two measured states of the ancilla qubit in acts (a) and (c) are both} _{|^^, this indicates an erasure error. The ancilla qubit states should instead measure |^^} ^{then |^^ in acts (a) and (c), respectively, when the single photon is in cavity 201 (state} _{|0^^). The ancilla qubit states should instead measure |^^ then |^^ in acts (a) and (c),} respectively, when the single photon is in cavity 202 (state |1^_^). [0061] The acts (a), (b), (c) and (d) may be repeated in this order one or more times (denoted as “n rounds” in FIG.3A) to perform repeated parity measurements and determining the dual-rail qubit logical state based on the collection of these measurements (e.g., by taking the majority rule result). [0062] The different measurement outcomes of a parity measurement sequence as described above are summarized in FIG.3B. The table denotes each state’s input probability, where ; is the probability of a photon loss event prior to the measurement, which would leave both cavities in the state |0^. Depending on the input state, the outcome of a parity measurement is either correctly assigned to the actual input state (check marks), incorrectly flagged as an erasure (<), or incorrectly assigned to the logical states (cross) resulting in a Pauli error. [0063] The probability of mistaking |0^_^ for |1^_^ (and vice versa) should be exceedingly small even in the presence of decoherence. First-order errors in the transmon and cavities give measurement outcomes that may be flagged as additional erasure errors. A combination of at least two errors is necessary for logical misassignment, a probability that is estimated to be below 10^&= for realistic coherence times and transmon readout fidelities . [0064] According to some embodiments, single qubit gates of the dual-rail qubit 200 may be performed by with resonant beamsplitter operation, as described above. To perform a single qubit gate the coupling element 203 is driven at a frequency equal to the difference in frequencies between the resonant frequencies of the cavities 201 and 202, with the beamsplitter drive amplitude, phase and duration control the type of single qubit gate. In some cases, for instance, the strength and duration of the beamsplitter drive control the polar angle of a rotation around the Bloch sphere, whereas the phase of the beamsplitter drive controls rotations around the Z-axis of the Bloch sphere. [0065] An important advantage of the dual-rail qubit described herein is the ability of perform QND detection of leakage out of the logical subspace caused by loss or gain of a photon in either or both cavities. For QEC protocols, the knowledge about the exact location of the error enables conversion of these otherwise pernicious leakage events into erasure errors, which are much easier to correct than Pauli errors. This leakage detection scheme may be referred to herein as an “erasure check,” or as “erasure detection.” [0066] According to some embodiments, an erasure check comprises measuring the joint photon number parity of the cavities 201 and 202 in dual-rail qubit 200. Since both logical states |0^_^ for |1^_^ have odd joint parity (i.e., there is one photon in the cavities combined in each case), the gain or the loss of a photon in either cavity changes the joint parity to even. [0067] Measuring the joint photon number parity of the cavities 201 and 202 in dual-rail qubit 200 may comprise applying a first on-resonant pulse to the ancilla qubit 204, then applying a beamsplitter operation to the coupling element 203, then applying a second on-resonant pulse to the ancilla qubit 204. This sequence of operations may cause photon number information to be mapped onto the ancilla, which is then measured. [0068] In some embodiments, the ancilla qubit 204 may be a three-level ancilla having a ground state |^^, first excited state |^^ and second excited state |^^. In this case, two of these states may be utilized to detect erasure errors caused by decay of the ancilla during the joint parity measurements as well as erasure due to photon loss. For example, ^{the resulting state of the ancilla after this sequence of operations may be measured to be:} _{|^^ when the joint parity is odd; |^^ when the joint parity is even; and |^^ when the state} of the ancilla decayed during the sequence of operations. Since the cavities of the dual- rail qubit should have an odd joint parity, the ancilla states |^^ and |^^ both indicate an erasure error, and |^^ indicates a successfully measure odd photon number parity. [0069] According to some embodiments, measuring the joint photon number parity of the cavities 201 and 202 in dual-rail qubit 200 may comprise driving the ancilla qubit 204 to perform a Y^^? 1^ rotation within the ancilla qubit’s ^-^ manifold (operation 411), then later performing another Y^^? 1^ rotation within the ancilla qubit’s ^-^ manifold (operation 413). In the period between the two operations performed on the ancilla qubit, the dispersive interaction between the ancilla qubit and cavity 201 maps photon number information to the ancilla state. A beamsplitter operation 412 is also applied during this period to exchange the states of the cavities, causing photon number information of both cavities to be mapped to the ancilla qubit. As one example, the beamsplitter drive strength may be ^_@A = _√3^/2 with a frequency detuned ( = ^/2 from resonance, which causes joint photon number parity information to be mapped to the ancilla qubit’s state, while allowing the cavities to return to their initial states at the end of the sequence (up to a deterministic cavity phase shift that can be tracked). [0070] An illustrative process for implementing a dual-rail qubit with single qubit gates and erasure checks is depicted in FIG.4, according to some embodiments. In the example of FIG.4, one or more single qubit gates 402 may be performed by driving the coupling element 203 of the depicted dual-rail qubit 200 with a beamsplitter drive ^_@A as described above. Subsequent to performing the one or more single qubit gates, an erasure check 404 may be performed through the joint photon number parity sequence described above and shown in inset 405. [0071] An erasure error may be detected by mapping the joint photon number parity onto the ancilla in a variety of instances. First, idling errors of the dual-rail qubit (dominantly photon loss) may be converted to erasure errors. Second, errors resulting from state preparation may be converted to erasure errors. Third, errors resulting during single-qubit gates may be detected by mapping the joint parity onto the ancilla and measuring the state of the ancilla subsequent to performing the single-qubit gate. [0072] An alternative scheme for erasure detection is depicted in FIGs.5A-5B, according to some embodiments. The drive scheme shown in the example of FIGs.5A- 5B differs from that of FIG.4 in that the scheme of FIGs.5A-5B only detects whether or not the system is in the zero photon state (i.e., where both cavities are in |0^), and involves simultaneously driving the ancilla qubit 204 and the coupling element 203. In addition, the ancilla qubit 204 is only driven once. In this scheme, the ancilla qubit 204 is excited to the |^^ state only when the cavities are both in the |0^ state, while the cavities states remain unchanged. [0073] In the example of FIGs.5A-5B, a drive ^_@A is applied to the coupling element 203 with a strength ^_@A > ^_9: a frequency detuned ( = ^_9:/2 from the resonance condition as described above. ^_9: is the strength of the dispersive interaction between the ancilla qubit 204 in the ^^-manifold. Concurrent with this drive, another drive is applied to the ancilla qubit 204 at a frequency that is the difference between the ^{ancilla qubit’s |^^ and |^^ states when the cavities contain no photons, and with a phase} _{8. As a result, the beamsplitter drive becomes resonant only if there are no photons in} the cavities. The two approaches to this erasure detection scheme shown in FIGs.5A and 5B differ in the duration of the pulses and in the pulse shapes of the ancilla drive. In the example of FIG.5A, the ancilla qubit drive is applied for a time period of approximately 28/^_9:, whereas in the example of FIG.5B, the ancilla qubit drive is applied for a time period of approximately 48/^_9:. [0074] One potential advantage to the erasure scheme of FIGs.5A-5B is that, since the detection result is based on only the |^^ and |^^ states of the ancilla qubit, a three level ancilla qubit may not be needed in the dual-rail qubit 200. This differs from the approach shown in FIG.4, where a three level ancilla qubit is used to distinguish between different outcomes. [0075] FIG.6 depicts one aspect of the evolution of the dual-rail qubit over time during application of the drives shown in FIGs.5A-5B, according to some embodiments. In the example of FIG.6, the time period over which the ancilla qubit drive is applied is shown along the horizontal axis, and the probability of measuring the ancilla qubit to be in |^^ is shown along the vertical axis. Each of the four curves represent a different state of the dual-rail qubit, with the cavity states are written as a two digit state where |11^ refers to both cavities being in the |1^ state, etc. It may be noted from FIG.6 that the dual-rail states ^|01^ + |10^^/_√2 and ^|01^ − |10^^/_√2 are associated with unchanged probabilities by the time the ancilla drive ends at 601, as does the state |11^. However, the state |00^, which this scheme is designed to detect, is associated with a probability at or close to 1 by the end of the ancilla drive at 601. [0076] Entangling gates between dual-rail qubits can also be constructed so that both cavity and transmon errors can also be detected and converted to erasures, as will be described below. The inventors have recognized that interactions with only a single cavity of a dual-rail qubit are necessary to realize the logical Z operator of the dual-rail. By driving a beamsplitter interaction between a pair of cavities, one from each of the two dual-rail qubits, a single transmon can effectively act as a control on both dual-rail qubits. A suitable system for performing such operations is shown in FIG.7A, according to some embodiments. [0077] In the example of FIG.7A, a pair of dual-rail logical qubits 701 and 702 are depicted coupled to one another by a coupling element 703. Dual-rail qubit 701 includes cavities 711 and 712 (e.g., microwave cavities), which are coupled together via coupling element 713; and dual-rail qubit 702 includes cavities 721 and 722 (e.g., microwave cavities), which are coupled together via coupling element 723. Each of cavities 712 and 722 is coupled to a respective ancilla qubit 714 or 724 (each may for instance be a transmon qubit) coupled to a respective readout resonator. As with the coupling element 203 in dual-rail qubit 200, each of the coupling elements 703, 713, and 723 may be a superconducting nonlinear asymmetric inductive element (SNAIL), a superconducting quantum interference device (SQUID) or some other non-linear element. [0078] Two-qubit gates as described above may be performed on the two dual-rail logical qubits by directing energy to the coupling element 703 between the dual-rail qubits (instead of, for instance, the coupling element 203 between the two logical qubits implemented by cavities 201 and 202 as in the example of FIG.2A). When performing two qubit gates on a pair of dual-rail qubits, the ancilla qubit 724 that is coupled to cavity 722, which is coupled to cavity 711 of the other dual-rail qubit via coupling element 703 may be operated as the ancilla qubit in a two-qubit gate scheme. As such, two-qubit operations may be applied to ancilla qubit 724, and any errors that occur during performance of the two-qubit gate can be detected by measuring the state of the ancilla qubit 724 and determining whether the ancilla qubit is in the state |^^, |^^ or |^^. The ^{couplings depicted in FIG. 7A are further illustrated in FIG. 7B, indicating that modes} c_{omprise logical qubit 701 and} _{$%1' comprise logical qubit 702.} [0079] One two-qubit entangling gate is the ^^^^^ gate, which can be written as acting on the two-qubit logical subspace. When ^ = 8/2 this gate is locally equivalent to a CNOT or CZ gate . A ^^^^^ gate acting on the subspace {|00^, |01^, |10^, |11^} as if each cavity were encoded in the Fock {|0^, |1^} code will also realize the ^^^^^ for the dual-rail code, provided each of these cavities belongs to a different dual-rail qubit. [0080] One motivation for implementing the ^^^^^ gate is that it can be performed with the exact same hardware required for the erasure checks discussed in the previous section. Similar to the error detection of the erasure check, it is important to detect ancilla qubit errors that happen while performing the gate. The arrangement of FIGs.7A- 7B allows first-order transmon errors to be detected when the transmon is measured at the end of the gate, erasing both dual-rail qubits if the transmon is not in |^^. Once again, this operation is designed such that Pauli errors are only introduced from second-order hardware errors, when two decoherence events happen during a single gate. This approach is depicted in FIG.8, according to some embodiments; ^{[0081] In the example of FIG. 8, dual-rail qubits 701 and 702 are depicted, and a} _{^^^^^ gate may be performed on these dual-rail qubits through the sequence of} operations shown in inset 801. The circuit depicted in inset 801 allows for detection of a single ancilla dephasing error in addition to ancilla decay events during the ^^^{^}^^{^} gate by measuring the state of the ancilla qubit in operation 816. The state of the ancilla qubit 724 acts as a flag to indicated whether or not the gate represented by operations 811, 812, 813, 814 and 815 was performed without ancilla dephasing or ancilla decay errors. In particular, if the state of the ancilla qubit 724 is the ground state |^^ after performing operations 811, 812, 813, 814 and 815, this indicates no such error occurred. Otherwise, if the state of the ancilla qubit is the first excited state |^^ or the second excited state |^^, this indicates that at least one such error occurred while performing operations 811, 812, 813, 814 and 815. [0082] Operation 811 is a Y^^? 1^ rotation within the ancilla qubit 724 ^-^ manifold, operation 815 is a Y^− ^? 1^ rotation within the ancilla qubit 724 ^-^ manifold, and operation 813 is a X^^^ rotation within the ancilla qubit 724 ^-^ manifold, which parameterizes the ^^^{^}^^{^} gate (e.g., when ^ = 8/2 this is locally equivalent to a CNOT or CZ gate). In operations 812 and 814, as with operation 412, a beamsplitter operation is applied to coupling element 703 to exchange the states of the cavities 711 and 722, causing photon number information of both these cavities to be mapped to the ancilla qubit 724. [0083] Photon loss during a two-qubit gate is detectable via separate erasure checks on the two dual-rail qubits after each gate, but modifications can also be made to the ^^^{^}^^{^} gate pulse sequence to simultaneously perform an erasure check in which the transmon can be mapped to the |^^ level if one of the input dual-rail qubits was in a leakage state outside the logical subspace. [0084] A different approach to a particular type of entangling gate – the CPHASE^^^ gate – is depicted in FIGs.9A and 9B. The drive scheme shown in the example of FIGs.9A-9B involves simultaneously driving the ancilla qubit 724 and the coupling element 703, with two distinct pulses being applied to the ancilla 724 while a single drive is applied to the coupling element 703. In each case, two back-to-back 8 ^{pulses are applied to the ancilla qubit 724. If the cavities 711 and 722 both begin in the} _{|0^ state, the 8 pulses produce a phase shift on the ancilla qubit 724, and by controlling} the phase of the second 8 pulse, a CPHASE^{^}^^{^} gate is performed between the dual-rail ^{qubits 701 and 702. In particular the phase of the second 8 pulse is parameterized by} _{^ = 8 + L.} [0085] The examples of FIGs.9A and 9B differ in that FIG.9A depicts an approach where the ancilla qubit is a two level qubit, and FIG.9B an approach where the ancilla qubit is a three level qubit. These two approaches will be described separately below. [0086] In the example of FIG.9A, in terms of ancilla qubit errors, if ancilla dephasing occurs during the CPHASE^{^}^^{^} gate, the ancilla qubit will be measured to be in |^^ instead of |^^ at the end of the gate. If instead ancilla qubit decay occurs, the ancilla qubit will be measured to be in |^^ 50% of the time. In either case, an ancilla qubit error may be detected subsequent to performing this gate by measuring the state of the ancilla qubit afterwards, and affirmatively determining that an error occurred when the ancilla is measured to be in |^^. [0087] In the example of FIG.9A, a drive ^_@A is applied to the coupling element 703 with a strength ^_@A > ^_9: a frequency detuned ( = ^_9:/2 from the resonance condition as described above. ^_9: is the strength of the dispersive interaction between the ancilla qubit 724 and the cavity 722 in the ^^-manifold. Concurrent with this drive, two back-to-back drives are applied to the ancilla qubit 724 at a frequency that is the difference between the ancilla qubit’s |^^ and |^^ states when the cavities contain no photons. The first of these two drives has a phase of zero, and the second has a phase ^ = 8 + L relative to the first drive. In the first (square pulse) example of FIG. ^{9A, the two ancilla qubit drives are each applied for a time period of approximately} _{28/^9:, whereas in the second (Gaussian pulse) example of FIG. 9A, the two ancilla} qubit drives are each applied for a time period of approximately 48/^_9:. In some embodiments, the two back-to-back ancilla qubit drives applied in the CPHASE gate are applied for an equal amount of time. [0088] In the example of FIG.9B, ancilla qubit decays can be detected through use of a three level ancilla qubit. In particular, an ancilla qubit error may be detected by measuring the state of the ancilla qubit after applying one of the drive sequences shown in FIG.9B, and affirmatively determining that an error occurred when the ancilla is measured to be in |^^ or |^^. In particular, measuring the ancilla qubit to be in the |^^ state indicates that the CPHASE gate failed due to decay of the ancilla qubit state, whereas measuring the ancilla qubit to be in the |^^ state indicates that the CPHASE gate failed due to ancilla qubit dephasing. Measuring the ancilla qubit to be in the ^|^^{^} state indicates the CPHASE gate was performed successfully. [0089] In the example of FIG.9B, a drive ^_@A is applied to the coupling element 703 with a strength ^_@A > ^_9M a frequency detuned ( = ^_9M/2 from the resonance condition as described above. ^_9M is the strength of the dispersive interaction between the ancilla qubit 724 and the cavity 722 in the ^^-manifold. Concurrent with this drive, two back-to-back drives are applied to the ancilla qubit 724 at a frequency that is the difference between the ancilla qubit’s |^^ and |^^ states when the cavities contain no photons. The first of these two drives has a phase of zero, and the second has a phase ^ = 8 + L relative to the first drive. In the first (square pulse) example of FIG. ^{9B, the two ancilla qubit drives are each applied for a time period of approximately} _{28/^9:, whereas in the second (Gaussian pulse) example of FIG. 9B, the two ancilla} qubit drives are each applied for a time period of approximately 48/^_9:. In some embodiments, the two back-to-back ancilla qubit drives applied in the CPHASE gate are applied for an equal amount of time. [0090] The above-described dual-rail qubit and techniques for preparing states, performing single dual-rail qubit gates, performing entangling gates on two dual-rail qubits, and detecting erasure errors may be extended to a system that comprises more than two dual-rail qubits. [0091] In some embodiments, a plurality of dual-rail qubits may be arranged in an array, network. grid, lattice, or other repeating configuration. The dual-rail qubits may be arranged in this manner with one or more types of structures providing connectivity between dual-rail qubits, which may include other dual-rail qubits and/or other structures such as cavities. In some embodiments, a ‘unit cell’ may be repeated a plurality of times to produce a lattice arrangement. A suitable unit cell may for instance comprise a dual- rail qubit (comprising two superconducting cavities, an ancilla qubit such as a transmon, and a coupling element between the two cavities such as a SNAIL) and two additional coupling elements to interface with neighboring unit cells. [0092] One illustrative example of a network of dual-rail qubits is shown in FIG. 10. In FIG.10, dual-rail qubits (of which 1001 are two examples) are represented as a pair of coupled cavities (circles with a connecting line) surrounded by a rectangular box. The dual-rail qubits are coupled to one another via measure qubits (of which 1002 is an example), represented by solid circles. Each of the measure qubits comprises a single cavity coupled to an ancilla qubit, in addition to a readout cavity. The lines between the measure qubits and dual-rail qubits represent coupling elements (e.g., SNAILs). [0093] In some embodiments, a network of dual-rail qubits may be operated so that a plurality of the dual-rail qubits are operated as a single logical qubit. For instance, a plurality of dual-rail qubits may be entangled together using a suitable sequence of quantum gates, with subsequent measurements of the entanglement states providing a means for error correction and error detection. The set of physical dual-rail qubits entangled in this way is thereby used to define a logical qubit, which may be expected to have better performance than a single dual-rail qubit. [0094] One way to entangle dual-rail qubits in this way is to use the so-called ‘surface code,’ in which a group of dual-rail qubits are selected to be either data qubits, which store computational quantum states, or measurement qubits, which are used to stabilize and manipulate the quantum state of the data qubits. The stabilizer codes used in the surface code are described for instance in “Surface codes: Towards practical large-scale quantum computation,” by Austin Fowler et al., Phys. Rev. A 86, 032324 (2012), which is hereby incorporated by reference in its entirety. [0095] One example of a physical arrangement of dual-rail qubits that may be grouped and operated as a logical qubit using the surface code (or any other suitable code) is depicted in FIG.11, according to some embodiments. In the example of FIG. 11, each dual-rail qubit is coupled to four neighboring dual-rail qubits; each data dual- rail qubit is coupled to four neighboring measurement dual-rail qubits, and each measurement dual-rail qubit is coupled to four neighboring data dual-rail qubits. [0096] One challenge with this type of arrangement may be that, although it is scalable, it is not modular which may make it difficult to fabricate simply. Moreover, it may lack flexibility in the spatial arrangement of the dual-rail qubits (e.g., it may require the dual-rail qubits to be aligned in a flat plane). [0097] A more modular and scalable approach to that shown in FIG.11 is depicted in FIG.12A, according to some embodiments. In the example of FIG.12A, each dual- rail qubit data qubit is coupled to four dual-rail qubit measurement qubits (and vice versa) as before, however the dual-rail qubits are arranged in modules, the boundaries of which are denoted in FIG.12A by the light grey rectangles. Coupling between adjacent cavities within a module (intra-module links) are implemented with a non-linear coupling element as described above, whereas coupling between cavities across the module boundaries (inter-module links) are instead implemented with a quantum bus, examples of which are described below. [0098] As a result, a plurality of dual-rail qubit modules may be fabricated that are nominally identical, and which include a plurality of couplings (and/or coupling ports) that couple cavities of the module to an external quantum bus. In the example of FIG. 12A, each such module would include 9 dual-rail qubit data qubits and 4 dual-rail qubit measurement qubits, in addition to 12 cavities that are each arranged to be coupled to a quantum bus. It may be noted that the cavities coupled to a quantum bus form a dual-rail qubit measurement qubit when coupled to another cavity in an adjacent module. That is, the resulting system comprises a plurality of dual-rail qubit measurement qubits where two cavities are coupled by a quantum bus (which acts as the coupling element 103). [0099] It may further be noted that this approach results in a “quad-rail” qubit at the corners between four modules, where four cavities are coupled end-to-end in a loop via four quantum buses (in addition to each being coupled to a respective cavity of a data qubit via a non-linear coupling element). [00100] FIG.12B depicts an illustrative quantum bus that may act as an inter- module link in the example of FIG.12A, according to some embodiments. In the example of FIG.12B, the quantum bus is implemented as a standing wave transport mechanism. For example, the quantum bus may comprise a microwave coaxial cable, flexicable and/or other kind of photonic microwave link that contains a bus mode for performing SWAP operations and/or 50-50 beamsplitter operations between cavities coupled to opposing ends of the bus. [00101] Having thus described several aspects of at least one embodiment of this invention, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art. [00102] Such alterations, modifications, and improvements are intended to be part of this disclosure, and are intended to be within the spirit and scope of the invention. Further, though advantages of the present invention are indicated, it should be appreciated that not every embodiment of the technology described herein will include every described advantage. Some embodiments may not implement any features described as advantageous herein and in some instances one or more of the described features may be implemented to achieve further embodiments. Accordingly, the foregoing description and drawings are by way of example only. [00103] Aspects of the present disclosure may include, but are not limited to: [00104] Aspect 1. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to operate the at least one energy source to initialize the dual-rail qubit in a 0 or 1 logical state by: when the dual- rail qubit is to be initialized in the 0 logical state, operating the at least one energy source to arrange the first quantum oscillator in a single photon state and the second quantum oscillator in its ground state; or when the dual-rail qubit is to be initialized in the 1 logical state, operating the at least one energy source to arrange the first quantum oscillator in its ground state and the second quantum oscillator in a single photon state. [00105] Aspect 2. The dual-rail qubit of aspect 1, wherein arranging the first quantum oscillator or the second quantum oscillator in a single photon state comprises operating the at least one energy source to perform a plurality of optimal control pulses and/or cavity-ancilla sideband drives. [00106] Aspect 3. The dual-rail qubit of aspect 1, wherein arranging the first quantum oscillator in the single photon state comprises operating the at least one energy source to arrange the first quantum oscillator in a |1^ Fock state. [00107] Aspect 4. The dual-rail qubit of aspect 1, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator. [00108] Aspect 5. The dual-rail qubit of aspect 1, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL). [00109] Aspect 6. The dual-rail qubit of aspect 1, wherein the ancilla qubit is not coupled to the second quantum oscillator. [00110] Aspect 7. The dual-rail qubit of aspect 1, wherein the ancilla qubit is a transmon qubit. [00111] Aspect 8. The dual-rail qubit of aspect 1, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity. [00112] Aspect 9. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to operate the at least one energy source to perform a single-qubit rotation of a logical state of the dual-rail qubit by operating the at least one energy source to direct energy to the coupling element one or more times. [00113] Aspect 10. The dual-rail qubit of aspect 9, wherein the at least one controller is further configured to drive the coupling element at a frequency equal to a difference between a resonant frequency of the first quantum oscillator and a resonant frequency of the second quantum oscillator. [00114] Aspect 11. The dual-rail qubit of aspect 9, wherein the at least one controller is further configured to select an amplitude, duration and phase for said operation of the at least one energy source based on rotation angles of the single-qubit rotation. [00115] Aspect 12. The dual-rail qubit of aspect 11, wherein the amplitude and duration are selected based on a polar angle of the single-qubit rotation, and wherein the phase is selected based on an angle of rotation about the Bloch sphere Z axis in the single-qubit rotation. [00116] Aspect 13. The dual-rail qubit of aspect 9, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator. [00117] Aspect 14. The dual-rail qubit of aspect 9, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL). [00118] Aspect 15. The dual-rail qubit of aspect 9, wherein the ancilla qubit is not coupled to the second quantum oscillator. [00119] Aspect 16. The dual-rail qubit of aspect 9, wherein the ancilla qubit is a transmon qubit. [00120] Aspect 17. The dual-rail qubit of aspect 9, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity. [00121] Aspect 18. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator and having a ground state |^^, a first excited state |^^ and a second excited state |^^; at least one energy source; and at least one controller configured to operate the at least one energy source to: (a) direct energy to the ancilla qubit to perform a first rotation of the quantum state of the ancilla qubit; (b) subsequent to (a), direct energy to the coupling element to perform a beamsplitter operation on the first quantum oscillator and the second quantum oscillator; and (c) subsequent to (b), direct energy to the ancilla qubit to perform a second rotation of the quantum state of the ancilla qubit. [00122] Aspect 19. The dual-rail qubit of aspect 18, wherein the first and second rotations of the state of the ancilla qubit are rotations within a manifold between the ground state |^^and the second excited state |^^ of the ancilla qubit. [00123] Aspect 20. The dual-rail qubit of aspect 18, wherein the first and second rotations of the state of the ancilla qubit are π/2 rotations about the Bloch sphere Y axis. [00124] Aspect 21. The dual-rail qubit of aspect 18, wherein the at least one controller is further configured to operate the at least one energy source to measure a state of the ancilla qubit subsequent to (c). [00125] Aspect 22. The dual-rail qubit of aspect 18, wherein the at least one controller is further configured to identify an erasure error in the dual-rail qubit when the state of the ancilla qubit is detected to not be in the ground state |^^. [00126] Aspect 23. The dual-rail qubit of aspect 18, wherein the at least one controller is further configured to identify an erasure error in the dual-rail qubit when the state of the ancilla qubit is detected to be in the first excited state |^^ or the second excited state |^^. [00127] Aspect 24. The dual-rail qubit of aspect 18, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator. [00128] Aspect 25. The dual-rail qubit of aspect 18, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL). [00129] Aspect 26. The dual-rail qubit of aspect 18, wherein the ancilla qubit is not coupled to the second quantum oscillator. [00130] Aspect 27. The dual-rail qubit of aspect 18, wherein the ancilla qubit is a transmon qubit. [00131] Aspect 28. The dual-rail qubit of aspect 18, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity. [00132] Aspect 29. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to: operate the at least one energy source to measure a parity state of the first quantum oscillator; operate the at least one energy source to measure a parity state of the second quantum oscillator; and determine a logical state of the dual-rail qubit based on the measured parity states of the first and second quantum oscillators. [00133] Aspect 30. The dual-rail qubit of aspect 29, wherein, when the parity state of the first quantum oscillator is measured to be opposite to the parity state of the second quantum oscillator, the determined logical state of the dual-rail qubit is the 0 logical state or the 1 logical state. [00134] Aspect 31. The dual-rail qubit of aspect 29, wherein the at least one controller is further configured to identify an erasure error when the parity states of the first and second quantum oscillators are both measured to be even. [00135] Aspect 32. The dual-rail qubit of aspect 29, wherein the at least one controller is configured to operate the at least one energy source to measure the parity state of the first quantum oscillator a plurality of times, to operate the at least one energy source to measure the parity state of the second quantum oscillator a plurality of times, thereby producing a plurality of parity state measurements for each of the first and second quantum oscillators. [00136] Aspect 33. The dual-rail qubit of aspect 29, wherein the at least one controller is configured to determine the logical state of the dual-rail qubit based on a majority result of a plurality of parity state measurements for the first quantum oscillator and based on a majority result of the plurality of parity state measurements for the second quantum oscillator. [00137] Aspect 34. The dual-rail qubit of aspect 29, wherein operating the at least one energy source to measure the parity state of the first quantum oscillator comprises operating the at least one energy source to measure a state of the ancilla qubit and identifying the parity state of the first quantum oscillator based on the measured state of the ancilla qubit. [00138] Aspect 35. The dual-rail qubit of aspect 29, wherein the at least one controller is further configured to direct energy to the coupling element to perform a beamsplitter operation on the first quantum oscillator and the second quantum oscillator subsequent to operating the at least one energy source to measure the parity state of the first quantum oscillator and prior to operating the at least one energy source to measure the parity state of the second quantum oscillator. [00139] Aspect 36. The dual-rail qubit of aspect 29, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator. [00140] Aspect 37. The dual-rail qubit of aspect 29, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL). [00141] Aspect 38. The dual-rail qubit of aspect 29, wherein the ancilla qubit is not coupled to the second quantum oscillator. [00142] Aspect 39. The dual-rail qubit of aspect 29, wherein the ancilla qubit is a transmon qubit. [00143] Aspect 40. The dual-rail qubit of aspect 29, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity. [00144] Aspect 41. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator and having a ground state |^^, a first excited state |^^ and a second excited state |^^; at least one energy source; and at least one controller configured to: operate the at least one energy source to perform one or more gates and/or operations at least in part on the dual- rail qubit; detect an erasure error subsequent to performing the one or more gates and/or operations; and in response to detecting the erasure error, operate the at least one energy source to initialize the dual-rail qubit in a new logical state. [00145] Aspect 42. The dual-rail qubit of aspect 41, wherein detecting the erasure error comprises determining that a quantum state of the ancilla qubit is not in the ground state |^^. [00146] Aspect 43. The dual-rail qubit of aspect 42, wherein detecting the erasure error comprises determining that the quantum state of the ancilla qubit is in the first excited state |^^ or the second excited state |^^. [00147] Aspect 44. The dual-rail qubit of aspect 41, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator. [00148] Aspect 45. The dual-rail qubit of aspect 41, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL). [00149] Aspect 46. The dual-rail qubit of aspect 41, wherein the ancilla qubit is not coupled to the second quantum oscillator. [00150] Aspect 47. The dual-rail qubit of aspect 41, wherein the ancilla qubit is a transmon qubit. [00151] Aspect 48. The dual-rail qubit of aspect 41, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity. [00152] Aspect 49. A system comprising: a first dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and to the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; a second dual-rail qubit comprising: a third quantum oscillator; a fourth quantum oscillator; a second coupling element coupled to the third quantum oscillator and to the fourth quantum oscillator; and a second ancilla qubit coupled to the third quantum oscillator; and a third coupling element coupled to the second quantum oscillator and to the third quantum oscillator. [00153] Aspect 50. The system of aspect 49, further comprising: at least one energy source; and at least one controller configured to operate the at least one energy source to perform a two-qubit gate on logical states of the first dual-rail qubit and the second dual- rail qubit. [00154] Aspect 51. The system of aspect 50, wherein the two-qubit gate is a ZZ(θ) gate. [00155] Aspect 52. The system of aspect 50, wherein operating the at least one energy source to perform the two-qubit gate comprises directing energy to the third coupling element. [00156] Aspect 53. The system of aspect 52, wherein operating the at least one energy source to perform the two-qubit gate comprises performing one or more qubit rotations of the second ancilla qubit. [00157] Aspect 54. The system of aspect 53, wherein operating the at least one energy source to perform the two-qubit gate comprises: (a) performing a first qubit rotation of the second ancilla qubit; (b) subsequent to (a), directing energy to the third coupling element; and (c) subsequent to (b), performing a second qubit rotation of the second ancilla qubit. [00158] Aspect 55. The system of aspect 50, wherein the at least one controller is further configured to, subsequent to performing the two-qubit gate, identify an erasure error when a quantum state of the second ancilla qubit is detected to not be in a ground state |^^. [00159] Aspect 56. The system of aspect 55, wherein the at least one controller is further configured to, subsequent to performing the two-qubit gate, identify an erasure error when the quantum state of the second ancilla qubit is detected to be in a first excited state |^^ or a second excited state |^^. [00160] Aspect 57. The system of aspect 50, wherein: the first coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; the second coupling element is dispersively coupled to the third quantum oscillator and to the fourth quantum oscillator; the first ancilla qubit is dispersively coupled to the first quantum oscillator; the second ancilla qubit is dispersively coupled to the third quantum oscillator; and the third coupling element is dispersively coupled to the second quantum oscillator and to the third quantum oscillator. [00161] Aspect 58. A system comprising: a plurality of dual-rail qubits, each dual- rail qubit of the plurality of dual-rail qubits comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and to the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; a plurality of measurement qubits, each measurement qubit of the plurality of measurement qubits comprising: a first quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator of the measure qubit, wherein each of the plurality of measurement qubits is coupled to four of the plurality of dual-rail qubits, with each coupling provided via a respective coupling element. [00162] Aspect 59. The system of aspect 58, wherein each coupling of the measurement qubits to one of the plurality of dual-rail qubits couples the first quantum oscillator of the measurement qubit to the first or second quantum oscillator of the one of the plurality of dual-rail qubits. [00163] Aspect 60. The system of aspect 58, wherein the first coupling element of each dual-rail qubit of the plurality of dual-rail qubits is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL). [00164] Aspect 61. The system of aspect 58, wherein the first ancilla qubit of each dual-rail qubit of the plurality of dual-rail qubits is a transmon qubit. [00165] Aspect 62. A module comprising: a plurality of dual-rail qubits, each dual- rail qubit of the plurality of dual-rail qubits comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and to the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; and a plurality of cavities each coupled to: one of the plurality of dual-rail qubits via a respective coupling element; and an external port configured for coupling to a quantum bus. [00166] Aspect 63. The module of aspect 62, wherein the external port comprises a coaxial connector. [00167] Aspect 64. A system comprising a plurality of the modules of aspect 62 coupled together via a plurality of quantum bus connections between external ports of the modules. [00168] Aspect 65. The module of aspect 62, wherein the first coupling element of each dual-rail qubit of the plurality of dual-rail qubits is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL). [00169] Aspect 66. The module of aspect 62, wherein the first ancilla qubit of each dual-rail qubit of the plurality of dual-rail qubits is a transmon qubit. [00170] The above-described embodiments of the technology described herein can be implemented in any of numerous ways. For example, the controller of any of the embodiments, including controller 106 shown in FIG.1, may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers. Such processors may be implemented as integrated circuits, with one or more processors in an integrated circuit component, including commercially available integrated circuit components known in the art by names such as CPU chips, GPU chips, microprocessor, microcontroller, or co-processor. Alternatively, a processor may be implemented in custom circuitry, such as an ASIC, or semi-custom circuitry resulting from configuring a programmable logic device. As yet a further alternative, a processor may be a portion of a larger circuit or semiconductor device, whether commercially available, semi-custom or custom. As a specific example, some commercially available microprocessors have multiple cores such that one or a subset of those cores may constitute a processor. Though, a processor may be implemented using circuitry in any suitable format. [00171] Various aspects of the present invention may be used alone, in combination, or in a variety of arrangements not specifically described in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings. For example, aspects described in one embodiment may be combined in any manner with aspects described in other embodiments. [00172] Also, the invention may be embodied as a method, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments. [00173] Use of ordinal terms such as “first,” “second,” “third,” etc., in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements. [00174] The terms “approximately” and “about” may be used to mean within ±20% of a target value in some embodiments, within ±10% of a target value in some embodiments, within ±5% of a target value in some embodiments, and yet within ±2% of a target value in some embodiments. The terms “approximately” and “about” may include the target value. The term “substantially equal” may be used to refer to values that are within ±20% of one another in some embodiments, within ±10% of one another in some embodiments, within ±5% of one another in some embodiments, and yet within ±2% of one another in some embodiments. [00175] The term “substantially” may be used to refer to values that are within ±20% of a comparative measure in some embodiments, within ±10% in some embodiments, within ±5% in some embodiments, and yet within ±2% in some embodiments. For example, a first direction that is “substantially” perpendicular to a second direction may refer to a first direction that is within ±20% of making a 90° angle with the second direction in some embodiments, within ±10% of making a 90° angle with the second direction in some embodiments, within ±5% of making a 90° angle with the second direction in some embodiments, and yet within ±2% of making a 90° angle with the second direction in some embodiments. [00176] Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” or “having,” “containing,” “involving,” and variations thereof herein, is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.

Claims

CLAIMS What is claimed is: 1. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to operate the at least one energy source to initialize the dual-rail qubit in a 0 or 1 logical state by: when the dual-rail qubit is to be initialized in the 0 logical state, operating the at least one energy source to arrange the first quantum oscillator in a single photon state and the second quantum oscillator in its ground state; or when the dual-rail qubit is to be initialized in the 1 logical state, operating the at least one energy source to arrange the first quantum oscillator in its ground state and the second quantum oscillator in a single photon state.

2. The dual-rail qubit of claim 1, wherein arranging the first quantum oscillator or the second quantum oscillator in a single photon state comprises operating the at least one energy source to perform a plurality of optimal control pulses and/or cavity-ancilla sideband drives.

3. The dual-rail qubit of claim 1, wherein arranging the first quantum oscillator in the single photon state comprises operating the at least one energy source to arrange the first quantum oscillator in a |1^ Fock state.

4. The dual-rail qubit of claim 1, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

5. The dual-rail qubit of claim 1, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL).

6. The dual-rail qubit of claim 1, wherein the ancilla qubit is not coupled to the second quantum oscillator.

7. The dual-rail qubit of claim 1, wherein the ancilla qubit is a transmon qubit.

8. The dual-rail qubit of claim 1, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

9. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to operate the at least one energy source to perform a single-qubit rotation of a logical state of the dual-rail qubit by operating the at least one energy source to direct energy to the coupling element one or more times.

10. The dual-rail qubit of claim 9, wherein the at least one controller is further configured to drive the coupling element at a frequency equal to a difference between a resonant frequency of the first quantum oscillator and a resonant frequency of the second quantum oscillator.

11. The dual-rail qubit of claim 9, wherein the at least one controller is further configured to select an amplitude, duration and phase for said operation of the at least one energy source based on rotation angles of the single-qubit rotation.

12. The dual-rail qubit of claim 11, wherein the amplitude and duration are selected based on a polar angle of the single-qubit rotation, and wherein the phase is selected based on an angle of rotation about the Bloch sphere Z axis in the single-qubit rotation.

13. The dual-rail qubit of claim 9, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

14. The dual-rail qubit of claim 9, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL).

15. The dual-rail qubit of claim 9, wherein the ancilla qubit is not coupled to the second quantum oscillator.

16. The dual-rail qubit of claim 9, wherein the ancilla qubit is a transmon qubit.

17. The dual-rail qubit of claim 9, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

18. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; a^{n ancilla qubit coupled to the first quantum oscillator and having a ground state} _{|^^, a first excited state |^^ and a second excited state |^^;} at least one energy source; and at least one controller configured to operate the at least one energy source to: (a) direct energy to the ancilla qubit to perform a first rotation of the quantum state of the ancilla qubit; (b) subsequent to (a), direct energy to the coupling element to perform a beamsplitter operation on the first quantum oscillator and the second quantum oscillator; and (c) subsequent to (b), direct energy to the ancilla qubit to perform a second rotation of the quantum state of the ancilla qubit.

19. The dual-rail qubit of claim 18, wherein the first and second rotations of the state of the ancilla qubit are rotations within a manifold between the ground state |^^ and the second excited state |^^ of the ancilla qubit.

20. The dual-rail qubit of claim 18, wherein the first and second rotations of the state of the ancilla qubit are π/2 rotations about the Bloch sphere Y axis.

21. The dual-rail qubit of claim 18, wherein the at least one controller is further configured to operate the at least one energy source to measure a state of the ancilla qubit subsequent to (c).

22. The dual-rail qubit of claim 18, wherein the at least one controller is further configured to identify an erasure error in the dual-rail qubit when the state of the ancilla qubit is detected to not be in the ground state |^^.

23. The dual-rail qubit of claim 18, wherein the at least one controller is further configured to identify an erasure error in the dual-rail qubit when the state of the ancilla qubit is detected to be in the first excited state |^^ or the second excited state |^^.

24. The dual-rail qubit of claim 18, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

25. The dual-rail qubit of claim 18, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL).

26. The dual-rail qubit of claim 18, wherein the ancilla qubit is not coupled to the second quantum oscillator.

27. The dual-rail qubit of claim 18, wherein the ancilla qubit is a transmon qubit.

28. The dual-rail qubit of claim 18, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

29. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to: operate the at least one energy source to measure a parity state of the first quantum oscillator; operate the at least one energy source to measure a parity state of the second quantum oscillator; and determine a logical state of the dual-rail qubit based on the measured parity states of the first and second quantum oscillators.

30. The dual-rail qubit of claim 29, wherein, when the parity state of the first quantum oscillator is measured to be opposite to the parity state of the second quantum oscillator, the determined logical state of the dual-rail qubit is the 0 logical state or the 1 logical state.

31. The dual-rail qubit of claim 29, wherein the at least one controller is further configured to identify an erasure error when the parity states of the first and second quantum oscillators are both measured to be even.

32. The dual-rail qubit of claim 29, wherein the at least one controller is configured to operate the at least one energy source to measure the parity state of the first quantum oscillator a plurality of times, to operate the at least one energy source to measure the parity state of the second quantum oscillator a plurality of times, thereby producing a plurality of parity state measurements for each of the first and second quantum oscillators.

33. The dual-rail qubit of claim 29, wherein the at least one controller is configured to determine the logical state of the dual-rail qubit based on a majority result of a plurality of parity state measurements for the first quantum oscillator and based on a majority result of the plurality of parity state measurements for the second quantum oscillator.

34. The dual-rail qubit of claim 29, wherein operating the at least one energy source to measure the parity state of the first quantum oscillator comprises operating the at least one energy source to measure a state of the ancilla qubit and identifying the parity state of the first quantum oscillator based on the measured state of the ancilla qubit.

35. The dual-rail qubit of claim 29, wherein the at least one controller is further configured to direct energy to the coupling element to perform a beamsplitter operation on the first quantum oscillator and the second quantum oscillator subsequent to operating the at least one energy source to measure the parity state of the first quantum oscillator and prior to operating the at least one energy source to measure the parity state of the second quantum oscillator.

36. The dual-rail qubit of claim 29, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

37. The dual-rail qubit of claim 29, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL).

38. The dual-rail qubit of claim 29, wherein the ancilla qubit is not coupled to the second quantum oscillator.

39. The dual-rail qubit of claim 29, wherein the ancilla qubit is a transmon qubit.

40. The dual-rail qubit of claim 29, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

41. A dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and to the second quantum oscillator; a^{n ancilla qubit coupled to the first quantum oscillator and having a ground state} _{|^^, a first excited state |^^ and a second excited state |^^;} at least one energy source; and at least one controller configured to: operate the at least one energy source to perform one or more gates and/or operations at least in part on the dual-rail qubit; detect an erasure error subsequent to performing the one or more gates and/or operations; and in response to detecting the erasure error, operate the at least one energy source to initialize the dual-rail qubit in a new logical state.

42. The dual-rail qubit of claim 41, wherein detecting the erasure error comprises determining that a quantum state of the ancilla qubit is not in the ground state |^^.

43. The dual-rail qubit of claim 42, wherein detecting the erasure error comprises determining that the quantum state of the ancilla qubit is in the first excited state |^^ or the second excited state |^^.

44. The dual-rail qubit of claim 41, wherein: the coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

45. The dual-rail qubit of claim 41, wherein the coupling element is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL).

46. The dual-rail qubit of claim 41, wherein the ancilla qubit is not coupled to the second quantum oscillator.

47. The dual-rail qubit of claim 41, wherein the ancilla qubit is a transmon qubit.

48. The dual-rail qubit of claim 41, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

49. A system comprising: a first dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and to the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; a second dual-rail qubit comprising: a third quantum oscillator; a fourth quantum oscillator; a second coupling element coupled to the third quantum oscillator and to the fourth quantum oscillator; and a second ancilla qubit coupled to the third quantum oscillator; and a third coupling element coupled to the second quantum oscillator and to the third quantum oscillator.

50. The system of claim 49, further comprising: at least one energy source; and at least one controller configured to operate the at least one energy source to perform a two-qubit gate on logical states of the first dual-rail qubit and the second dual- rail qubit.

51. The system of claim 50, wherein the two-qubit gate is a ZZ(θ) gate.

52. The system of claim 50, wherein operating the at least one energy source to perform the two-qubit gate comprises directing energy to the third coupling element.

53. The system of claim 52, wherein operating the at least one energy source to perform the two-qubit gate comprises performing one or more qubit rotations of the second ancilla qubit.

54. The system of claim 53, wherein operating the at least one energy source to perform the two-qubit gate comprises: (a) performing a first qubit rotation of the second ancilla qubit; (b) subsequent to (a), directing energy to the third coupling element; and (c) subsequent to (b), performing a second qubit rotation of the second ancilla qubit.

55. The system of claim 50, wherein the at least one controller is further configured to, subsequent to performing the two-qubit gate, identify an erasure error when a quantum state of the second ancilla qubit is detected to not be in a ground state |^^.

56. The system of claim 55, wherein the at least one controller is further configured to, subsequent to performing the two-qubit gate, identify an erasure error when the quantum state of the second ancilla qubit is detected to be in a first excited state |^^ or a second excited state |^^.

57. The system of claim 50, wherein: the first coupling element is dispersively coupled to the first quantum oscillator and to the second quantum oscillator; the second coupling element is dispersively coupled to the third quantum oscillator and to the fourth quantum oscillator; the first ancilla qubit is dispersively coupled to the first quantum oscillator; the second ancilla qubit is dispersively coupled to the third quantum oscillator; and the third coupling element is dispersively coupled to the second quantum oscillator and to the third quantum oscillator.

58. A system comprising: a plurality of dual-rail qubits, each dual-rail qubit of the plurality of dual-rail qubits comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and to the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; a plurality of measurement qubits, each measurement qubit of the plurality of measurement qubits comprising: a first quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator of the measure qubit, wherein each of the plurality of measurement qubits is coupled to four of the plurality of dual-rail qubits, with each coupling provided via a respective coupling element.

59. The system of claim 58, wherein each coupling of the measurement qubits to one of the plurality of dual-rail qubits couples the first quantum oscillator of the measurement qubit to the first or second quantum oscillator of the one of the plurality of dual-rail qubits.

60. The system of claim 58, wherein the first coupling element of each dual-rail qubit of the plurality of dual-rail qubits is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL).

61. The system of claim 58, wherein the first ancilla qubit of each dual-rail qubit of the plurality of dual-rail qubits is a transmon qubit.

62. A module comprising: a plurality of dual-rail qubits, each dual-rail qubit of the plurality of dual-rail qubits comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and to the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; and a plurality of cavities each coupled to: one of the plurality of dual-rail qubits via a respective coupling element; and an external port configured for coupling to a quantum bus.

63. The module of claim 62, wherein the external port comprises a coaxial connector.

64. A system comprising a plurality of the modules of claim 62 coupled together via a plurality of quantum bus connections between external ports of the modules.

65. The module of claim 62, wherein the first coupling element of each dual-rail qubit of the plurality of dual-rail qubits is a transmon qubit or a superconducting nonlinear asymmetric inductive element (SNAIL).

66. The module of claim 62, wherein the first ancilla qubit of each dual-rail qubit of the plurality of dual-rail qubits is a transmon qubit.