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

Abstract

A technique for quantum error correction is described that utilizes dual-rail encoding of a single excitation in a pair of coupled quantum oscillators. The dual-rail qubit can be implemented in a cQED (circuit quantum electrodynamics) system that includes a pair of coupled resonators. In this dual-rail encoding, an excitation is stored in one of two resonators. The excitation of the first resonator is treated as a logic 0, and the excitation of the other resonator is treated as a logic 1. Thus, together, the two resonators store a logic state |0> L =|01> and |1> L =|10>, forming a single logical dual-rail qubit.

Description

[Technical Field]

[0001] Government financial support This invention was made with government support under W911NF-18-1-0212 awarded by the Army Research Office (ARO) and DE-SC0012704 awarded by the Department of Energy (DOE). The government has certain rights in this invention. [Background technology]

[0002] background Quantum information processing techniques perform computations by manipulating one or more quantum objects. These techniques are sometimes referred to as "quantum computing." To perform computations, quantum information processors utilize quantum objects to reliably store and retrieve information. Some quantum information processing approaches have developed a quantum analog to classical computational "bits" (equal to 1 or 0), called quantum bits or "qubits." A qubit can consist of any quantum system that has two distinct states (which may be thought of as a 1 state and a 0 state), but also has the special property that the system can be placed in quantum superposition, thereby potentially existing in both of these states at once. Summary of the Invention

[0003] overview In some aspects, the technology described herein relates to a dual-rail qubit that includes: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and 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 to a 0 or 1 logical state by operating the at least one energy source to align the first quantum oscillator to a single-photon state and the second quantum oscillator to its ground state when the dual-rail qubit is initialized to a 0 logical state; or by operating the at least one energy source to align the first quantum oscillator to its ground state and the second quantum oscillator to the single-photon state when the dual-rail qubit is initialized to a 1 logical state.

[0004] In some aspects, the technology described herein relates to a dual-rail qubit that includes: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and 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 the logical state of the dual-rail qubit by directing energy to the coupling element one or more times.

[0005] In some aspects, the technology described herein relates to a dual-rail qubit that includes: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator and having a ground state |g>, a first excited state |e>, and a second excited state |f>; 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) after (a), direct energy to the coupling element to perform a beam splitter operation on the first quantum oscillator and the second quantum oscillator; and (c) after (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 technology described herein relates to a dual-rail qubit that includes: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and 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 technology described herein relates to a dual-rail qubit that includes: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator and having a ground state |g〉, a first excited state |e〉, and a second excited state |f〉; at least one energy source; and at least one controller configured to operate the at least one energy source to perform, at least in part, one or more gates and / or operations on the dual-rail qubit; detect an erasure error after performing the one or more gates and / or operations; and, in response to detecting the erasure error, operate the at least one energy source and initialize the dual-rail qubit to a new logical state.

[0008] In some aspects, the technology described herein relates to a system that includes: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first dual-rail qubit including a first ancilla qubit coupled to the first quantum oscillator; a third quantum oscillator; a fourth quantum oscillator; a second coupling element coupled to the third quantum oscillator and the fourth quantum oscillator; and a second dual-rail qubit including a second ancilla qubit coupled to the third quantum oscillator; and a third coupling element coupled to the second quantum oscillator and the third quantum oscillator.

[0009] In some aspects, the technology described herein relates to a system that includes: a plurality of dual-rail qubits, wherein each dual-rail qubit of the plurality of dual-rail qubits includes: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; and a plurality of measurement qubits, wherein each measurement qubit of the plurality of measurement qubits includes: a first quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator of the measurement qubit, wherein each of the plurality of measurement qubits is coupled to four of the plurality of dual-rail qubits, each coupling provided via a respective coupling element.

[0010] In some aspects, the technology described herein relates to a module including: a plurality of dual-rail qubits, where each dual-rail qubit of the plurality of dual-rail qubits includes: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and 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 aspects may be implemented by any suitable combination of the aspects, features, and acts described above or in more detail below. These and other aspects, aspects, and features of the present teachings may be more fully understood from the following description taken in conjunction with the accompanying drawings. [Brief explanation of the drawings]

[0012] BRIEF DESCRIPTION OF THE DRAWINGS Various aspects and embodiments are described with reference to the following drawings. It should be understood that the figures are not necessarily drawn to scale. In the drawings, each identical or nearly identical component shown in various figures is represented by a like numeral. For clarity, not every component may be labeled in every figure. [Figure 1] FIG. 1 illustrates an example system suitable for implementing the techniques described herein, according to some embodiments. [Figure 2] Figure 2A illustrates an example implementation of the system of Figure 1 including a microwave cavity, according to some embodiments. Figure 2B illustrates couplings between elements of Figure 2A, according to some embodiments. Figure 2C illustrates logical states of the dual-rail qubit of Figure 2A, according to some embodiments. [Figure 3] 3A illustrates a parity measurement of a dual-rail qubit, according to some embodiments. FIG. 3B summarizes different measurement outcomes of the parity measurement sequence of FIG. 3A, according to some embodiments. [Figure 4] FIG. 4 illustrates an exemplary process for implementing a dual-rail qubit with single-qubit gates and vanishing checks, according to some embodiments. [Figure 5] 5A-5B illustrate an alternative scheme for erasure detection, according to some embodiments. [Figure 6] FIG. 6 illustrates one aspect of the evolution of a dual-rail qubit over time during application of the drive shown in FIGS. 5A-5B, according to some embodiments. [Figure 7] 7A illustrates an example implementation of the system of FIG. 1 including two dual-rail logic qubits implemented by microwave cavities, according to some embodiments. FIG. 7B illustrates coupling between the elements of FIG. 7A, according to some embodiments. [Figure 8] FIG. 8 illustrates an exemplary implementation of a ZZ(θ) gate between two dual-rail qubits, according to some embodiments. [Figure 9]9A-9B illustrate an example drive for performing a CPHASE(θ) gate between two dual-rail qubits, according to some embodiments. [Figure 10] FIG. 10 illustrates an illustrative example of a network of dual-rail qubits, according to some embodiments. [Figure 11] FIG. 11 shows an illustrative example of a physical alignment of dual-rail qubits that may be grouped and operated as a logical qubit, according to some embodiments. [Figure 12-1] FIG. 12A illustrates multiple modules, each including multiple dual-rail qubits that can be grouped and operated as a logical qubit, according to some embodiments. [Figure 12-2] FIG. 12B illustrates an exemplary quantum bus that may serve as an inter-module link in the example of FIG. 12A, according to some embodiments. DETAILED DESCRIPTION OF THE INVENTION

[0013] Detailed Description Quantum multi-level systems, such as superconducting qubits, exhibit quantum states that decoherence in approximately 100 μs, based on current experimental implementations. While experimental techniques will undoubtedly improve upon this, resulting in qubits with longer decoherence times, it may nevertheless be beneficial to couple a multi-level system to another system that exhibits significantly longer decoherence times. Systems composed of boson modes may be particularly desirable for coupling to multi-level systems. Through this coupling, the state of the multi-level system may instead be represented by a boson mode(s), thereby maintaining the same information in a state with a longer lifetime than would otherwise exist in the multi-level system alone. When used in this manner, boson systems are often referred to as "logical" qubits.

[0014] Nevertheless, quantum information stored in bosonic modes may still have a limited lifetime, and errors may still occur in bosonic systems. Therefore, it may be desirable to manipulate a bosonic system when errors occur in its state to effectively correct these errors and thereby restore the system's previous state. If a wide range of classes of errors can be corrected, it may be possible to maintain the state of a bosonic system indefinitely (or at least for a long time) by correcting any kind of error that may occur.

[0015] Nevertheless, quantum information stored in bosonic modes may still have a limited lifetime, and errors still occur within the bosonic system. It is desirable to manipulate the bosonic system when errors occur within its state to effectively correct these errors and thereby restore the system's previous state. In some cases, simply detecting that an error occurred during manipulation may be sufficient. The desire to detect and / or correct errors in a quantum system requires that several requirements be simultaneously addressed, such as the number of physical qubits used to encode logical information, an appropriate scheme for performing gates and measurements on the quantum system, and the precision of the system's physical components.

[0016] The inventors recognize and understand that not all types of errors occur equally in a given quantum error correction architecture, nor are they equally detrimental to the logical information stored in a quantum system. Some types of errors, for example, require that a specific error syndrome be measured to pinpoint the location of the error and to obtain the necessary information to properly repair the error. This does not always result in successful error correction, but often there is not enough information to unambiguously determine the error. On the other hand, erasure errors—errors in which information in a qubit is erased in a way that allows the qubit to be identified—can be handled fairly easily.

[0017] The inventors further recognize and understand techniques for error correction that utilize dual-rail encoding of a single excitation in a pair of coupled quantum oscillators. In particular, a dual-rail qubit can be implemented in a cQED (circuit quantum electrodynamics) system that includes a pair of coupled resonators. In this dual-rail encoding, an excitation is stored in one of two resonators. The excitation of the first resonator is treated as a logic 0, while the excitation of the other resonator is treated as a logic 1. Thus, the two resonators together produce a logic state |0> L =|01> and |1> L =|10>. This type of logical qubit has several advantages, including: First, photon loss manifests as an erasure error, which, as noted above, is a type of error that is easier to correct; Second, the single-photon state is the lowest energy state of the cavity and therefore has the lowest error rate of any state in the cavity. Thus, dual-rail encoding minimizes the rate of erasure errors; Third, photon gain or loss is easily detectable by measuring the joint parity of the cavity.

[0018] According to some embodiments, a dual-rail qubit can be implemented as a cQED system including two quantum resonators (e.g., microwave cavities) coupled together via a suitable nonlinear 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 resonators can be coupled to an ancilla qubit (e.g., a transmon qubit). The ancilla qubit couples to only one of the boson modes of the system due to the beam-splitter interaction (described below) provided by the coupling element, but both boson modes interact with the ancilla qubit, enabling various dual-rail operations. For example, dual-rail states can be prepared, and gates can be performed upon the dual-rail states via the application of energy (e.g., microwave pulses) to the ancilla qubit and / or the coupling element, as described further below.

[0019] According to some embodiments, a logical quantum state of a dual-rail qubit is prepared by adjusting a single excitation of one of the two quantum oscillators of the dual-rail qubit. For example, |0> L To prepare a |1> state (logic 0 state), both oscillators can be prepared in their respective ground states |0>, with the first oscillator excited to its |1> state and the second oscillator remaining in its |0> state. L To prepare a |0> state (logic 1 state), both oscillators can be prepared in their respective ground states, |0>, with the second oscillator excited to its |1> state and the first oscillator remaining in its |0> state. In the case of a cQED dual-rail qubit containing two quantum resonators, for example, one of the resonators can be prepared in the |1> state by loading a single photon into the resonator.

[0020] According to some embodiments, the logic state of a dual-rail qubit can be measured via one or more parity measurements of either and / or both of the two quantum oscillators (e.g., by performing one or more quantum non-demolition (QND) measurements). Because dual-rail logic states have odd concatenated parity, a change in parity indicates an error, which is either an erasure error resulting in concatenated state |00> or an excitation to one of states |11>, |20>, or |02>. When implementing a dual-rail qubit as a cQED system including two quantum resonators, parity measurements can be performed by applying one or more drives to one of the resonators and / or to an ancilla qubit coupled to a nonlinear coupling element that couples the resonators to each other, examples of which are described below.

[0021] According to some aspects, multiple dual-rail qubits may be aligned in a repeating arrangement, such as an array or lattice. Such alignment may provide improved response to errors by treating multiple dual-rail qubits as a single logical group. For example, multiple dual-rail qubits may be coupled and operated in a surface code architecture in which formally identical dual-rail qubits serve as either data qubits or measurement qubits on which stabilizer measurements are performed as an error correction process.

[0022] A more detailed description of various concepts related to the implementation and operation of dual-rail qubits, and aspects thereof, follows below. It should be understood that the various aspects described herein can be implemented in any of numerous ways. Examples of specific implementations are provided herein for illustrative purposes only. Also, the various aspects described in the following aspects can be used alone or in any combination, and are not limited to the combinations expressly described herein.

[0023] According to some embodiments, an exemplary system suitable for implementing the techniques described herein is shown in Figure 1. In system 100, oscillators 101 and 102 are coupled to each other via coupling element 103. Oscillator 101 is also coupled to an ancilla qubit 104. Energy source 105 can be operated by controller 106 to direct energy to ancilla qubit 104, coupling element 103, and / or readout resonator 107.

[0024] According to some embodiments, oscillator 101 and oscillator 102 each include a cavity that supports a quantum state of microwave photons. For example, in some embodiments, first oscillator 101 and second oscillator 102 may each include a resonator, such as a transmission line resonator or a three-dimensional cavity, formed of a superconducting material, e.g., aluminum.

[0025] Coupling element 103 may include a nonlinear element coupled to oscillators 101 and 102. Coupling element 103 mediates coupling between the quantum states of the two oscillators, enabling interaction between first oscillator 101 and oscillator 102. In some embodiments, coupling element 103 may be a superconducting nonlinear asymmetric inductive element (SNAIL), a superconducting quantum interference device (SQUID), a Josephson junction, or some other nonlinear element. In some embodiments, coupling element 103 may include a transmon qubit dispersively coupled to both first oscillator 101 and second oscillator 102.

[0026] In some embodiments, an ancilla qubit 104 may be a transmon qubit, a SNAIL, a SQUID, or some other nonlinear element.

[0027] An exemplary implementation of system 100 is shown in FIG. 2A as dual-rail qubit 200. In this implementation, oscillators are implemented as boson modes stored in cavities 201 and 202 (e.g., microwave cavities). A microwave source (not shown in FIG. 2A) may be configured as energy source 105 in this system and may be configured to direct microwave pulses of a desired amplitude, frequency, and phase to an ancilla qubit 204 (e.g., a transmon qubit), coupling element 203, and / or readout resonator 207. Such a microwave source may be coupled to the ancilla qubit and coupling element. The coupling between the microwave source and these components provides a path for the microwave source to apply microwave irradiation to the components.

[0028] In some embodiments, energy source 105 may be capacitively coupled to each of ancilla qubit 204, coupling element 203, and readout resonator 207. In some embodiments, multiple instances of the dual-rail qubit shown in FIG. 2A may be aligned to be coupled with one or more instances of energy source 105 and configured by one or more controllers 106 to perform operations on the multiple dual-rail qubits.

[0029] In the example of FIG. 2A , a microwave source (not shown) can be operated to read out the state of the ancilla qubit 204. For example, the readout resonator 207 can be aligned so that its resonant frequency (e.g., ∼GHz) is away from the transition frequency of the ancilla qubit 204 (e.g., dispersively coupled). Coupling between the ancilla qubit and the readout resonator means that there is a shift in the resonator frequency that depends on the state of the qubit. This shift is small compared to the resonator's resonant frequency (e.g., ∼MHz). Consequently, transmitting a tone near the resonant frequency to the readout resonator 207 is 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 nondestructively probed by transmitting a probe tone to the readout resonator 207, which is dispersively coupled to the qubit.

[0030] The dual-rail qubit 200 has a logical codeword encoded in the single-photon subspace of the qubit, |0> L =|01> and |1> L =|10>, for which the beam-splitter interaction is sufficient to facilitate any single-qubit rotation. These states are shown as "logical subspaces" in Figure 2C. By encoding the logical state of the qubit in the single-photon subspace of the connected-cavity Hilbert space with odd connected-photon-number parity, a dominant error in the cavity system, i.e., a single-photon loss to the common ground state |00>, can be transformed into an annihilation error detectable by a connected-parity measurement. Also, photon gain events in the cavity, which give rise to states |02>, |11>, or |20> and are generally rare at thermal equilibrium, can also be detected by a connected-parity measurement. This leaves cavity dephasing as a dominant Pauli error in the system.

[0031] 2A-2B, the Hamiltonian of dual-rail qubit 200 combines the beam splitter interaction between cavities 201 and 202 and the dispersive interaction between the modes of ancilla qubit 204 and cavity 201. This Hamiltonian is:

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[0032] In the example of Figures 2A-2B, the term

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[0033] Hamiltonian

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[0034] According to some embodiments, the coupling element 203 may include a nonlinear asymmetric inductive element (SNAIL), which provides tunneling energy E J and multiple (e.g., three) Josephson junctions with E JA superconducting circuit includes a loop with a single Josephson junction on the other arm of the loop with a tunneling energy less than 100 kHz, and two large capacitor pads are connected to the two arms of the loop. In this approach, the SNAIL is aligned between cavities 201 and 202, creating capacitive coupling between the SNAIL mode and each of the cavity modes. Magnetic flux also passes through the SNAIL loop and biases it (e.g., using a superconducting flux converter) to an operating point where it has third-order nonlinearity. SNAILs are described in U.S. Pat. No. 11,737,376, which is incorporated herein by reference in its entirety.

[0035] According to some embodiments, performing an operation on dual-rail qubit 200 may include pumping the coupling element to initiate a so-called "beam splitter" operation. In particular, the nonlinearity of the coupling element may enable three- or four-wave mixing, which is used to perform frequency-converting bilinear coupling between the first and second cavities. In the case of a coupling element that is a SNAIL, for example, the beam splitter operation may include three-wave mixing, and in the case of a coupling element that is a SQUID or transmon, the beam splitter operation may include four-wave mixing.

[0036] According to some embodiments, the beam splitter operation adjusts the microwave drive to a frequency matching condition ω1-ω2=ω p2 -ω p1 where ω is the resonant frequency of the first cavity, ω is the resonant frequency of the second cavity, and ω p1 is the frequency of the first pump associated with mode c, and ω p2 is the frequency of the second pump associated with mode d. This frequency matching condition is relevant in the case of four-wave mixing (for example, when 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 ω1-ω2=ω pThat is, if the coupling element 203 is a SNAIL, a drive frequency equal to the difference between the resonant frequencies of the cavities 201 and 202 may be applied to the SNAIL. In general, the drive frequency ω p is detuned from this frequency, ω p = ω1 - ω2 - Δ. When Δ = 0, the drive performs what is hereafter referred to as a "resonant beam splitter operation," which can be applied to exchange photons between the two cavities. When Δ ≠ 0, the drive performs what is hereafter referred to as a "detuned beam splitter operation," which can be applied to perform an entanglement gate between dual-rail qubits, as described below. The amplitude and phase of the microwave drive can be selected to select the strength and phase, respectively, of the beam splitter operation.

[0037] Beam splitter drive BS The length of time that (t) is applied to the coupling element 203 determines the particular operation applied to the two cavities 201 and 202. In particular, the integral of the beam splitter drive over time t is:

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[0038] According to some embodiments, state preparation of a dual-rail qubit may include loading a single photon into one of two cavities, where the selected cavity has |0> L or |1> L Whether the dual-rail qubit logic state is prepared (e.g., a single photon is loaded into cavity 201 and |0> L The state can be prepared or a single photon can be loaded into the cavity 202 to L In some embodiments, single-photon loading can be performed via optimal control pulses or cavity-ancilla sideband driving.

[0039] The logical state of the dual-rail qubit is determined by whether a single photon is in cavity 201 (state |0> L ) or in the cavity 202 (state |1> LIf instead it is determined that no photons are in either cavity, this indicates a leakage event due to photon loss.

[0040] According to some embodiments, the logical state of a dual-rail qubit can be determined by parity measurements of the two cavities, as follows: In some cases, the parity measurements can be performed after state preparation to verify that the state of the dual-rail qubit was prepared as intended. For example, after loading a single photon into one of the two cavities of the dual-rail qubit, the parity of each cavity in the dual-rail qubit is measured, and the results of the parity measurements are examined to determine whether the state of the dual-rail qubit was prepared as intended.

[0041] According to some embodiments, a parity measurement of a dual-rail qubit is shown in FIG. 3A. One exemplary method for performing a parity measurement of cavity 201 is to align an ancilla qubit 204 in its ground state |g> and drive the ancilla qubit with a π / 2 pulse to prepare the ancilla qubit in state |g>+|e> (normalization factor omitted). After the π / 2 pulse, ancilla qubit 204 has a π / χ ge where χ is not driven for time ge is the strength of the dispersive interaction between the ancilla qubit 204 and the cavity 201 in the ge manifold. At this time, the ancilla qubit 204 and the cavity 201 are in a state where their dispersive interaction

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[0042] The parity of one cavity can be measured in this manner, and then a beam splitter operation can be performed to swap the two cavity states as described above. The parity of the other cavity can then be measured in the same manner, and the cavity states are swapped again, with another beam splitter operation to return the states to their original configuration. For example, the sequence for parity measurements could be: a) measuring the parity in cavity 201 by driving an ancillary beam 204 with a π / 2 pulse, at time π / χ ge a) driving the ancillar 204 with a π / 2 pulse and measuring the state of the ancillar qubit 204; b) driving the coupling element 203 to perform a beam splitter operation to exchange the states in cavities 201 and 202; c) measuring the parity in cavity 201 by driving a π / 2 pulse into the ancillar 204, at time π / χ ge , driving the ancillar 204 with a π / 2 pulse, and measuring the state of the ancillar qubit 204; and d) driving the coupling element 203 to perform a beam splitter operation that swaps the states in the cavities 201 and 202.

[0043] If the two measured states of the ancilla qubit in acts (a) and (c) are both |g>, this indicates an annihilation error. Instead, the ancilla qubit state is the same as when a single photon is in cavity 201 (state |0> L), we should measure |e> then |g> in acts (a) and (c), respectively. Instead, the ancilla qubit state is the state |1> when a single photon is in cavity 202 (state |1> L ), |e> then |g> should be measured in each of actions (a) and (c).

[0044] Acts (a), (b), (c), and (d) may be repeated one or more times in this order (denoted as “n rounds” in FIG. 3A ) to perform repeated parity measurements and determine the dual-rail qubit logic state based on the set of these measurements (e.g., by adopting a majority rule outcome).

[0045] The different measurement outcomes of the parity measurement sequence described above are summarized in Figure 3B. The table shows the probability of inputting each state, where p is the probability of a photon loss event before the measurement, which leaves both cavities in state |0>. Depending on the input state, the outcome of the parity measurement will either be correctly aligned with the actual input state (checkmark), incorrectly flagged as an erasure (ε), or incorrectly aligned with a logic state (cross), resulting in a Pauli error.

[0046] |1> L Mistakes about |0> L The probability of (and vice versa) should be extremely small even in the presence of decoherence. First-order errors in the transmon and cavity give rise to measurements that can be flagged as further erasure errors. Logical misalignment, 10 for the actual coherence time -4 A combination of at least two errors is required for the probability and transmon readout fidelity to be estimated to be less than .

[0047] According to some embodiments, single-qubit gates of dual-rail qubit 200 can be implemented by resonant beam splitter operation as described above. To implement a single-qubit gate, coupling element 203 is driven at a frequency equal to the difference in frequency between the resonant frequencies of cavities 201 and 202, and the beam splitter drive amplitude, phase, and duration control the type of single-qubit gate. In some cases, for example, the strength and duration of the beam splitter drive control the polar angle of rotation around the Bloch sphere, and the phase of the beam splitter drive controls the rotation around the Z-axis of the Bloch sphere.

[0048] A key advantage of the dual-rail qubit described herein is its ability to perform QND detection of leakage from the logical subspace caused by the loss or gain of photons in either or both cavities. For QEC protocols, knowledge of the exact location of the error allows for the conversion of these otherwise fatal leakage events into erasure errors, which are significantly easier to correct than Pauli errors. This leakage detection scheme is sometimes referred to herein as "erasure checking" or "erasure detection."

[0049] According to some embodiments, the vanishing check includes measuring the joint photon number parity of cavities 201 and 202 in dual-rail qubit 200. L For both logic states |0> L has odd joint parity (i.e., there is one photon in the combined cavities in each case), so gaining or losing a photon in either cavity changes the joint parity to even.

[0050] Measuring the joint photon number parity of cavities 201 and 202 in dual-rail qubit 200 may involve applying a first on-resonant pulse to ancilla qubit 204, then applying a beam splitter operation to coupling element 203, and then applying a second on-resonant pulse to ancilla qubit 204. This sequence of operations may allow photon number information to be mapped onto the ancilla and then measured.

[0051] In some embodiments, the ancilla qubit 204 may be a three-level ancilla having a ground state |g>, a first excited state |e>, and a second excited state |f>. In this case, two of these states may be utilized to detect erasure errors caused by decay of the ancilla during joint parity measurement as well as erasures due to photon loss. For example, the resulting state of the ancilla after this sequence of operations may be measured to be: |g> if the joint parity is odd; |f> if the joint parity is even; and |e> if the ancilla state decays during the sequence of operations. Because the dual-rail qubit cavity should have odd joint parity, the ancilla states |f> and |e> both indicate erasure errors, and |g> indicates successful measurement of odd photon number parity.

[0052] According to some embodiments, measuring the joint photon number parity of cavities 201 and 202 of dual-rail qubit 200 involves driving an ancilla qubit 204 to measure the parity of the joint photon number of cavities 201 and 202 during the ancilla qubit gf manifold (operation 411).

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[0053] According to some embodiments, an exemplary process for implementing a dual-rail qubit with single-qubit gates and vanishing checks is shown in Figure 4. In the example of Figure 4, one or more single-qubit gates 402 couple coupling element 203 of the illustrated dual-rail qubit 200 to a beam splitter drive g as described above. bs After performing one or more single-qubit gates, the erasure check 404 can be performed by the concatenated photon number parity sequence described above and shown in inset 405.

[0054] Erasure errors can be detected by mapping the joint photon number parity to an ancillary element in various examples. First, idling errors (dominant photon loss) of dual-rail qubits can be converted to erasure errors. Second, errors arising from state preparation can be converted to erasure errors. Third, errors occurring during single-qubit gates can be detected by mapping the joint parity to an ancillary element and measuring the state of the ancillary element after performing the single-qubit gate.

[0055] According to some embodiments, an alternative scheme for extinction detection is shown in Figures 5A-5B. The driving scheme shown in the example of Figures 5A-5B differs from that of Figure 4 in that the scheme of Figures 5A-5B only detects whether the system is in a 0-photon state (i.e., both cavities are in |0>), and involves simultaneously driving ancilla qubit 204 and coupling element 203. Also, ancilla qubit 204 is driven only once. In this scheme, ancilla qubit 204 is excited to the |e> state only if both cavities are in the |0> state, and the cavity states remain unchanged.

[0056] In the example of FIGS. 5A-5B, the driving g bs is the intensity g bs >χ ge and as mentioned above the frequency is detuned from the resonance condition Δ=χ ge / 2. χ ge is the strength of the dispersive interaction between the ancilla qubits 204 in the ge manifold. Concurrently with this drive, another drive is applied to the ancilla qubit 204 at a frequency that is the difference between the |e> and |g> states of the ancilla qubit when the cavity contains no photons and has phase π. Consequently, the beam splitter drive is resonant only when there are no photons in the cavity. The two approaches for this annihilation detection scheme, shown in Figures 5A and 5B, differ in the pulse duration and pulse shape of the ancilla drive. In the example of Figure 5A, the ancilla qubit drive is approximately 2π / χ ge is applied for a time, and in the example of FIG. 5B, the ancilla qubit drive is approximately 4π / χ ge The time applies.

[0057] 5A-5B is that a three-level ancilla qubit may not be required in dual-rail qubit 200 because the detection outcome is based only on the |g> and |e> states of the ancilla qubit. This differs from the approach shown in FIG. 4, where a three-level ancilla qubit is used to distinguish between different outcomes.

[0058] According to some embodiments, FIG. 6 illustrates one aspect of the evolution of a dual-rail qubit over time during application of the drive shown in FIGS. 5A-5B. In the example of FIG. 6, the time at which the ancilla qubit drive is applied is shown along the horizontal axis, and the probability of measuring that the ancilla qubit is in |e> is shown along the vertical axis. Each of the four curves represents a different state of the dual-rail qubit, and the cavity states are described as two-number states, such as |11> referring to both cavities being in the |1> state. From FIG. 6, the dual-rail states

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[0059] Entanglement gates between dual-rail qubits can also be constructed such that errors in both cavities and transmons can be detected and converted to vanishings as described below. The inventors recognize that to realize the dual-rail logical Z operator, interaction of a dual-rail qubit with only a single cavity is required. By driving a beam-splitter interaction between a pair of cavities, one single transmon from each of the two dual-rail qubits can effectively act as a control for both dual-rail qubits. According to some embodiments, a suitable system for performing such an operation is shown in FIG. 7A.

[0060] 7A , a pair of dual-rail logic qubits 701 and 702 are shown coupled to one another by coupling element 703. Dual-rail qubit 701 includes cavities 711 and 712 (e.g., microwave cavities) coupled together via coupling element 713; dual-rail qubit 702 includes cavities 721 and 722 (e.g., microwave cavities) coupled together via coupling element 723. Each of cavities 712 and 722 is coupled to a respective ancilla qubit 714 or 724 (e.g., each may be a transmon qubit), which is coupled to a respective readout resonator. Similar to coupling element 203 in dual-rail qubit 200, each of coupling elements 703, 713, and 723 may be a superconducting nonlinear asymmetric induction element (SNAIL), a superconducting quantum interference device (SQUID), or some other nonlinear element.

[0061] The two-qubit gates described above can be performed on two dual-rail logical qubits by directing energy to coupling element 703 between the dual-rail qubits (e.g., alternatively, coupling element 203 between two logical qubits implemented by cavities 201 and 202 as in the example of FIG. 2A ). When performing a two-qubit gate on a pair of dual-rail qubits, an ancilla qubit 724 coupled to cavity 722, which is coupled to cavity 711 of the other dual-rail qubit via coupling element 703, can be operated similarly to the ancilla qubit in a two-qubit gate scheme. In this way, a two-qubit operation can be applied to ancilla qubit 724, and any error that occurs in performing a two-qubit gate can be detected by measuring the state of ancilla qubit 724 and determining whether the ancilla qubit is in state |g>, |e>, or |f>. The coupling shown in FIG. 7A is further illustrated in FIG. 7B, which shows the modes

number

number

[0062] One two-qubit entanglement gate is a ZZ(θ) gate,

number

[0063] One motivation for implementing the ZZ(θ) gate is that it can be performed using the exact same hardware required for the erasure check described in the previous section. Similar to error detection for the erasure check, it is important to detect an ancillary qubit error that occurs when performing the gate. The alignment of Figures 7A-7B allows for a first-order transmon error to be detected when the transmon is measured at the end of the gate, and causes both dual-rail qubits to disappear when the transmon is not in |g>. Once again, this operation is designed so that only Pauli error is introduced from second-order hardware errors when two decoherence events occur during a single gate. According to some embodiments, this approach is illustrated in Figure 8.

[0064] In the example of Figure 8, dual-rail qubits 701 and 702 are shown, and ZZ(θ) gates can be performed on these dual-rail qubits through the sequence of operations shown in inset 801. The circuitry shown in inset 801 enables detection of single ancilla dephasing errors in addition to ancilla damping events during ZZ(θ) gates by measuring the state of the ancilla qubit in operation 816. The state of ancilla qubit 724 serves as a flag to indicate whether the gates shown in operations 811, 812, 813, 814, and 815 were performed without ancilla dephasing or ancilla damping errors. In particular, if the state of ancilla qubit 724 is the ground state |g> after performing operations 811, 812, 813, 814, and 815, this indicates that no such errors occurred. Otherwise, if the state of the ancilla qubit is the first excited state |e> or the second excited state |f>, this indicates that at least one such error occurred in performing operations 811, 812, 813, 814 and 815.

[0065] Operation 811 is the ancillary qubit 724 gf manifold

number

number

[0066] While photon loss between two-qubit gates is detectable via separate vanishing checks on the two dual-rail qubits after each gate, modifications can also be made to the ZZ(θ) gate pulse sequence to simultaneously perform vanishing checks that can map transmons to the |f> level if one of the input dual-rail qubits is in a leaky state outside the logical subspace.

[0067] A different approach for a particular type of entanglement gate—the CPHASE(θ) gate—is illustrated in Figures 9A and 9B. The drive scheme illustrated in the example of Figures 9A-9B involves simultaneously driving the ancilla qubit 724 and the coupling element 703, with two separate pulses applied to the ancilla 724 and a single drive applied to the coupling element 703. In each case, two back-to-back π pulses are applied to the ancilla qubit 724. If both cavities 711 and 722 begin in the |0> state, the π pulses produce a phase shift on the ancilla qubit 724, and by controlling the phase of the second π pulse, a CPHASE(θ) gate is implemented between the dual-rail qubits 701 and 702. In particular, the phase of the second π pulse is parameterized by θ = π + φ.

[0068] The examples of Figures 9A and 9B differ in that Figure 9A shows an approach in which the ancilla qubit is a two-level qubit, and Figure 9B shows an approach in which the ancilla qubit is a three-level qubit. These two approaches are described separately below.

[0069] In the example of Figure 9A, with respect to an ancilla qubit error, if ancilla dephasing occurs during the CPHASE(θ) gate, the ancilla qubit will be measured at |e> rather than |g> at the end of the gate. If ancilla qubit decay occurs instead, the ancilla qubit will be measured at |e> 50% of the time. In either case, the ancilla qubit error can be detected after performing this gate by measuring the later state of the ancilla qubit, and affirmatively determining that an error occurred if the ancilla is measured at |e>.

[0070] In the example of FIG. 9A, the drive g bs is the intensity g bs >χ ge and, as mentioned above, the frequency is detuned from the resonance condition Δ=χ ge / 2. χ ge is the strength of the dispersive interaction between ancilla qubit 724 and cavity 722 in the ge manifold. Simultaneously with this drive, two back-to-back drives are applied to ancilla qubit 724 at frequencies that are the difference between the |e> and |g> states of the ancilla qubit when the cavity contains no photons. The first of these two drives is of phase 0, and the second has a phase θ = π + φ relative to the first drive. In the first (square pulse) example of Figure 9A, the two ancilla qubit drives are each approximately 2π / χ ge In the second (Gaussian pulse) example of FIG. 9A, the two ancilla qubit drives are each approximately 4π / χ ge In some embodiments, the two back-to-back ancilla qubit drives applied in the CPHASE gate are applied for equal amounts of time.

[0071] In the example of FIG. 9B, an ancilla qubit decay can be detected through the use of a three-level ancilla qubit. In particular, an ancilla qubit error can be detected by measuring the state of the ancilla qubit after applying one of the drive sequences shown in FIG. 9B, and determining that an error has occurred if the ancilla is measured to be |e> or |f> in the affirmative. In particular, measuring the ancilla qubit to be in the |e> state indicates that the CPHASE gate has failed due to decay of the ancilla qubit state, and measuring the ancilla qubit to be in the |f> state indicates that the CPHASE gate has failed due to ancilla qubit dephasing. Measuring the ancilla qubit to be in the |g> state indicates that the CPHASE gate has been successfully executed.

[0072] In the example of FIG. 9B, the drive g bs is the intensity g bs >χ gf and, as mentioned above, the frequency is detuned from the resonance condition Δ=χ gf / 2. χ gf is the strength of the dispersive interaction between ancilla qubit 724 and cavity 722 in the gf manifold. Concurrently with this drive, two back-to-back drives are applied to ancilla qubit 724 at frequencies that are the difference between the |g> and |f> states of the ancilla qubit when the cavity contains no photons. The first of these two drives has a phase of 0, and the second has a phase θ = π + φ relative to the first drive. In the first (square pulse) example of Figure 9B, the two ancilla qubit drives are each approximately 2π / χ ge In the second (Gaussian pulse) example of Figure 9B, the two ancilla qubit drives are each approximately 4π / χ ge In some embodiments, the two back-to-back ancilla qubit drives applied in the CPHASE gate are applied for equal amounts of time.

[0073] The dual-rail qubits described above and techniques for preparing states, performing single dual-rail qubit gates, performing entanglement gates on two dual-rail qubits, and detecting erasure errors can be extended to systems including more than two dual-rail qubits.

[0074] In some embodiments, multiple dual-rail qubits can be arranged in an array, network, grid, lattice, or other repeating configuration. Dual-rail qubits can be arranged in this manner using one or more structures that provide connectivity between the dual-rail qubits, which may include other dual-rail qubits and / or other structures, such as cavities. In some embodiments, a "unit cell" can be repeated multiple times to generate a lattice configuration. A suitable unit cell can include, for example, a dual-rail qubit (including two superconducting cavities, an ancilla qubit, such as a transmon, and a coupling element, such as a SNAIL, between the two cavities) and two additional coupling elements to interface adjacent unit cells.

[0075] One illustrative example of a network of dual-rail qubits is shown in Figure 10. In Figure 10, dual-rail qubits (two examples of which are 1001) are represented as a pair of coupled cavities (circles with connecting lines) surrounded by a rectangular box. The dual-rail qubits are coupled to each other via measurement qubits (two examples of which are 1002), represented by solid circles. Each of the measurement qubits includes a single cavity coupled to an ancilla qubit, plus a readout cavity. The line between the measurement qubits and the dual-rail qubits represents a coupling element (e.g., a SNAIL).

[0076] In some embodiments, a network of dual-rail qubits can be operated such that multiple dual-rail qubits are operated as a single logical qubit. For example, multiple dual-rail qubits can be entangled together using an appropriate sequence of quantum gates, and subsequent measurement of the entangled state provides a means for error correction and detection. A set of physical dual-rail qubits entangled in this way is thereby used to define a logical qubit, which can be expected to have better performance than a single dual-rail qubit.

[0077] One way to entangle dual-rail qubits in this way is to use so-called "surface codes," in which a group of dual-rail qubits is selected to be either data qubits, which store computational quantum states or measurement qubits, and which are used to stabilize and manipulate the quantum states of the data qubits. Stabilizer codes used in surface codes are described, for example, in "Surface codes: Towards practical large-scale quantum computation," Austin Fowler et al., Phys. Rev. A 86, 032324 (2012), which is incorporated herein by reference in its entirety.

[0078] According to some embodiments, an example of a physical arrangement of dual-rail qubits that may be grouped and manipulated as logical qubits using a surface code (or any other suitable code) is shown in Figure 11. In the example of Figure 11, each dual-rail qubit is coupled to four adjacent dual-rail qubits; each data dual-rail qubit is coupled to four adjacent measurement dual-rail qubits, and each measurement dual-rail qubit is coupled to four adjacent data dual-rail qubits.

[0079] One challenge with this type of arrangement may be that, although scalable, it is not modular, which may make it difficult to simply fabricate. Additionally, it may lack flexibility in the spatial placement of the dual-rail qubits (e.g., it may require the dual-rail qubits to be aligned on a flat surface).

[0080] According to some embodiments, a more modular and scalable approach to that shown in Figure 11 is shown in Figure 12A. In the example of Figure 12A, each dual-rail qubit data qubit is coupled to four dual-rail qubit measurement qubits (and vice versa) as described above, but the dual-rail qubits are aligned in modules, the boundaries of which are shown in Figure 12A by light grey rectangles. Coupling between adjacent cavities within a module (intra-module coupling) is performed using nonlinear coupling elements as described above, while coupling between cavities across module boundaries (inter-module coupling) is instead performed by a quantum bus, an example of which is described below.

[0081] Consequently, multiple formally identical dual-rail qubit modules can be fabricated that include multiple couplings (and / or coupling ports) that couple the module's cavities to an external quantum bus. In the example of Figure 12A, each such module includes nine dual-rail qubit data qubits and four dual-rail qubit measurement qubits, plus 12 cavities that are aligned so that each is coupled to the quantum bus. It may be noted that a cavity that is coupled to a quantum bus forms a dual-rail qubit measurement qubit when coupled to another cavity in an adjacent module. That is, the resulting system includes multiple dual-rail qubit measurement qubits, where two cavities are coupled by a quantum bus (acting as coupling element 103).

[0082] It may be further noted that this approach yields "quad-rail" qubits at the corners of the four modules, where the 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 the data qubit via a nonlinear coupling element).

[0083] Figure 12B shows an exemplary quantum bus that can serve as an inter-module link in the example of Figure 12A, according to some embodiments. In the example of Figure 12B, the quantum bus is implemented as a standing wave transport mechanism. For example, the quantum bus can include microwave coaxial cable, flexicable, and / or other types of photonic microwave links that include bus modes for performing SWAP and / or 50-50 beam splitter operations between cavities coupled to opposite ends of the bus.

[0084] Having thus described several aspects of at least one embodiment of this invention, it will be appreciated that various alterations, modifications, and improvements will be apparent to those skilled in the art.

[0085] Such changes, modifications, and improvements are intended to be part of this disclosure and are intended to be within the spirit and scope of the invention. Moreover, while advantages of the invention are set forth, it should be understood that not all aspects of the technology described herein include all of the described advantages. Some aspects may not implement any of the features described as advantageous herein, and in some instances, additional aspects may be achieved by implementing one or more of the described features. Accordingly, the foregoing description and drawings are by way of example only.

[0086] Aspects of the present disclosure may include, but are not limited to, the following.

[0087] 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 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 to a 0 or 1 logical state by operating the at least one energy source to align the first quantum oscillator to a single-photon state and the second quantum oscillator to its ground state when the dual-rail qubit is initialized to a 0 logical state; or by operating the at least one energy source to align the first quantum oscillator to its ground state and the second quantum oscillator to the single-photon state when the dual-rail qubit is initialized to a 1 logical state.

[0088] Aspect 2. The dual-rail qubit of Aspect 1, wherein aligning the first quantum oscillator or the second quantum oscillator to a single-photon state includes manipulating at least one energy source to perform a plurality of optimal control pulses and / or cavity-ancilla sideband drives.

[0089] Aspect 3. The dual-rail qubit of Aspect 1, wherein aligning the first quantum oscillator to a single-photon state includes manipulating at least one energy source to align the first quantum oscillator to a |1> Fock state.

[0090] Aspect 4. The dual-rail qubit of Aspect 1, wherein the coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

[0091] 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).

[0092] Phase 6. The dual-rail qubit of Phase 1, where the ancilla qubit is not coupled to the second quantum oscillator.

[0093] Phase 7. Dual-rail qubit of Phase 1, where the ancyla qubit is a transmon qubit.

[0094] Phase 8. The dual-rail qubit of Phase 1, wherein the first quantum oscillator is a first microwave cavity and the second quantum oscillator is a second microwave cavity.

[0095] 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 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 the logical state of the dual-rail qubit by directing energy to the coupling element one or more times.

[0096] 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 the difference between the resonant frequency of the first quantum oscillator and the resonant frequency of the second quantum oscillator.

[0097] 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 manipulation of the at least one energy source based on a rotation angle of the single qubit rotation.

[0098] Phase 12. The dual-rail qubit of Phase 11, wherein the amplitude and duration are selected based on the polar angle of the single-qubit rotation, and the phase is selected based on the angle of rotation about the Bloch sphere Z axis during the single-qubit rotation.

[0099] Aspect 13. The dual-rail qubit of Aspect 9, wherein the coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

[0100] 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).

[0101] Phase 15. The dual-rail qubit of Phase 9, where the ancilla qubit is not coupled to the second quantum oscillator.

[0102] Phase 16. Dual-rail qubit of Phase 9, where the ancyla qubit is a transmon qubit.

[0103] Aspect 17. The dual-rail qubit of Aspect 9, wherein the first quantum oscillator is a first microwave cavity and the second quantum oscillator is a second microwave cavity.

[0104] Aspect 18. A dual-rail qubit, including: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator and having a ground state |g>, a first excited state |e>, and a second excited state |f>; 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 a quantum state of the ancilla qubit; (b) after (a), direct energy to the coupling element to perform a beam splitter operation on the first quantum oscillator and the second quantum oscillator; and (c) after (b), direct energy to the ancilla qubit to perform a second rotation of a quantum state of the ancilla qubit.

[0105] Aspect 19. The dual-rail qubit of Aspect 18, wherein the first and second rotations of the state of the ancilla qubit are rotations in a manifold between the ground state |g> and the second excited state |f> of the ancilla qubit.

[0106] Phase 20. Dual-rail qubit of phase 18, where the first and second rotations of the ancilla qubit's state are π / 2 rotations around the Y axis of the Bloch sphere.

[0107] 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 the state of the ancilla qubit after (c).

[0108] 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 be not in a basis state |g>.

[0109] 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 |e> or the second excited state |f>.

[0110] Aspect 24. The dual-rail qubit of Aspect 18, wherein the coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

[0111] 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).

[0112] Phase 26. The dual-rail qubit of Phase 18, where the ancilla qubit is not coupled to the second quantum oscillator.

[0113] Aspect 27. A dual-rail qubit in aspect 18, where the ancyla qubit is a transmon qubit.

[0114] Aspect 28. The dual-rail qubit of Aspect 18, wherein the first quantum oscillator is a first microwave cavity and the second quantum oscillator is a second microwave cavity.

[0115] Aspect 29. A dual-rail qubit, including: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and 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 logic state of the dual-rail qubit based on the measured parity states of the first and second quantum oscillators.

[0116] Aspect 30. The dual-rail qubit of Aspect 29, wherein the determined logic state of the dual-rail qubit is a 0 logic state or a 1 logic state when the parity state of the first quantum oscillator is measured to be opposite to the parity state of the second quantum oscillator.

[0117] Aspect 31. The dual-rail qubit of Aspect 29, wherein the at least one controller is further configured to identify an erasure error if both parity states of the first and second quantum oscillators are measured to be even.

[0118] 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 multiple times and to operate the at least one energy source to measure the parity state of the second quantum oscillator multiple times, thereby generating multiple parity state measurements for each of the first and second quantum oscillators.

[0119] Aspect 33. The dual-rail qubit of Aspect 29, wherein the at least one controller is configured to determine a logic state of the dual-rail qubit based on a majority outcome of the plurality of parity state measurements for the first quantum oscillator and based on a majority outcome of the plurality of parity state measurements for the second quantum oscillator.

[0120] Aspect 34. The dual-rail qubit of Aspect 29, wherein manipulating the at least one energy source to measure the parity state of the first quantum oscillator includes manipulating the at least one energy source to measure the state of the ancilla qubit and identifying the parity state of the first quantum oscillator based on the measured state of the ancilla qubit.

[0121] Aspect 35. The dual-rail qubit of Aspect 29, wherein the at least one controller is further configured to, after operating the at least one energy source to measure a parity state of the first quantum oscillator and before operating the at least one energy source to measure a parity state of the second quantum oscillator, direct energy to a coupling element to perform a beam splitter operation on the first quantum oscillator and the second quantum oscillator.

[0122] Aspect 36. The dual-rail qubit of Aspect 29, wherein the coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

[0123] 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).

[0124] Phase 38. The dual-rail qubit of Phase 29, where the ancilla qubit is not coupled to the second quantum oscillator.

[0125] Aspect 39. A dual-rail qubit in aspect 29, where the ancyla qubit is a transmon qubit.

[0126] Aspect 40. The dual-rail qubit of Aspect 29, wherein the first quantum oscillator is a first microwave cavity and the second quantum oscillator is a second microwave cavity.

[0127] Aspect 41. A dual-rail qubit including: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator and having a ground state |g>, a first excited state |e>, and a second excited state |f>; 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 after performance of the one or more gates and / or operations; and operate the at least one energy source to initialize the dual-rail qubit to a new logical state in response to detecting the erasure error.

[0128] Aspect 42. The dual-rail qubit of Aspect 41, wherein detecting the erasure error includes determining that the quantum state of the ancilla qubit is not in a ground state |g>.

[0129] Aspect 43. The dual-rail qubit of Aspect 42, wherein detecting the erasure error includes determining that the quantum state of the ancilla qubit is in a first excited state |e> or a second excited state |f>.

[0130] Aspect 44. The dual-rail qubit of Aspect 41, wherein the coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and the ancilla qubit is dispersively coupled to the first quantum oscillator.

[0131] 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).

[0132] Phase 46. The dual-rail qubit of Phase 41, where the ancilla qubit is not coupled to the second quantum oscillator.

[0133] Aspect 47. A dual-rail qubit in aspect 41, where the ancyla qubit is a transmon qubit.

[0134] Aspect 48. The dual-rail qubit of Aspect 41, wherein the first quantum oscillator is a first microwave cavity and the second quantum oscillator is a second microwave cavity.

[0135] Aspect 49. A system including a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first dual-rail qubit including a first ancilla qubit coupled to the first quantum oscillator; a third quantum oscillator; a fourth quantum oscillator; a second coupling element coupled to the third quantum oscillator and the fourth quantum oscillator; and a second dual-rail qubit including a second ancilla qubit coupled to the third quantum oscillator; and a third coupling element coupled to the second quantum oscillator and the third quantum oscillator.

[0136] Aspect 50. The system of Aspect 49, further including 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 the logic states of the first dual-rail qubit and the second dual-rail qubit.

[0137] Phase 51. The system of Phase 50, where the two-qubit gate is a ZZ(θ) gate.

[0138] Aspect 52. The system of Aspect 50, wherein manipulating at least one energy source to perform a two-qubit gate includes directing energy to a third coupling element.

[0139] Aspect 53. The system of Aspect 52, wherein manipulating at least one energy source to perform a two-qubit gate includes performing one or more qubit rotations of a second ancilla qubit.

[0140] Aspect 54. The system of Aspect 53, wherein operating at least one energy source to perform a two-qubit gate includes: (a) performing a first qubit rotation of a second ancilla qubit; (b) after (a), directing energy to a third coupling element; and (c) after (b), performing a second qubit rotation of the second ancilla qubit.

[0141] Aspect 55. The system of Aspect 50, wherein the at least one controller is further configured to identify an erasure error if, after performing the two-qubit gate, the quantum state of the second ancilla qubit is detected to be not in a basis state |g>.

[0142] Aspect 56. The system of Aspect 55, wherein the at least one controller is further configured to identify an erasure error if, after performing the two-qubit gate, the quantum state of the second ancilla qubit is detected to be in the first excited state |e> or the second excited state |f>.

[0143] Aspect 57. The system of Aspect 50, wherein the first coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; the second coupling element is dispersively coupled to the third quantum oscillator and 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 the third quantum oscillator.

[0144] Aspect 58. A system comprising: a plurality of dual-rail qubits, wherein each dual-rail qubit of the plurality of dual-rail qubits includes: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator; and a plurality of measurement qubits, wherein each measurement qubit of the plurality of measurement qubits includes: a first quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator of the measurement qubit; wherein each of the plurality of measurement qubits is coupled to four of the plurality of dual-rail qubits, each coupling provided via a respective coupling element.

[0145] Aspect 59. The system of Aspect 58, wherein each coupling of the measurement qubit to one of the plurality of dual-rail qubits couples a first quantum oscillator of the measurement qubit to a first or second quantum oscillator of one of the plurality of dual-rail qubits.

[0146] 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).

[0147] 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.

[0148] Aspect 62. A module including a plurality of dual-rail qubits, wherein each dual-rail qubit of the plurality of dual-rail qubits includes: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and 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.

[0149] Aspect 63. The module of Aspect 62, wherein the external port includes a coaxial connector.

[0150] Aspect 64. A system including a plurality of modules of Aspect 62 coupled together via a plurality of quantum bus links between the external ports of the modules.

[0151] 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).

[0152] 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.

[0153] The above-described aspects of the technology described herein can be implemented in any of many ways. For example, a controller of any of the aspects, such as controller 106 shown in FIG. 1, can 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 on a single computer or distributed among multiple computers. Such processors can be implemented as commercially available integrated circuit components known in the art under names such as CPU chips, GPU chips, microprocessors, microcontrollers, or coprocessors, having one or more processors in the integrated circuit component. Alternatively, the processor can be implemented in custom circuitry such as an ASIC or semi-custom circuitry resulting from constructing a programmable logic device. As a still further alternative, the processor can be part 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, and one or a subset of these cores can constitute the processor. Nevertheless, the processor can be implemented using circuitry in any suitable form.

[0154] The various aspects of the present invention may be used alone, in combination, or in various arrangements not specifically described in the embodiments described above, and are therefore not limited in their application to the details and arrangements 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.

[0155] The present invention may also be embodied as a method, examples of which are provided. The acts performed as part of the method may be ordered in any suitable manner. Thus, although shown as sequential acts in the exemplary embodiments, embodiments may be constructed in which acts are performed in an order different from that shown, which may include performing some acts simultaneously.

[0156] The use of ordinal terms, e.g., "first," "second," "third," etc., in the claims to modify claim elements does not, in itself, imply any priority, precedence, or order of one claim element over another, or the temporal order in which acts of a method are performed, but is merely used as a marker to distinguish between multiple claim elements, distinguishing one claim element having a particular name from another element having the same name (other than the use of ordinal terms).

[0157] The terms "approximately" and "about" can be used to mean, in some embodiments, within ±20% of a target value, in some embodiments, within ±10% of a target value, in some embodiments, within ±5% of a target value, and even in some embodiments, within ±2% of a target value. The terms "approximately" and "about" can include the target value. The term "substantially equal" can be used to refer to values ​​that are, in some embodiments, within ±20% of each other, in some embodiments, within ±10% of each other, in some embodiments, within ±5% of each other, and even in some embodiments, within ±2% of each other.

[0158] The term "substantially" can be used to refer to a value that is within ±20% of a reference point in some embodiments, within ±10%, within ±5%, and even within ±2% in some embodiments. For example, a first direction that is "substantially" perpendicular to a second direction can refer to a first direction that is within ±20% of forming a 90° angle with the second direction in some embodiments, within ±10% of forming a 90° angle with the second direction in some embodiments, within ±5% of forming a 90° angle with the second direction in some embodiments, and even within ±2% of forming a 90° angle with the second direction in some embodiments.

[0159] 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," "having," "containing," "involving," and variations thereof herein is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.

Claims

1. First quantum oscillator; Second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and operating at least one energy source to align the first quantum oscillator to a single-photon state and the second quantum oscillator to its ground state when the dual-rail qubit is initialized to a 0 logic state; or When the dual-rail qubit is initialized to a one logic state, by operating at least one energy source to align the first quantum oscillator to its ground state and the second quantum oscillator to the single-photon state; at least one controller configured to operate the at least one energy source to initialize the dual-rail qubit to a 0 or 1 logical state; Dual-rail qubits, including:

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

3. 10. The dual-rail qubit of claim 1, wherein aligning the first quantum oscillator to the single-photon state comprises manipulating at least one energy source to align the first quantum oscillator to the |1> Fock state.

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

5. 10. 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 the second quantum oscillator is a second microwave cavity.

9. First quantum oscillator; Second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and and at least one controller configured to operate the at least one energy source to perform a single-qubit rotation of the 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. Dual-rail qubits, including:

10. 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 the difference between the resonant frequency of the first quantum oscillator and the resonant frequency of the second quantum oscillator.

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

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

13. a coupling element dispersively coupled to the first quantum oscillator and the second quantum oscillator; The ancilla qubit is dispersively coupled to the first quantum oscillator; 10. The dual-rail qubit of claim 9.

14. 10. 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. 10. The dual-rail qubit of claim 9, wherein the first quantum oscillator is a first microwave cavity and the second quantum oscillator is a second microwave cavity.

18. First quantum oscillator; Second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an ancilla qubit coupled to a first quantum oscillator and having a ground state |g>, a first excited state |e>, and a second excited state |f>; at least one energy source; and Manipulate at least one energy source to: (a) directing energy to the ancilla qubit to perform a first rotation of the quantum state of the ancilla qubit; (b) after (a), directing energy to a coupling element to perform a beam splitter operation on the first quantum oscillator and the second quantum oscillator; (c) after (b), at least one controller configured to direct energy to the ancilla qubit to perform a second rotation of the quantum state of the ancilla qubit; Dual-rail qubits, including:

19. 20. The dual-rail qubit of claim 18, wherein the first and second rotations of the state of the ancilla qubit are rotations in a manifold between a ground state |g> and a second excited state |f> of the ancilla qubit.

20. 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 Y axis of the Bloch sphere.

21. 20. 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 the state of the ancilla qubit after (c).

22. 20. 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 not in a basis state |g>.

23. 20. 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 |e> or the second excited state |f>.

24. a coupling element dispersively coupled to the first quantum oscillator and the second quantum oscillator; The ancilla qubit is dispersively coupled to the first quantum oscillator; 20. The dual-rail qubit of claim 18.

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

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

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

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

29. First quantum oscillator; Second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an ancilla qubit coupled to the first quantum oscillator; at least one energy source; and operating at least one energy source to measure a parity state of the first quantum oscillator; operating at least one energy source to measure the parity state of the second quantum oscillator; and at least one controller configured to determine a logic state of the dual-rail qubit based on the measured parity states of the first and second quantum oscillators. Dual-rail qubits, including:

30. 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 logic state of the dual-rail qubit is a 0 logic state or a 1 logic state.

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

32. 30. 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 multiple times and to operate the at least one energy source to measure the parity state of the second quantum oscillator multiple times, thereby generating multiple parity state measurements for each of the first and second quantum oscillators.

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

34. 30. The dual-rail qubit of claim 29, wherein operating the at least one energy source to measure a 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. 30. 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 beam splitter operation on the first quantum oscillator and the second quantum oscillator after operating the at least one energy source to measure a parity state of the first quantum oscillator and before operating the at least one energy source to measure a parity state of the second quantum oscillator.

36. a coupling element dispersively coupled to the first quantum oscillator and the second quantum oscillator; The ancilla qubit is dispersively coupled to the first quantum oscillator; 30. The dual-rail qubit of claim 29.

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

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

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

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

41. First quantum oscillator; Second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an ancilla qubit coupled to a first quantum oscillator and having a ground state |g>, a first excited state |e>, and a second excited state |f>; at least one energy source; and operating at least one energy source to perform one or more gates and / or operations at least in part on the dual-rail qubit; Detecting an erasure error after performing one or more gates and / or operations; and at least one controller configured to operate the at least one energy source to initialize the dual-rail qubit to a new logic state in response to detecting the erasure error. Dual-rail qubits, including:

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

43. 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 a first excited state |e> or a second excited state |f>.

44. a coupling element dispersively coupled to the first quantum oscillator and the second quantum oscillator; The ancilla qubit is dispersively coupled to the first quantum oscillator; 42. The dual-rail qubit of claim 41.

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

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

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

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

49. First quantum oscillator; Second quantum oscillator; a first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and A first ancillary qubit coupled to a first quantum oscillator a first dual-rail qubit, including: Third quantum oscillator; The fourth quantum oscillator; a second coupling element coupled to the third quantum oscillator and the fourth quantum oscillator; and A second ancillary qubit coupled to a third quantum oscillator a second dual-rail qubit, including: a third coupling element coupled to the second quantum oscillator and the third quantum oscillator; Including, the system.

50. 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 the logic states of the first dual-rail qubit and the second dual-rail qubit; 50. The system of claim 49, further comprising:

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

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

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

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

55. 51. The system of claim 50, wherein the at least one controller is further configured to identify an erasure error if, after performing the two-qubit gate, the quantum state of the second ancilla qubit is detected to be not in a basis state |g>.

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

57. a first coupling element dispersively coupled to the first quantum oscillator and the second quantum oscillator; a second coupling element dispersively coupled to the third quantum oscillator and the fourth quantum oscillator; a first ancilla qubit dispersively coupled to a first quantum oscillator; a second ancilla qubit dispersively coupled to a third quantum oscillator; a third coupling element dispersively coupled to the second quantum oscillator and the third quantum oscillator; 51. The system of claim 50.

58. A plurality of dual-rail qubits, where each dual-rail qubit of the plurality of dual-rail qubits: First quantum oscillator; Second quantum oscillator; a first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and A first ancillary qubit coupled to a first quantum oscillator Includes; A plurality of measurement qubits, where each measurement qubit of the plurality of measurement qubits: a first quantum oscillator; and a first ancilla qubit coupled to the first quantum oscillator of the measurement qubit A system comprising: A system in which each of the plurality of measurement qubits is coupled to four of the plurality of dual-rail qubits, each coupling provided via a respective coupling element.

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

60. 59. 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. 59. 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 plurality of dual-rail qubits, where each dual-rail qubit of the plurality of dual-rail qubits: First quantum oscillator; Second quantum oscillator; a first coupling element coupled to the first quantum oscillator and 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 a plurality of dual-rail qubits via a respective coupling element; and External port configured for coupling to a quantum bus Contains the module.

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

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

65. 63. 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. 63. 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.

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