Techniques for dual track coding of qubits and related systems and methods

Abstract

Techniques for quantum error correction are described that utilize dual track coding of a single excitation in a pair of coupled quantum oscillators. The dual track qubit may be implemented in a cQED (Circuit Quantum Electrodynamics) system that includes a pair of coupled resonators. In this dual track coding, the excitation is stored in one of the two resonators. The excitation of the first resonator is regarded as logic 0, while the excitation of the other resonator is regarded as logic 1. Thus, the two resonators together form a single logical dual track qubit with logical states of 0 > L = 01 > and 1 > L = 10 >.

Description

[0001] Government funding

[0002] 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 Art

[0003] Quantum information processing techniques perform calculations by manipulating one or more quantum objects. These techniques are sometimes referred to as "quantum computing". In order to perform calculations, quantum information processors use quantum objects to reliably store and retrieve information. According to some quantum information processing methods, quantum simulations of classical computing "bits" (equal to 1 or 0) have been developed, which are called quantum bits (quantum bits) or "qubits". A qubit can be composed of any quantum system with two distinguishable states (which can be considered as a 1 state and a 0 state), but also has the following special properties: the system can be placed in a quantum superposition and therefore potentially exist in these two states at the same time. Summary of the Invention

[0004] In some aspects, the technology described herein relates to 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 auxiliary 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 logic state by: operating the at least one energy source to arrange the first quantum oscillator into a single-photon state and the second quantum oscillator into its ground state when the dual-rail qubit is to be initialized to a 0 logic state; or operating the at least one energy source to arrange the first quantum oscillator into its ground state and the second quantum oscillator into a single-photon state when the dual-rail qubit is to be initialized to a 1 logic state.

[0005] In some aspects, the technology described herein relates to 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 auxiliary qubit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to operate the at least one energy source to perform a single-qubit rotation of a logical state of the dual-rail qubit by operating the at least one energy source one or more times to direct energy to the coupling element.

[0006] In some aspects, the technology described herein relates to 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 auxiliary 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 auxiliary qubit to perform a first rotation of the quantum state of the auxiliary qubit; (b) subsequent to (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) subsequent to (b), direct energy to the auxiliary qubit to perform a second rotation of the quantum state of the auxiliary qubit.

[0007] In some aspects, the technology described herein relates to 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 auxiliary 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 quantum oscillator and the second quantum oscillator.

[0008] In some aspects, the technology described herein relates to 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 auxiliary 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 performing the one or more gates and / or operations; and in response to detecting the erasure error, operate the at least one energy source to initialize the dual-rail qubit to a new logical state.

[0009] In some aspects, the technology described herein relates to a system comprising: a first dual-rail qubit, the first dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element, the first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first auxiliary qubit, the first auxiliary qubit coupled to the first quantum oscillator; a second dual-rail qubit, the second dual-rail qubit comprising: a third quantum oscillator; a fourth quantum oscillator; a second coupling element, the second coupling element coupled to the third quantum oscillator and the fourth quantum oscillator; and a second auxiliary qubit coupled to the third quantum oscillator; and a third coupling element coupled to the second quantum oscillator and the third quantum oscillator.

[0010] In some aspects, the technology described herein relates to a system comprising: a plurality of dual-rail qubits, each dual-rail qubit in the plurality of dual-rail qubits comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first ancillary qubit coupled to the first quantum oscillator; and a plurality of measurement qubits, each measurement qubit in the plurality of measurement qubits comprising: a first quantum oscillator; and a first ancillary qubit coupled to the first quantum oscillator of the measurement qubit, wherein each of the plurality of measurement qubits is coupled to four dual-rail qubits in the plurality of dual-rail qubits, wherein each coupling is provided via a corresponding coupling element.

[0011] In some aspects, the technology described herein relates to a module comprising: a plurality of dual-rail qubits, each dual-rail qubit in the plurality of dual-rail qubits comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first auxiliary qubit coupled to the first quantum oscillator; and a plurality of cavities, each of the plurality of cavities being: coupled to one of the dual-rail qubits in the plurality of dual-rail qubits via a respective coupling element; and coupled to an external port configured for coupling to a quantum bus.

[0012] The aforementioned device and method embodiments can be implemented with any appropriate combination of the aspects, features, and actions described above or in more detail below. These and other aspects, embodiments, and features of the present teachings can be more fully understood from the following description in conjunction with the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Various aspects and embodiments will be described with reference to the following drawings. It should be understood that the drawings are not necessarily drawn to scale. In the drawings, each identical or nearly identical component shown in various figures is represented by the same reference numeral. For clarity, not every component may be labeled in every drawing.

[0014] Figure 1 depicts an illustrative system suitable for practicing the techniques described herein, according to some embodiments;

[0015] Figure 2A Depicted is a microwave cavity according to some embodiments. Figure 1 illustrative implementation of a system;

[0016] Figure 2B Depicted according to some embodiments Figure 2A Coupling between components;

[0017] Figure 2C Depicted according to some embodiments Figure 2A The logical state of the dual-track quantum bit;

[0018] Figure 3A represents a parity measurement of a dual-rail qubit according to some embodiments;

[0019] Figure 3B Summarizes the following embodiments Figure 3A Different measurement results of the parity measurement sequence;

[0020] Figure 4 depicts an illustrative process for implementing a dual-rail qubit with single-qubit gates and erasure checking, according to some embodiments;

[0021] Figures 5A to 5B depicts an alternative approach for erasure detection according to some embodiments;

[0022] Figure 6 Depicts the application of some embodiments Figures 5A to 5B One aspect of the temporal evolution of a dual-rail qubit during actuation is shown in ;

[0023] Figure 7A Depicts a system including two dual-rail logical qubits implemented using a microwave cavity according to some embodiments. Figure 1 illustrative implementation of a system;

[0024] Figure 7B Depicted according to some embodiments Figure 7A Coupling between components;

[0025] Figure 8depicts an illustrative implementation of a ZZ(θ) gate between two dual-rail qubits according to some embodiments;

[0026] Figures 9A to 9B depicts an illustrative driver for performing a CPHASE(θ) gate between two dual-rail qubits, according to some embodiments;

[0027] Figure 10 depicts an illustrative example of a network of dual-rail qubits in accordance with some embodiments;

[0028] Figure 11 depicts an illustrative example of a physical arrangement of dual-rail qubits that can be grouped and operated as logical qubits, according to some embodiments;

[0029] Figure 12A depicts a plurality of modules, each module comprising a plurality of dual-rail qubits that can be grouped and operated as logical qubits, according to some embodiments; and

[0030] Figure 12B Depicts a device that can serve as Figure 12A Illustrative quantum bus for intermodule links in the example. DETAILED DESCRIPTION

[0031] Quantum multilevel systems, such as superconducting qubits, exhibit quantum states that decohere in about ~100 μs based on current experimental practice. While experimental techniques will undoubtedly improve on this and produce qubits with longer decoherence times, it may be beneficial to couple the multilevel system with another system that exhibits a longer decoherence time. Systems configured with bosonic modes may be particularly desirable for coupling to multilevel systems. Through this coupling, the state of the multilevel system can be represented instead by the bosonic mode, thereby maintaining the same information in a state that is even longer-lived than would otherwise exist in the multilevel system alone. When used in this way, bosonic systems are sometimes referred to as "logical" qubits.

[0032] Nevertheless, the quantum information stored in the bosonic mode may still have a finite lifetime, so that errors will still occur within the bosonic system. Therefore, when errors occur in the state of the bosonic system, it may be desirable to manipulate the bosonic system to effectively correct these errors and thus restore the previous state of the system. If a wide range of errors can be corrected, the state of the bosonic system can be preserved indefinitely (or at least for a long period of time) by correcting any type of error that may occur.

[0033] Nevertheless, the quantum information stored in the bosonic modes may still have a finite lifetime, so that errors will still occur within the bosonic system. When errors occur in the state of the bosonic system, it is desirable to manipulate the bosonic system to effectively correct these errors and thereby restore the previous state of the system. In some cases, simply detecting the errors that occurred during operation may be sufficient. The desire to detect and / or correct errors in quantum systems requires simultaneously addressing a number of requirements, including the number of physical qubits used to encode logical information, a suitable scheme for performing gates and measurements on the quantum system, and the precision of the physical components of the system.

[0034] The inventors have recognized and understood that not all types of errors are equally likely to occur in a given quantum error correction architecture, nor are all types of errors equally harmful to the logical information stored in the quantum system. For example, some types of errors require measuring specific error syndromes to precisely determine the error and obtain the information needed to correctly repair the error. However, this does not always lead to successful error correction because sometimes there is not enough information to unambiguously determine the error. On the other hand, erasure errors—errors in which the information in a qubit is erased but in a way that allows the qubit to be identified—can be more easily resolved.

[0035] The inventors have further recognized and appreciated the following 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 as a cQED (circuit quantum electrodynamics) system comprising a pair of coupled resonators. In this dual-rail encoding, an excitation is stored in one of the two resonators. The excitation of the first resonator is considered a logical 0, while the excitation of the other resonator is considered a logical 1. Thus, the two resonators together form a single logical dual-rail qubit, where the logical state is |0> L =|01> and |1> L =|10>. This type of logical qubit has several benefits, including the following. First, photon loss manifests as an erasure error, which, as mentioned above, is one of the easier error types to correct. Second, the single-photon state is the lowest energy state of the cavity and therefore has the lowest error rate of any cavity state. Therefore, such dual-rail encoding minimizes the loss error rate. Third, by measuring the joint parity of the cavity, photon gain or loss can be easily detected.

[0036] According to some embodiments, a dual-rail qubit can be implemented as a cQED system comprising two quantum resonators (e.g., microwave cavity resonators) coupled together via a suitable nonlinear coupling element (e.g., a superconducting transmon qubit or a superconducting nonlinear asymmetric inductor element (SNAIL) or a superconducting quantum interference device (SQUID)). One of the quantum oscillators can be coupled to an auxiliary qubit (e.g., a superconducting transmon qubit). Although the auxiliary qubit is only coupled to one of the bosonic modes of the system, due to the beam splitter interaction provided by the coupling element (described below), both bosonic modes interact with the auxiliary qubit, thereby enabling various dual-rail operations. For example, by applying energy (e.g., a microwave pulse) to the auxiliary qubit and / or to the coupling element, a dual-rail state can be prepared, and a gate can be performed on the dual-rail state, as further described below.

[0037] According to some embodiments, the logical quantum state of a dual-rail qubit is prepared by preparing a single excitation of one of the two quantum oscillators of the dual-rail qubit. For example, to prepare |0> L state (logical 0 state), the two oscillators can be prepared in their respective ground states |0>, and the first oscillator is excited to its |1> state while the second oscillator remains in its |0> state. Similarly, to prepare |1> L To achieve a state (logical 1 state), two oscillators can be prepared into their respective ground states |0>, and the second oscillator is excited to its |1> state while the first oscillator remains in its |0> state. For example, in the case of a cQED dual-rail qubit comprising two quantum resonators, one of the resonators can be prepared into the |1> state by loading a single photon into the resonator.

[0038] According to some embodiments, the logic state of a dual-rail qubit can be measured by one or more parity measurements (e.g., by performing one or more quantum non-destructive (QND) measurements) on one or both of the two quantum oscillators. Because the dual-rail logic state has an odd joint parity, a change in parity indicates an error, either an erasure error that produces the joint state |00> or an excitation to one of the states |11>, |20>, or |02>. When the dual-rail qubit is implemented as a cQED system comprising two quantum resonators, parity measurements can be performed by applying one or more drives to an auxiliary qubit coupled to one of the resonators and / or to a nonlinear coupling element coupling the resonators to each other. Examples of such parity measurements are described below.

[0039] According to some embodiments, multiple dual-rail qubits can be arranged in a repeating arrangement (e.g., an array or lattice). Such an arrangement can provide improved response to errors by treating the multiple dual-rail qubits as a single logical group. For example, multiple dual-rail qubits can be coupled and operated in a surface code architecture, where nominally identical dual-rail qubits function as data qubits or measurement qubits on which stabler measurements are performed as an error correction process.

[0040] Below is a more detailed description of various concepts and embodiments related to the implementation and operation of dual-rail qubits. It should be understood that the various aspects described herein can be implemented in any of a variety of ways. Examples of specific implementations are provided herein for illustrative purposes only. In addition, the various aspects described in the following embodiments can be used alone or in any combination and are not limited to the combinations explicitly described herein.

[0041] Figure 1 An illustrative system suitable for practicing the techniques described herein, according to some embodiments, is shown in FIG. In system 100, oscillators 101 and 102 are coupled to each other via coupling element 103. Oscillator 101 is also coupled to an auxiliary qubit 104. An energy source 105 can be operated by a controller 106 to direct energy to the auxiliary qubit 104, coupling element 103, and / or a readout resonator 107.

[0042] According to some embodiments, oscillator 101 and oscillator 102 each include a cavity that supports quantum states of microwave photons. For example, in some embodiments, first oscillator 101 and second oscillator 102 can each include a resonator, such as a transmission line resonator or a three-dimensional cavity formed from a superconducting material (e.g., aluminum).

[0043] 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, thereby allowing interaction between first oscillator 101 and oscillator 102. In some embodiments, coupling element 103 may be a superconducting nonlinear asymmetric inductor 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 superconducting transport qubit that is dispersively coupled to both first oscillator 101 and second oscillator 102.

[0044] In some embodiments, ancillary qubit 104 may be a superconducting transporter qubit, a SNAIL, a SQUID, or some other nonlinear element.

[0045] exist Figure 2AIn FIG, an illustrative implementation of system 100 is shown as a dual-rail qubit 200. In this implementation, the oscillators are implemented as bosonic modes stored in cavities 201 and 202 (e.g., microwave cavities). Figure 2A 204 (e.g., a superconducting transport qubit), coupling element 203, and / or readout resonator 207. Such a microwave source can be coupled to the auxiliary qubit and coupling element. The coupling between the microwave source and these components provides a way for the microwave source to apply microwave radiation to the components.

[0046] In some embodiments, energy source 105 can be capacitively coupled to each of auxiliary qubit 204, coupling element 203, and readout resonator 207. In some embodiments, Figure 2A The multiple instances of the dual-rail qubit shown in can be arranged to be coupled with one or more instances of an energy source 105 and with one or more controllers 106, the one or more controllers being configured to perform operations on the multiple dual-rail qubits.

[0047] exist Figure 2A In an example of , a microwave source (not shown) can be operated to read out the state of the auxiliary qubit 204. For example, the readout resonator 207 can be arranged so that its resonant frequency (e.g., ~ GHz) is away from the transition frequency of the auxiliary qubit 204 (e.g., dispersive coupling). The coupling between the auxiliary qubit and the readout resonator means that there is an offset in the resonator frequency that depends on the state of the qubit. This offset is small compared to the resonant frequency of the resonator (e.g., ~ MHz). Therefore, sending a tone near the resonant frequency to the readout resonator 207 will be reflected by the resonator, and the form of the reflected tone (also referred to herein as the "readout signal") can be analyzed to determine the state of the auxiliary qubit. In this way, the state of the auxiliary qubit 204 can be non-destructively probed by sending a probe tone to the readout resonator 207, which is dispersively coupled to the qubit.

[0048] The dual-rail qubit 200 has a logical code word encoded in the single-photon subspace of the qubit, where |0> L =|01> and |1> L =|10>, for this logical code word, the beam splitter interaction is sufficient to promote arbitrary single qubit rotations. These states are Figure 2Cdenoted as the "logical subspace" in [1]. By encoding the logical state of the qubit in a single-photon subspace of the joint cavity Hilbert space (with odd joint photon number parity), the dominant error in the cavity system (i.e., the loss of a single photon to the common ground state |00>) can be converted into a detectable erasure error by means of the joint parity measurement. Furthermore, photon gain events in the cavity, which lead to the states |02>, |11>, or |20> and are typically rare in thermal equilibrium, can also be detected by the joint parity measurement. This makes cavity dephasing the dominant Pauli error in the system.

[0049] exist Figures 2A to 2B In the example of , the Hamiltonian of the dual-rail qubit 200 combines the beam splitter interaction between cavities 201 and 202 and the dispersion interaction between the auxiliary qubit 204 and the mode of cavity 201. This Hamiltonian can be written as:

[0050]

[0051] in

[0052]

[0053] and is a Pauli Z-operator in a two-level subspace defined by the auxiliary |g> and |f> levels. In this example, a three-level auxiliary qubit is considered, which may have the benefit of allowing the use of the |e> level for detecting a single auxiliary decay event. However, in some implementations described below, a two-level auxiliary qubit may be implemented.

[0054] exist Figures 2A to 2B In the example, the item represents the beam splitter coupling between cavities 201 and 202, which is generated by coupling element 203, and the term represents the dispersion coupling between cavity 201 and auxiliary qubit 204. These couplings are Figure 2B In the Hamiltonian , annihilation operator and The Bosonic modes acting on cavities 201 and 202, g BS (t) is the complex amplitude of the beam splitter interaction between cavities 201 and 202, Δ(t) is the effective detuning between the modes of the two cavities, and χ is the mode of the auxiliary qubit 204 (in the gf manifold) with cavity 201 The Hamiltonian Written in the dispersion interaction symmetry system, according to the auxiliary state The frequency shift is ±χ / 2.

[0055] The Hamiltonian can be controlled and varied by appropriately selecting the microwave drive signals applied to the coupling element 203 and the auxiliary qubit 204. For example, the coupling strength g BS (t), its phase and detuning Δ(t) can all be rapidly changed via microwave drive technology. As described below, in this system, these operations can be designed by actuating the microwave drive while designing these parameters, where different values ​​of the parameters correspond to different operations.

[0056] According to some embodiments, the coupling element 203 may include a nonlinear asymmetric inductor element (SNAIL), which is a superconducting circuit including a loop with a tunneling energy E in one arm of the loop. J multiple (e.g., three) Josephson junctions, and having a tunneling energy less than E on the other arm of the loop J A single Josephson junction is formed, and two large capacitor pads are connected to the two arms of the loop. In this approach, a SNAIL is placed between cavities 201 and 202 to generate capacitive coupling between the SNAIL mode and each of the cavity modes. Magnetic flux is also passed through the loop of the SNAIL to bias the SNAIL to an operating point where it has third-order nonlinearity (e.g., using a superconducting flux transformer). The SNAIL is described in U.S. Patent No. 11,737,376, which is incorporated herein in its entirety.

[0057] According to some embodiments, performing operations on the dual-rail qubit 200 may include pumping the coupling element to enable so-called "beam splitter" operations. In particular, the nonlinearity of the coupling element may enable three-wave mixing or four-wave mixing, which is used to perform frequency-converting bilinear coupling between the first cavity and the second cavity. For example, when the coupling element is a SNAIL, the beam splitter operation may include three-wave mixing, while when the coupling element is a SQUID or a superconducting transport, the beam splitter operation may include four-wave mixing.

[0058] According to some embodiments, the frequency matching condition ω1-ω2=ω can be satisfied by applying the frequency matching condition ω1-ω2=ω p2 -ω p1 (where ω1 is the resonant frequency of the first cavity, ω2 is the resonant frequency of the second cavity, ω p1 is the frequency of the first pump associated with mode c, and ω p2The beam splitter operation is performed by microwave drive (where ω is the frequency of the second pump associated with mode d). This frequency matching condition is relevant to the four-wave mixing case (for example, where the coupling element is a SQUID or a superconducting transport). In the case where the coupling element is a SNAIL, the frequency matching condition is ω1 = ω2 = ω p That is, when the coupling element 203 is a SNAIL, a driving frequency equal to the difference between the resonant frequencies of the cavities 201 and 202 can be applied to the SNAIL. p Can be relative to the frequency ω p =ω1-ω2-Δdetuning. When Δ=0, the drive implements an operation, hereinafter referred to as a "resonant beam splitter operation," which can be applied to exchange photons between two cavities. When Δ≠0, the drive implements an operation, hereinafter referred to as a "detuned beam splitter operation," which can be applied to perform an entanglement gate between two-rail qubits as described below. The amplitude and phase of the microwave drive can be selected to select the strength and phase of the beam splitter operation, respectively.

[0059] Beam splitter drive g BS The length of time during which (t) is applied to the coupling element 203 determines the specific operation applied to the two cavities 201 and 202. Specifically, the integral of the beam splitter drive during time period t1 is written as:

[0060]

[0061] This allows for parameterization of the type of beamsplitter operation. When θ = π and Δ = 0, the beamsplitter unitary operator implements a SWAP operation that exchanges states between the cavity modes of cavities 201 and 202. When Δ = 0 and θ = ±π / 2, the operation resulting from the resonant beamsplitter operation corresponds to a 50 / 50 beamsplitter. Such operations can be referred to as "beamsplitter" operations because they can be viewed as playing the same role and implementing the same unitary operation as beamsplitters do in linear optical quantum computing (LOQC) schemes, although here the beamsplitter operation is implemented by pumping a nonlinear coupling element that couples the two cavities together—a completely different configuration and process than an optical beamsplitter. As described below, beamsplitter operations can be performed as part of various operations on dual-rail qubits, such as photon number parity measurements, single-qubit gates, entanglement gates between two dual-rail qubits, and so on.

[0062] According to some embodiments, the state preparation of a dual-rail qubit may include loading a single photon into one of two cavities, where the cavity selected depends on the state of the qubit being prepared. L Still|1> L Dual-rail qubit logic states (e.g., single photons can be loaded into cavity 201 to prepare |0> Lstate, or a single photon can be loaded into the cavity 202 to prepare |1> L In some embodiments, single photon loading can be performed via optimally controlled pulses or via cavity-assisted sideband driving.

[0063] The logical state of the dual-track qubit can be determined by determining whether a single photon is in cavity 201 (state |0> L ) or in cavity 202 (state |1> L Alternatively, if it is determined that the photon is not in either cavity, this indicates a leakage event due to photon loss.

[0064] According to some embodiments, the logical state of a dual-rail qubit can be determined by measuring the parity of the two cavities, as described below. In some cases, the parity measurement can be performed after state preparation to verify that the state of the dual-rail qubit was prepared as expected. 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 measurement are checked to determine whether the state of the dual-rail qubit was prepared as expected.

[0065] According to some embodiments, the parity measurement of a dual-rail qubit is Figure 3A One illustrative way to perform a parity measurement of cavity 201 is to place auxiliary qubit 204 in its ground state |g> and drive the auxiliary qubit with a π / 2 pulse to prepare the auxiliary qubit into the state |g> + |e> (omitting the normalization factor). After the π / 2 pulse, auxiliary qubit 204 is in the time period π / χ ge The inner is not driven, where χ ge is the strength of the dispersion interaction between the auxiliary qubit 204 and the cavity 201 in the ge manifold. During this time, the auxiliary qubit 204 and the cavity 201 interact through their dispersion to interact. After this waiting period in which the auxiliary qubit is not driven, the auxiliary qubit is then driven with another π / 2 pulse. After this pulse, if there is an odd number of photons in cavity 201, auxiliary qubit 204 will be in its excited state |e>, and if there is an even number of photons in cavity 201, auxiliary qubit 204 will be in its ground state |g>. Therefore, measuring the state of the auxiliary qubit at this stage produces a measurement result that indicates the parity of the photon number in cavity 201.

[0066] 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 can be swapped again by another beam splitter operation to return the states to their original configuration. For example, a sequence of parity measurements can include: a) driving auxiliary 204 with a π / 2 pulse, waiting for a period of π / χ, and then ge , driving auxiliary 204 with a π / 2 pulse, and measuring the state of auxiliary qubit 204 to measure the parity of cavity 201; b) driving coupling element 203 to perform a beam splitter operation that exchanges the states of cavities 201 and 202; c) a) driving auxiliary 204 with a π / 2 pulse, waiting for a period of time π / χ ge , driving auxiliary 204 with a π / 2 pulse, and measuring the state of auxiliary qubit 204 to measure the parity of cavity 201; and d) driving coupling element 203 to perform a beam splitter operation that exchanges the states of cavities 201 and 202.

[0067] If the states of both the auxiliary qubits measured in actions (a) and (c) are |g>, this indicates an erasure error. Alternatively, when a single photon is in cavity 201 (state |0> L ), the auxiliary qubit state should be measured to |e> and then |g> in actions (a) and (c), respectively. Alternatively, when a single photon is in cavity 202 (state |1> L ), the auxiliary qubit state should be measured to |e> and then |g> in actions (a) and (c), respectively.

[0068] Actions (a), (b), (c) and (d) may be repeated one or more times in this order (in Figure 3A ) to perform repeated parity measurements and determine the dual-rail qubit logic state based on a collection of these measurements (e.g., by employing a majority rule outcome).

[0069] exist Figure 3B The different measurement outcomes of the parity measurement sequence described above are summarized in Figure 2. The table shows the input probabilities for each state, where p is the probability of a photon loss event before the measurement, which would leave both cavities in the state |0>. Depending on the input state, the result of the parity measurement is either correctly assigned to the actual input state (check mark), incorrectly labeled as an erasure (ε), or incorrectly assigned to a logical state (cross mark), resulting in a Pauli error.

[0070] Even in the presence of decoherence, L Mistakenly believed|1> LThe probability of a fault (or vice versa) should be very small. First-order errors in the superconducting transport and the cavity give measurements that can be labeled as additive erasure errors. A combination of at least two errors is required to obtain a logical error assignment, and for realistic coherence times and superconducting transport readout fidelity, the probability is estimated to be less than 10 -4 .

[0071] According to some embodiments, a single-qubit gate for dual-rail qubit 200 can be performed by operating with a resonant beam splitter, as described above. To perform a single-qubit gate, coupling element 203 is driven at a frequency equal to the difference between the resonant frequencies of cavities 201 and 202, where the beam splitter drive amplitude, phase, and duration control the type of single-qubit gate. In some cases, for example, the intensity and duration of the beam splitter drive control the polar angle of rotation about the Bloch sphere, while the phase of the beam splitter drive controls the rotation about the Z-axis of the Bloch sphere.

[0072] A key advantage of the dual-rail qubits described herein is the ability to perform QND detection for leakage outside the logical subspace caused by the loss or gain of photons in one or both cavities. For the QEC protocol, knowledge of the exact location of the error enables conversion of these otherwise harmful leakage events into erasure errors, which are easier to correct than Pauli errors. This leakage detection scheme may be referred to herein as "erasure checking" or "erasure detection."

[0073] According to some embodiments, the erasure check comprises measuring the joint photon number parity of cavities 201 and 202 in the dual-rail qubit 200. Since the two logical states |0> L or |1> L With an odd joint parity (ie, there is one photon in each cavity combined in each case), the addition or loss of a photon in either cavity changes the joint parity to even.

[0074] Measuring the joint photon number parity of cavities 201 and 202 in dual-rail qubit 200 may include applying a first on-resonant pulse to auxiliary qubit 204, then applying a beam splitter operation to coupling element 203, and then applying a second on-resonant pulse to auxiliary qubit 204. This sequence of operations may enable photon number information to be mapped onto the auxiliary, and then measured.

[0075] In some embodiments, the auxiliary qubit 204 can be a three-level auxiliary having a ground state |g>, a first excited state |e>, and a second excited state |f>. In this case, two of these states can be used to detect erasure errors caused by the decay of the auxiliary during the joint parity measurement and erasure due to photon loss. For example, after this sequence of operations, the resulting state of the auxiliary can be measured as: |g> when the joint parity is odd; |f> when the joint parity is even; and |e> when the state of the auxiliary decays during the sequence of operations. Since the cavity of the dual-rail qubit should have an odd joint parity, the auxiliary states |f> and |e> both indicate an erasure error, and |g> indicates a successful measurement of an odd photon number parity.

[0076] According to some embodiments, measuring the joint photon number parity of cavities 201 and 202 in dual-rail qubit 200 may include driving auxiliary qubit 204 to perform within the gf manifold of the auxiliary qubit Rotate (operation 411); then later, perform another Rotate (operation 413). In the period between the two operations performed on the auxiliary qubit, the dispersion interaction between the auxiliary qubit and the cavity 201 maps the photon number information to the auxiliary state. During this period, a beam splitter operation 412 is also applied to exchange the states of the cavities so that the photon number information of the two cavities is mapped to the auxiliary qubit. As an example, the beam splitter drive strength can be where the frequency is detuned relative to resonance by Δ=χ / 2, which enables the joint photon number parity information to be mapped to the state of the auxiliary qubit, while enabling the cavities to return to their initial state at the end of the sequence (up to a deterministic cavity phase shift that can be tracked).

[0077] exist Figure 4 An illustrative process for implementing a dual-rail qubit with single-qubit gates and erasure checking, according to some embodiments, is depicted in FIG. Figure 4 In the example above, the beam splitter can be used to drive g bs The coupling element 203 of the depicted dual-rail qubit 200 is driven to perform one or more single-qubit gates 402. After performing the one or more single-qubit gates, an erasure check 404 may be performed by the joint photon number parity sequence described above and shown in inset 405.

[0078] In various cases, erasure errors can be detected by mapping the joint photon number parity to the auxiliary. First, idler errors (primarily photon losses) of dual-rail qubits can be converted into erasure errors. Second, errors generated by state preparation can be converted into erasure errors. Third, errors generated during single-qubit gates can be detected by mapping the joint parity to the auxiliary and measuring the state of the auxiliary after executing the single-qubit gate.

[0079] exist Figures 5A to 5B An alternative approach for erasure detection according to some embodiments is depicted in . Figures 5A to 5B The drive scheme shown in the example is Figure 4 The difference between the drive schemes is: Figures 5A to 5B The proposed scheme only detects whether the system is in a zero-photon state (i.e., where both cavities are in |0>) and involves driving ancillary qubit 204 and coupling element 203 simultaneously. Furthermore, ancillary qubit 204 is driven only once. In this scheme, ancillary qubit 204 is excited to the |e> state only when both cavities are in the |0> state, while the cavity states remain unchanged.

[0080] exist Figures 5A to 5B In the example, g bs applied to the coupling element 203, where the intensity g bs >χ ge , the frequency is detuned relative to the resonance condition Δ = χ ge / 2 (as described above). ge is the strength of the dispersive interaction between auxiliary qubits 204 in the g-e manifold. Simultaneously with this drive, another drive with phase π is applied to auxiliary qubit 204 at a frequency that is the difference between the |e> state and the |g> state of the auxiliary qubit when the cavity contains no photons. Therefore, the beam splitter drive becomes resonant only when no photons are present in the cavity. Figure 5A and Figure 5B The two methods shown for the erase detection scheme differ in the duration of the pulse and the pulse shape of the auxiliary drive. Figure 5A In the example above, the auxiliary qubit is driven by approximately 2π / χ ge time period, and in Figure 5B In the example above, the auxiliary qubit is driven by approximately 4π / χ ge time period.

[0081] against Figures 5A to 5B One potential advantage of the erasure scheme is that since the detection result is based only on the |g> state and |e> state of the auxiliary qubit, three levels of auxiliary qubits may not be required in the dual-track qubit 200. This is different from Figure 4The method shown in where a three-level auxiliary qubit is used to distinguish between different outcomes.

[0082] Figure 6 Depicts the application of some embodiments Figures 5A to 5B One aspect of the time evolution of a two-track qubit during actuation is shown in . Figure 6 In the example of , the time period during which the auxiliary qubit drive is applied is shown along the horizontal axis, and the probability of measuring the auxiliary qubit to be in |e> is shown along the vertical axis. Each of the four curves represents a different state of the two-rail qubit, where the cavity state is written as a two-digit state, where |11> means both cavities are in the |1> state, etc. Figure 6 It can be noted that the dual-track state and The state |11> is associated with a constant probability when the auxiliary drive ends at 601. However, the state |00> that the scheme is designed to detect is associated with a probability equal to or close to 1 when the auxiliary drive ends at 601.

[0083] It is also possible to construct entanglement gates between the dual-rail qubits so that both cavity and superconducting transport errors can also be detected and converted to erasure, as will be described below. The inventors have recognized that to realize the logical Z operator for dual-rail, interaction with only a single cavity of the dual-rail qubit is necessary. By driving a beamsplitter interaction between a pair of cavities (one from each of the two dual-rail qubits), a single superconducting transport can effectively act as a control for both dual-rail qubits. Figure 7A A suitable system for performing such operations according to some embodiments is shown in .

[0084] exist Figure 7A In the example of FIG, a pair of dual-rail logical qubits 701 and 702 are depicted as coupled to each other via coupling element 703. Dual-rail qubit 701 includes cavities 711 and 712 (e.g., microwave cavities) coupled together via coupling element 713; and 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 corresponding ancillary qubit 714 or 724 (e.g., each of which can be a superconducting transport qubit), which is coupled to a corresponding readout resonator. As with coupling element 203 in dual-rail qubit 200, each of coupling elements 703, 713, and 723 can be a superconducting nonlinear asymmetric inductor element (SNAIL), a superconducting quantum interference device (SQUID), or some other nonlinear element.

[0085] This can be achieved by directing energy to the coupling element 703 between the dual-rail qubits (rather than, for example, Figure 2A A two-qubit gate as described above is performed on two dual-rail logical qubits using a coupling element 203 between the two logical qubits implemented by cavities 201 and 202 in the example of FIG. 2 . When a two-qubit gate is performed on a pair of dual-rail qubits, an auxiliary qubit 724 coupled to cavity 722, which is coupled to cavity 711 of another dual-rail qubit via coupling element 703, can be operated as an auxiliary qubit in the two-qubit gate scheme. Therefore, a two-qubit operation can be applied to the auxiliary qubit 724, and any errors occurring during the execution of the two-qubit gate can be detected by measuring the state of the auxiliary qubit 724 and determining whether the auxiliary qubit is in the state |g>, |e>, or |f>. In Figure 7B It is further shown in Figure 7A The coupling depicted in the figure indicates the mode including a logical qubit 701, and Includes logical quantum bits 702.

[0086] A two-qubit entanglement gate is a ZZ(θ) gate, which can be written as

[0087]

[0088] It operates on a two-qubit logical subspace. When θ = π / 2, this gate is locally equivalent to a CNOT or CZ gate. A ZZ(θ) gate operating on the subspace {|00>, |01>, |10>, |11>} (as if each cavity were encoded according to the Fock {|0>, |1>} code) will also implement ZZ(θ) for the dual-rail encoding, provided that each of these cavities belongs to a different dual-rail qubit.

[0089] One motivation for implementing the ZZ(θ) gate is that it can be performed using exactly the same hardware required for the erasure check discussed in the previous section. Similar to error detection for erasure checking, it is important to detect auxiliary qubit errors that occur when executing the gate. 7A to 7B The arrangement enables detection of first-order superconducting transport errors when measurements are made on the superconducting transport at the end of the gate, erasing both dual-rail qubits if the superconducting transport is not in |g>. Again, this operation is designed so that Pauli errors are introduced only from second-order hardware errors when two decoherence events occur during a single gate. Figure 8 This method according to some embodiments is depicted in .

[0090] exist Figure 8In the example of FIG, dual-rail qubits 701 and 702 are depicted, and a ZZ(θ) gate can be performed on these dual-rail qubits by the sequence of operations shown in illustration 801. The circuit depicted in illustration 801 allows detection of a single auxiliary dephasing error, in addition to an auxiliary decay event, during the ZZ(θ) gate by measuring the state of the auxiliary qubit in operation 816. The state of the auxiliary qubit 724 serves as a flag to indicate whether the gate represented by operations 811, 812, 813, 814, and 815 was performed without an auxiliary dephasing or auxiliary decay error. In particular, if the state of the auxiliary qubit 724 after performing operations 811, 812, 813, 814, and 815 is the ground state |g>, this indicates that no such error occurred. Otherwise, if the state of the auxiliary qubit is the first excited state |e> or the second excited state |f>, this indicates that at least one such error occurred while performing operations 811, 812, 813, 814, and 815.

[0091] Operation 811 is within the gf manifold of auxiliary qubit 724 Rotation, operation 815 is within the gf manifold of auxiliary qubit 724 Rotation, and operation 813 is an X(θ) rotation within the gf manifold of the auxiliary qubit 724, which parameterizes the ZZ(θ) gate (for example, when θ=π / 2, this is locally equivalent to a CNOT or CZ gate). In operations 812 and 814, as in operation 412, a beam splitter operation is applied to the coupling element 703 to exchange the states of the cavities 711 and 722, so that the photon number information of the two cavities is mapped to the auxiliary qubit 724.

[0092] Photon losses during a two-qubit gate can be detected via separate erasure checks on the two dual-rail qubits after each gate, but the pulse sequence of the ZZ(θ) gate can also be modified to perform an erasure check simultaneously, where the superconducting transporter can be mapped to the |f> level if one of the input dual-rail qubits is in a leakage state outside the logical subspace.

[0093] exist Figure 9A and Figure 9B Different approaches for a specific type of entanglement gate, the CPHASE(θ) gate, are described in

[15] . Figures 9A to 9BThe driving scheme shown in the example of [ 1 ] involves simultaneously driving auxiliary qubit 724 and coupling element 703, with two distinct pulses applied to auxiliary qubit 724 and a single drive applied to coupling element 703. In each case, two back-to-back π pulses are applied to auxiliary qubit 724. If both cavities 711 and 722 begin in the |0> state, the π pulses produce a phase shift on auxiliary qubit 724, and by controlling the phase of the second π pulse, a CPHASE(θ) gate is executed between dual-rail qubits 701 and 702. In particular, the phase of the second π pulse is parameterized by θ = π + φ.

[0094] Figure 9A and Figure 9B The difference between the examples is that Figure 9A depicts a method in which the auxiliary qubit is a two-level qubit, and Figure 9B A method in which the auxiliary qubit is a three-level qubit is described. The two methods are described below.

[0095] exist Figure 9A In the example of an auxiliary qubit error, if auxiliary dephasing occurs during the CPHASE(θ) gate, the auxiliary qubit will be measured as being at |e> rather than |g> at the end of the gate. Alternatively, if auxiliary qubit decay occurs, the auxiliary qubit will be measured as being at |e> 50% of the time. In either case, an auxiliary qubit error can be detected after executing the gate by subsequently measuring the state of the auxiliary qubit and determining with certainty that an error occurred when the auxiliary was measured as being at |e>.

[0096] exist Figure 9A In the example, g bs applied to the coupling element 703, where the intensity g bs >χ ge , the frequency is detuned relative to the resonance condition Δ = χ ge / 2 (as described above). ge is the strength of the dispersion interaction between the auxiliary qubit 724 and the cavity 722 in the g-e manifold. Simultaneously with this drive, two back-to-back drives are applied to the auxiliary qubit 724 at a frequency that is the difference between the |e> state and the |g> state of the auxiliary qubit when the cavity contains no photons. The first of these two drives has zero phase, and the second drive has a phase θ = π + φ relative to the first drive. Figure 9A In the first (square pulse) example, the two auxiliary qubit drives are each applied to approximately 2π / χ ge time period, and in Figure 9A In the second (Gaussian pulse) example, the two auxiliary qubit drives are each applied with a value of approximately 4π / χ geIn some embodiments, the two back-to-back auxiliary qubit drives applied during a CPHASE gate are applied for equal amounts of time.

[0097] exist Figure 9B In the example of , the auxiliary qubit decay can be detected by using a three-level auxiliary qubit. In particular, it can be detected by applying Figure 9B An auxiliary qubit error is detected by measuring the state of the auxiliary qubit after one of the drive sequences shown in FIG. 1 and positively determining that an error occurred when the auxiliary qubit is measured to be in the state of |e> or |f>. Specifically, measuring the auxiliary qubit in the state of |e> indicates that the CPHASE gate failed due to decay of the auxiliary qubit state, while measuring the auxiliary qubit in the state of |f> indicates that the CPHASE gate failed due to dephasing of the auxiliary qubit. Measuring the auxiliary qubit in the state of |g> indicates that the CPHASE gate was successfully executed.

[0098] exist Figure 9B In the example, g bs applied to the coupling element 703, where the intensity g bs >χ gf , the frequency is detuned relative to the resonance condition Δ = χ gf / 2 (as described above). gf is the strength of the dispersion interaction between the auxiliary qubit 724 and the cavity 722 in the gf manifold. Simultaneously with this drive, two back-to-back drives are applied to the auxiliary qubit 724 at a frequency that is the difference between the |g> state and the |f> state of the auxiliary qubit when the cavity contains no photons. The first of these two drives has zero phase, and the second drive has a phase θ = π + φ relative to the first drive. Figure 9B In the first (square pulse) example, the two auxiliary qubit drives are each applied to approximately 2π / χ ge time period, and in Figure 9B In the second (Gaussian pulse) example, the two auxiliary qubit drives are each applied to approximately 4π / χ ge In some embodiments, the two back-to-back auxiliary qubit drives applied during a CPHASE gate are applied for equal amounts of time.

[0099] The dual-rail qubits and techniques described above 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.

[0100] In some embodiments, multiple dual-rail qubits may be arranged in an array, network, grid, lattice, or other repeating configuration. The dual-rail qubits may be arranged in such a manner that connectivity between the dual-rail qubits is provided by one or more types of structures, which may include other dual-rail qubits and / or other structures such as cavities. In some embodiments, a "unit cell" may be repeated multiple times to create a lattice arrangement. A suitable unit cell may, for example, include: a dual-rail qubit (including two superconducting cavities, an auxiliary qubit (e.g., a superconducting transporter), and a coupling element (e.g., a SNAIL) between the two cavities) and two additional coupling elements for docking with adjacent unit cells.

[0101] exist Figure 10 An illustrative example of a two-rail qubit network is shown in . Figure 10 In Figure 1, dual-rail qubits (two examples of which are 1001) are represented as a pair of coupled cavities (circles with connecting lines) surrounded by rectangular boxes. The dual-rail qubits are coupled to each other via measurement qubits (one example of which is 1002), represented by solid circles. Each measurement qubit includes a single cavity coupled to an auxiliary qubit in addition to a readout cavity. The line between the measurement qubit and the dual-rail qubit represents a coupling element (e.g., a SNAIL).

[0102] In some embodiments, a network of dual-rail qubits can be manipulated so that multiple dual-rail qubits operate as a single logical qubit. For example, multiple dual-rail qubits can be entangled together using a suitable sequence of quantum gates, where subsequent measurement of the entangled state provides a means for error correction and error detection. Thus, using a collection of physical dual-rail qubits entangled in this manner to define a logical qubit can be expected to have better performance than a single dual-rail qubit.

[0103] One approach to entangle dual-rail qubits in this way is to use so-called "surface codes," in which a set of dual-rail qubits is selected as either data qubits, which store computational quantum states, or measurement qubits, which are used to stabilize and manipulate the quantum states of the data qubits. The stable subcodes used in surface codes are described, for example, in "Surface codes: Toward spractical large-scale quantum computation" (Austin Fowler et al., Phys. Rev. A, Vol. 86, 032324 (2012)), which is incorporated herein by reference in its entirety.

[0104] exist Figure 11An example of a physical arrangement of dual-rail qubits that can be grouped using surface codes (or any other suitable code) and operated as logical qubits according to some embodiments is depicted in FIG. Figure 11 In the example, 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.

[0105] One challenge with this type of arrangement may be that, while it is scalable, it is not modular, which may make it difficult to manufacture simply. Additionally, this type of arrangement may lack flexibility in the spatial arrangement of the two-rail qubits (for example, it may require the two-rail qubits to be aligned in a flat plane).

[0106] exist Figure 12A The diagram depicts a comparison according to some embodiments. Figure 11 A more modular and extensible approach is shown in . Figure 12A In the example above, each dual-rail qubit data qubit is coupled to four dual-rail qubit measurement qubits (and vice versa), but the dual-rail qubits are arranged in modules with the boundaries of the modules at Figure 12A The coupling between adjacent cavities within a module (intra-module links) is achieved via nonlinear coupling elements as described above, while the coupling between cavities across module boundaries (inter-module links) is instead achieved via a quantum bus, an example of which is described below.

[0107] Thus, a plurality of nominally identical dual-rail qubit modules can be manufactured, and the plurality of dual-rail qubit modules include a plurality of couplings (and / or coupling ports) coupling the cavities of the modules to an external quantum bus. Figure 12A In the example of , each such module would include nine dual-rail qubit data qubits and four dual-rail qubit measurement qubits, in addition to twelve cavities each arranged to be coupled to a quantum bus. It can be noted that a cavity 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 via a quantum bus (which acts as coupling element 103).

[0108] It can also be noted that this approach produces "quad-rail" qubits at the corners between the four modules, where the four cavities are coupled end-to-end into a loop via four quantum buses (in addition, each cavity is coupled to the corresponding cavity of the data qubit via a nonlinear coupling element).

[0109] Figure 12BDepicted in accordance with some embodiments Figure 12A An illustrative quantum bus can serve as a link between modules in the example of Figure 12B In an example, a quantum bus is implemented as a standing wave transmission mechanism. For example, the quantum bus can include a microwave coaxial cable, a flexible cable, and / or other types of photonic microwave links that include a bus mode for performing a SWAP operation and / or a 50-50 beam splitter operation between cavities coupled to opposite ends of the bus.

[0110] Having thus described several aspects of at least one embodiment of this invention, it is to be appreciated various alterations, modifications, and improvements will readily occur to those skilled in the art.

[0111] 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. Furthermore, while the advantages of the invention are noted, it should be understood that not every embodiment of the technology described herein will include every described advantage. Some embodiments may not implement any of the features described herein as advantageous, and in some cases, may implement one or more of the described features to achieve further embodiments. Therefore, the foregoing description and accompanying drawings are intended to be examples only.

[0112] Aspects of the present disclosure may include, but are not limited to:

[0113] Aspect 1. A dual-rail quantum bit, comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an auxiliary quantum bit coupled to the first quantum oscillator; at least one energy source; and at least one controller, the at least one controller being configured to operate the at least one energy source to initialize the dual-rail quantum bit to a 0 or 1 logic state by: when the dual-rail quantum bit is to be initialized to a 0 logic state, operating the at least one energy source to arrange the first quantum oscillator into a single-photon state and the second quantum oscillator into its ground state; or when the dual-rail quantum bit is to be initialized to a 1 logic state, operating the at least one energy source to arrange the first quantum oscillator into its ground state and the second quantum oscillator into a single-photon state.

[0114] Aspect 2. A dual-rail quantum bit according to Aspect 1, wherein arranging the first quantum oscillator or the second quantum oscillator into a single-photon state includes: operating the at least one energy source to perform multiple optimal control pulses and / or cavity-assisted sideband driving.

[0115] Aspect 3. The dual-rail quantum bit according to aspect 1, wherein arranging the first quantum oscillator into the single-photon state comprises: operating the at least one energy source to arrange the first quantum oscillator into a |1> Fock state.

[0116] Aspect 4. The dual-rail qubit according to aspect 1, wherein: the coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and the auxiliary quantum bit is dispersively coupled to the first quantum oscillator.

[0117] Aspect 5. The dual-rail qubit according to Aspect 1, wherein the coupling element is a superconducting transport qubit or a superconducting nonlinear asymmetric inductor element (SNAIL).

[0118] Aspect 6. The dual-rail qubit according to aspect 1, wherein the auxiliary qubit is not coupled to the second quantum oscillator.

[0119] Aspect 7. The dual-rail qubit according to Aspect 1, wherein the auxiliary qubit is a superconducting transporter qubit.

[0120] Aspect 8. The dual-rail qubit of aspect 1, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

[0121] Aspect 9. A dual-rail quantum bit, comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an auxiliary quantum bit coupled to the first quantum oscillator; at least one energy source; and at least one controller configured to operate the at least one energy source to perform a single-qubit rotation of a logical state of the dual-rail quantum bit by operating the at least one energy source one or more times to direct energy to the coupling element.

[0122] 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.

[0123] 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 the operation of the at least one energy source based on a rotation angle of the single qubit rotation.

[0124] Aspect 12. The dual-rail qubit of aspect 11, wherein the amplitude and duration are selected based on the polar angle of the single qubit rotation, and wherein the phase is selected based on the angle of rotation about the Z axis of the Bloch sphere in the single qubit rotation.

[0125] 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 auxiliary quantum bit is dispersively coupled to the first quantum oscillator.

[0126] Aspect 14. The dual-rail qubit according to aspect 9, wherein the coupling element is a superconducting transport qubit or a superconducting nonlinear asymmetric inductor element (SNAIL).

[0127] Aspect 15. The dual-rail qubit of aspect 9, wherein the auxiliary qubit is not coupled to the second quantum oscillator.

[0128] Aspect 16. The dual-rail qubit according to Aspect 9, wherein the auxiliary qubit is a superconducting transporter qubit.

[0129] Aspect 17. The dual-rail qubit of aspect 9, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

[0130] Aspect 18. A dual-rail quantum bit, comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an auxiliary quantum bit 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 auxiliary quantum bit to perform a first rotation of the quantum state of the auxiliary quantum bit; (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 auxiliary quantum bit to perform a second rotation of the quantum state of the auxiliary quantum bit.

[0131] Aspect 19. The dual-rail qubit of aspect 18, wherein the first rotation and the second rotation of the state of the ancillary qubit are rotations within the manifold between the ground state |g> and the second excited state |f> of the ancillary qubit.

[0132] Aspect 20. The dual-rail qubit according to aspect 18, wherein the first rotation and the second rotation of the state of the auxiliary qubit are π / 2 rotations around the Y axis of the Bloch sphere.

[0133] 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 auxiliary qubit after (c).

[0134] 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 upon detecting that the state of the auxiliary qubit is not in the ground state |g>.

[0135] Aspect 23. The dual-rail qubit according to Aspect 18, wherein the at least one controller is further configured to: identify an erasure error in the dual-rail qubit when detecting that the state of the auxiliary qubit is in the first excited state |e> or the second excited state |f>.

[0136] 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 auxiliary quantum bit is dispersively coupled to the first quantum oscillator.

[0137] Aspect 25. The dual-rail qubit according to aspect 18, wherein the coupling element is a superconducting transport qubit or a superconducting nonlinear asymmetric inductor element (SNAIL).

[0138] Aspect 26. The dual-rail qubit of aspect 18, wherein the auxiliary qubit is not coupled to the second quantum oscillator.

[0139] Aspect 27. The dual-rail qubit according to aspect 18, wherein the auxiliary qubit is a superconducting transporter qubit.

[0140] Aspect 28. The dual-rail qubit of aspect 18, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

[0141] Aspect 29. A dual-rail quantum bit, comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an auxiliary quantum bit coupled to the first quantum oscillator; at least one energy source; and at least one controller, the at least one controller being configured to: operate the at least one energy source to measure the parity state of the first quantum oscillator; operate the at least one energy source to measure the parity state of the second quantum oscillator; and determine a logical state of the dual-rail quantum bit based on the measured parity states of the first quantum oscillator and the second quantum oscillator.

[0142] Aspect 30. The dual-rail qubit of aspect 29, wherein when the parity state of the first quantum oscillator is measured to be opposite to the parity state of the second quantum oscillator, the determined logic state of the dual-rail qubit is a 0 logic state or a 1 logic state.

[0143] Aspect 31. The dual-rail qubit of aspect 29, wherein the at least one controller is further configured to: identify an erasure error when the parity states of the first quantum oscillator and the second quantum oscillator are both measured to be even numbers.

[0144] Aspect 32. The dual-rail quantum bit of Aspect 29, wherein the at least one controller is configured to: operate the at least one energy source to measure the parity state of the first quantum oscillator a plurality of times, and operate the at least one energy source to measure the parity state of the second quantum oscillator a plurality of times, thereby generating a plurality of parity state measurements for each of the first quantum oscillator and the second quantum oscillator.

[0145] 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 a plurality of parity state measurements for the first quantum oscillator and based on a majority outcome of a plurality of parity state measurements for the second quantum oscillator.

[0146] Aspect 34. The dual-rail qubit of aspect 29, wherein operating the at least one energy source to measure the parity state of the first quantum oscillator comprises operating the at least one energy source to measure the state of the auxiliary qubit, and identifying the parity state of the first quantum oscillator based on the measured state of the auxiliary qubit.

[0147] Aspect 35. A dual-rail quantum bit according to Aspect 29, wherein the at least one controller is further configured to: after operating the at least one energy source to measure the parity state of the first quantum oscillator and before operating the at least one energy source to measure the parity state of the second quantum oscillator, direct energy to the coupling element to perform a beam splitter operation on the first quantum oscillator and the second quantum oscillator.

[0148] 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 auxiliary quantum bit is dispersively coupled to the first quantum oscillator.

[0149] Aspect 37. The dual-rail qubit according to aspect 29, wherein the coupling element is a superconducting transport qubit or a superconducting nonlinear asymmetric inductor element (SNAIL).

[0150] Aspect 38. The dual-rail qubit of aspect 29, wherein the auxiliary qubit is not coupled to the second quantum oscillator.

[0151] Aspect 39. The dual-rail qubit of aspect 29, wherein the ancillary qubit is a superconducting transporter qubit.

[0152] Aspect 40. The dual-rail qubit of aspect 29, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

[0153] Aspect 41. A dual-rail quantum bit, comprising: a first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an auxiliary quantum bit 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, the at least one controller being configured to: operate the at least one energy source to at least partially perform one or more gates and / or operations on the dual-rail quantum bit; 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 to initialize the dual-rail quantum bit to a new logical state.

[0154] Aspect 42. The dual-rail qubit of aspect 41, wherein detecting the erasure error comprises determining that the quantum state of the auxiliary qubit is not in the ground state |g>.

[0155] Aspect 43. The dual-rail qubit according to aspect 42, wherein detecting the erasure error comprises determining that the quantum state of the auxiliary qubit is in the first excited state |e> or the second excited state |f>.

[0156] 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 auxiliary quantum bit is dispersively coupled to the first quantum oscillator.

[0157] Aspect 45. The dual-rail qubit according to aspect 41, wherein the coupling element is a superconducting transport qubit or a superconducting nonlinear asymmetric inductor element (SNAIL).

[0158] Aspect 46. The dual-rail qubit of aspect 41, wherein the auxiliary qubit is not coupled to the second quantum oscillator.

[0159] Aspect 47. The dual-rail qubit of aspect 41, wherein the ancillary qubit is a superconducting transporter qubit.

[0160] Aspect 48. The dual-rail qubit of aspect 41, wherein the first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

[0161] Aspect 49. A system comprising: a first dual-rail qubit, the first dual-rail qubit comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element, the first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first auxiliary qubit, the first auxiliary qubit coupled to the first quantum oscillator; a second dual-rail qubit, the second dual-rail qubit comprising: a third quantum oscillator; a fourth quantum oscillator; a second coupling element, the second coupling element coupled to the third quantum oscillator and the fourth quantum oscillator; and a second auxiliary qubit, the second auxiliary qubit coupled to the third quantum oscillator; and a third coupling element, the third coupling element coupled to the second quantum oscillator and the third quantum oscillator.

[0162] Aspect 50. The system of Aspect 49 further comprises: 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 logical states of the first dual-rail qubit and the second dual-rail qubit.

[0163] Aspect 51. The system of aspect 50, wherein the two-qubit gate is a ZZ(θ) gate.

[0164] Aspect 52. The system of aspect 50, wherein operating the at least one energy source to perform the two-qubit gate comprises directing energy to the third coupling element.

[0165] Aspect 53. The system of aspect 52, wherein operating the at least one energy source to perform the two-qubit gate comprises performing one or more qubit rotations of the second ancillary qubit.

[0166] Aspect 54. A system according to Aspect 53, wherein operating the at least one energy source to perform the two-qubit gate includes: (a) performing a first qubit rotation of the second ancillary qubit; (b) after (a), directing energy to the third coupling element; and (c) after (b), performing a second qubit rotation of the second ancillary qubit.

[0167] Aspect 55. The system of aspect 50, wherein the at least one controller is further configured to: after executing the two-qubit gate, identify an erasure error when it is detected that the quantum state of the second auxiliary qubit is not in the ground state |g>.

[0168] Aspect 56. A system according to Aspect 55, wherein the at least one controller is further configured to: after executing the two-qubit gate, identify an erasure error when it is detected that the quantum state of the second auxiliary qubit is in the first excited state |e> or the second excited state |f>.

[0169] Aspect 57. A system according to 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 auxiliary quantum bit is dispersively coupled to the first quantum oscillator; the second auxiliary quantum bit 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.

[0170] Aspect 58. A system comprising: a plurality of dual-rail qubits, each of the plurality of dual-rail qubits comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element, the first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first auxiliary qubit, the first auxiliary qubit coupled to the first quantum oscillator; a plurality of measurement qubits, each of the plurality of measurement qubits comprising: a first quantum oscillator; and a first auxiliary qubit, the first auxiliary qubit coupled to the first quantum oscillator of the measurement qubit, wherein each of the plurality of measurement qubits is coupled to four dual-rail qubits of the plurality of dual-rail qubits, wherein each coupling is provided via a corresponding coupling element.

[0171] Aspect 59. The system of aspect 58, wherein each coupling of the measurement qubit to one of the plurality of dual-rail qubits couples the first quantum oscillator of the measurement qubit to the first quantum oscillator or the second quantum oscillator of the one of the plurality of dual-rail qubits.

[0172] Aspect 60. The system of aspect 58, wherein the first coupling element of each dual-rail qubit in the plurality of dual-rail qubits is a superconducting transporter qubit or a superconducting nonlinear asymmetric inductive element (SNAIL).

[0173] Aspect 61. The system of aspect 58, wherein the first ancillary qubit of each dual-rail qubit in the plurality of dual-rail qubits is a superconducting transporter qubit.

[0174] Aspect 62. A module comprising: a plurality of dual-rail qubits, each dual-rail qubit in the plurality of dual-rail qubits comprising: a first quantum oscillator; a second quantum oscillator; a first coupling element, the first coupling element coupled to the first quantum oscillator and the second quantum oscillator; and a first auxiliary qubit, the first auxiliary qubit coupled to the first quantum oscillator; and a plurality of cavities, each of the plurality of cavities: coupled to one dual-rail qubit in the plurality of dual-rail qubits via a corresponding coupling element; and coupled to an external port, the external port configured to couple to a quantum bus.

[0175] Aspect 63. The module of Aspect 62, wherein the external port comprises a coaxial connector.

[0176] Aspect 64. A system comprising a plurality of modules according to aspect 62, the plurality of modules according to aspect 62 being coupled together via a plurality of quantum bus connections between external ports of the modules.

[0177] Aspect 65. The module of aspect 62, wherein the first coupling element of each dual-rail qubit in the plurality of dual-rail qubits is a superconducting transporter qubit or a superconducting nonlinear asymmetric inductive element (SNAIL).

[0178] Aspect 66. The module of aspect 62, wherein the first ancillary qubit of each dual-rail qubit in the plurality of dual-rail qubits is a superconducting transporter qubit.

[0179] The above embodiments of the technology described herein can be implemented in any of a number of ways. For example, the controller of any of the embodiments (including Figure 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 set of processors, regardless of whether it is set in a single computer or distributed among multiple computers. Such a processor can be implemented as an integrated circuit, with one or more processors in an integrated circuit component, including commercially available integrated circuit components known in the art, such as CPU chips, GPU chips, microprocessors, microcontrollers, or coprocessors. Alternatively, the processor can be implemented in a customized circuit system such as an ASIC or in a semi-custom circuit system generated by configuring a programmable logic device. As another alternative, the processor can be part of a larger circuit or semiconductor device, whether commercially available, semi-custom, or customized. As a specific example, some commercially available microprocessors have multiple cores, so that one or a subset of those cores can constitute a processor. However, the processor can be implemented using a circuit system in any suitable format.

[0180] 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 therefore, their application is not limited to the arrangements and details of the components set forth in the foregoing description or shown in the accompanying drawings. For example, various aspects described in one embodiment may be combined in any manner with various aspects described in other embodiments.

[0181] Furthermore, the present invention can be implemented as a method, examples of which have been provided. The actions performed as part of the method can be ordered in any suitable manner. Thus, embodiments can be constructed in which the actions are performed in an order different from that shown (including performing some actions simultaneously), even though the actions are shown as sequential actions in an illustrative embodiment.

[0182] The use of ordinal terms such as "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 relative to another claim element or the temporal order of the acts of performing the method, but is merely used as a marker to distinguish one claim element with a particular name from another element with the same name (but using ordinal terms) to distinguish the claim elements.

[0183] The terms "approximately" and "about" may be used in some embodiments to mean within ±20% of a target value, in some embodiments to mean within ±10% of a target value, in some embodiments to mean within ±5% of a target value, and still in some embodiments to mean within ±2% of a target value. The terms "approximately" and "about" may include the target value. The term "substantially equal" may be used in some embodiments to mean values ​​that are within ±20% of each other, in some embodiments to mean values ​​that are within ±10% of each other, in some embodiments to mean values ​​that are within ±5% of each other, and still in some embodiments to mean values ​​that are within ±2% of each other.

[0184] The term "substantially" may be used in some embodiments to refer to values ​​within ±20% of a comparison measurement, in some embodiments to within ±10%, in some embodiments to within ±5%, and still in some embodiments to within ±2%. For example, a first direction that is "substantially" perpendicular to a second direction may, in some embodiments, refer to a direction that is 90° to the second direction.

[0185] In some embodiments, the first direction is within ±20% of an angle of 90° to the first direction, in some embodiments, the first direction is within ±10% of an angle of 90° to the second direction, in some embodiments, the first direction is within ±5% of an angle of 90° to the second direction, and in some embodiments, the first direction is within ±2% of an angle of 90° to the second direction.

[0186] In addition, the phraseology and terminology used herein are for the purpose of description and should not be regarded as limiting. The use of "including," "comprising," or "having," "containing," "involving," and variations thereof herein is intended to encompass the items listed thereafter and equivalents thereof as well as additional items.

Claims

1. A dual-track quantum bit, comprising: the first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an auxiliary qubit, the auxiliary qubit coupled to the first quantum oscillator; at least one energy source; as well as At least one controller configured to operate the at least one energy source to initialize the dual-rail qubit to a 0 or 1 logic state by: when the dual-rail qubit is to be initialized to the zero logic state, operating the at least one energy source to arrange the first quantum oscillator into a single-photon state and the second quantum oscillator into its ground state; or When the dual-rail qubit is to be initialized to the 1 logic state, the at least one energy source is operated to arrange the first quantum oscillator into its ground state and the second quantum oscillator into a single photon state.

2. The dual-rail quantum bit according to claim 1, wherein: Arranging the first quantum oscillator or the second quantum oscillator into a single-photon state includes operating the at least one energy source to perform a plurality of optimal control pulses and / or cavity-assisted sideband driving.

3. The dual-rail quantum bit according to claim 1, wherein: Arranging the first quantum oscillator into the single-photon state includes operating the at least one energy source to arrange the first quantum oscillator into a |1> Fock state.

4. The dual-rail quantum bit according to claim 1, wherein: The coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and The auxiliary quantum bit is coupled to the first quantum oscillator in a divergent manner.

5. The dual-rail quantum bit according to claim 1, wherein: The coupling element is a superconducting transport quantum bit or a superconducting nonlinear asymmetric inductor element (SNAIL).

6. The dual-rail quantum bit according to claim 1, wherein: The auxiliary qubit is not coupled to the second quantum oscillator.

7. The dual-rail quantum bit according to claim 1, wherein: The auxiliary qubit is a superconducting transporter qubit.

8. The dual-rail quantum bit according to claim 1, wherein: The first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

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

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

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

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

13. The dual-rail qubit of claim 9, wherein: The coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and The auxiliary quantum bit is coupled to the first quantum oscillator in a divergent manner.

14. The dual-rail quantum bit according to claim 9, wherein: The coupling element is a superconducting transport quantum bit or a superconducting nonlinear asymmetric inductor element (SNAIL).

15. The dual-rail quantum bit according to claim 9, wherein: The auxiliary qubit is not coupled to the second quantum oscillator.

16. The dual-rail quantum bit according to claim 9, wherein: The auxiliary qubit is a superconducting transporter qubit.

17. The dual-rail quantum bit according to claim 9, wherein: The first quantum oscillator is a first microwave cavity, and wherein the second quantum oscillator is a second microwave cavity.

18. A dual-track quantum bit, comprising: the first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an auxiliary 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; as well as at least one controller configured to operate the at least one energy source to: (a) directing energy to the auxiliary qubit to perform a first rotation of the quantum state of the auxiliary qubit; (b) after (a), directing energy to the coupling element to perform a beam splitter operation on the first quantum oscillator and the second quantum oscillator; as well as (c) Following (b), directing energy to the auxiliary qubit to perform a second rotation of the quantum state of the auxiliary qubit.

19. The dual-rail quantum bit according to claim 18, wherein: The first rotation and the second rotation of the state of the ancillary qubit are rotations within the manifold between the ground state |g> and the second excited state |f> of the ancillary qubit.

20. The dual-rail quantum bit according to claim 18, wherein: The first rotation and the second rotation of the state of the auxiliary quantum bit are π / 2 rotations around the Y axis of the Bloch sphere.

21. The dual-rail quantum bit according to 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 auxiliary qubit after (c).

22. The dual-rail quantum bit according to claim 18, wherein: The at least one controller is further configured to identify an erasure error in the dual-rail qubit upon detecting that the state of the auxiliary qubit is not in the ground state |g>.

23. The dual-rail quantum bit according to claim 18, wherein: The at least one controller is further configured to identify an erasure error in the dual-rail qubit upon detecting that the state of the auxiliary qubit is in the first excited state |e> or the second excited state |f>.

24. The dual-rail qubit of claim 18, wherein: The coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and The auxiliary quantum bit is coupled to the first quantum oscillator in a divergent manner.

25. The dual-rail quantum bit of claim 18, wherein: The coupling element is a superconducting transport quantum bit or a superconducting nonlinear asymmetric inductor element (SNAIL).

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

27. The dual-rail qubit of claim 18, wherein: The auxiliary qubit is a superconducting transporter qubit.

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

29. A dual-track quantum bit, comprising: the first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an auxiliary qubit, the auxiliary qubit coupled to the first quantum oscillator; at least one energy source; as well as at least one controller configured to: operating the at least one energy source to measure a parity state of the first quantum oscillator; operating the at least one energy source to measure a parity state of the second quantum oscillator; as well as A logical state of the dual-rail qubit is determined based on the measured parity states of the first quantum oscillator and the second quantum oscillator.

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 quantum bit is a 0 logic state or a 1 logic state.

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

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

33. The dual-rail qubit of claim 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 a plurality of parity state measurements for the first quantum oscillator and based on a majority outcome of a plurality of parity state measurements for the second quantum oscillator.

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

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

36. The dual-rail qubit of claim 29, wherein: The coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and The auxiliary quantum bit is coupled to the first quantum oscillator in a divergent manner.

37. The dual-rail qubit of claim 29, wherein: The coupling element is a superconducting transport quantum bit or a superconducting nonlinear asymmetric inductor element (SNAIL).

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

39. The dual-rail qubit of claim 29, wherein: The auxiliary qubit is a superconducting transporter qubit.

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

41. A dual-track quantum bit, comprising: the first quantum oscillator; a second quantum oscillator; a coupling element coupled to the first quantum oscillator and the second quantum oscillator; an auxiliary 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; as well as at least one controller configured to: operating the at least one energy source to perform, at least in part, one or more gates and / or operations on the dual-rail qubit; detecting an erase error after performing the one or more gates and / or operations; as well as In response to detecting the erasure error, operating the at least one energy source to initialize the dual-rail qubit to a new logical state.

42. The dual-rail qubit of claim 41 , wherein: Detecting the erasure error includes determining that the quantum state of the auxiliary quantum bit is not in the ground state |g>.

43. The dual-rail qubit of claim 42, wherein: Detecting the erasure error includes determining whether the quantum state of the auxiliary quantum bit is in the first excited state |e> or the second excited state |f>.

44. The dual-rail qubit of claim 41 , wherein: The coupling element is dispersively coupled to the first quantum oscillator and the second quantum oscillator; and The auxiliary quantum bit is coupled to the first quantum oscillator in a divergent manner.

45. The dual-rail qubit of claim 41 , wherein: The coupling element is a superconducting transport quantum bit or a superconducting nonlinear asymmetric inductor element (SNAIL).

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

47. The dual-rail qubit of claim 41 , wherein: The auxiliary qubit is a superconducting transporter qubit.

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

49. A system comprising: A first dual-track quantum bit, wherein the first dual-track quantum bit comprises: the 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 auxiliary qubit, the first auxiliary qubit being coupled to the first quantum oscillator; A second dual-track quantum bit, wherein the second dual-track quantum bit comprises: The 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 auxiliary qubit coupled to the third quantum oscillator; and A third coupling element is coupled to the second quantum oscillator and the third quantum oscillator.

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

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

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

53. The system of claim 52, wherein: Operating the at least one energy source to perform the two-qubit gate includes performing one or more qubit rotations of the second ancillary qubit.

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

55. The system of claim 50, wherein: The at least one controller is further configured to, after executing the two-qubit gate, identify an erasure error when it is detected that the quantum state of the second auxiliary qubit is not in a ground state |g>.

56. The system of claim 55, wherein: The at least one controller is further configured to, after executing the two-qubit gate, identify an erasure error when detecting that the quantum state of the second auxiliary qubit is in the first excited state |e> or the second excited state |f>.

57. The system of claim 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 auxiliary quantum bit is coupled to the first quantum oscillator in a divergent manner; The second auxiliary quantum bit is coupled to the third quantum oscillator in a divergent manner; and The third coupling element is dispersively coupled to the second quantum oscillator and the third quantum oscillator.

58. A system comprising: A plurality of dual-track quantum bits, each of the plurality of dual-track quantum bits comprising: the 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 auxiliary qubit, the first auxiliary qubit being coupled to the first quantum oscillator; a plurality of measurement qubits, each measurement qubit in the plurality of measurement qubits comprising: the first quantum oscillator; and a first ancillary qubit coupled to a first quantum oscillator of the measurement qubit, Wherein each of the plurality of measurement qubits is coupled to four dual-rail qubits of the plurality of dual-rail qubits, wherein each coupling is provided via a corresponding coupling element.

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

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

61. The system of claim 58, wherein: The first auxiliary qubit of each dual-rail qubit in the plurality of dual-rail qubits is a superconducting transporter qubit.

62. A module comprising: A plurality of dual-track quantum bits, each of the plurality of dual-track quantum bits comprising: the 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 ancillary qubit coupled to the first quantum oscillator; and a plurality of cavities, each of the plurality of cavities: coupled to one of the plurality of dual-rail qubits via a corresponding coupling element; and coupled to an external port, the external port being configured to be coupled to a quantum bus.

63. The module according to claim 62, wherein The external port includes a coaxial connector.

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

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

66. The module according to claim 62, wherein The first auxiliary qubit of each dual-rail qubit in the plurality of dual-rail qubits is a superconducting transporter qubit.

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