A quantum system for executing CNOT gates, and a quantum system for executing iterative codes using the same.

Abstract

The quantum system for executing the CNOT gate comprises a linearly connected command circuit providing radiation, a target cat qubit device, and a control qubit device, the target cat qubit comprising a nonlinear component called ATS, which serves two purposes: namely, to appropriately process the two-photon conversion Hamiltonian for cat qubit stabilization and to appropriately process the CNOT Hamiltonian for executing the CNOT gate by the control qubit device. At any given time, the ATS plays either the role of cat qubit stabilization or the role of the CNOT gate.

Description

[Technical Field]

[0001] This invention relates to the execution of quantum gates, and more specifically, to the use of such gates in the context of cat qubits. [Background technology]

[0002] To extract interrelated information about several data qubits, conventional quantum circuits perform a sequence of two-qubit gates (typically CNOT or CZ gates) between an ancilla qubit and several data qubits before measuring the ancilla qubit. This operation is usually called a syndrome measurement.

[0003] The realization of these quantum gates is critically important for error detection and therefore for implementing quantum error correction code (QECC), which is currently considered the only way to build a reliable and usable quantum processor. The schematic idea behind QECC is to encode logical qubits using several (at least two) physical data qubits, and the error detection scheme is designed to verify that the information of some physical data qubits does not change over time. If an error is detected, error correction is performed either by changing the state of the physical data qubits themselves or by post-processing the results of a quantum algorithm on these qubits.

[0004] The properties used to evaluate the quality of a quantum gate are its execution time (the time the gate is in operation) and the associated error probability. For a cat qubit, this error probability depends on the ratio κ1 / κ2, where κ1 is the single-photon loss rate (error rate) of the qubit used to execute the gate, and κ2 is the two-photon loss rate (correction or stabilization rate) of the qubit used to execute the gate.

[0005] One of the goals pursued in current quantum hardware experiments is to achieve quantum gates that allow iterative codes to become more effective as the distance of the error correction code used to perform the error correction scheme (which is related to the number of physical qubits used to encode the logical qubits) increases. Currently, 5*10 -3 A ratio κ1 / κ2 below this is required. In previous literature, proposed CNOT dynamics for cat qubit QECCs are based on the annihilation operator q and resonant frequency f. q The definition of a control qubit characterized by and a target qubit consisting of a cat qubit stabilized by two-photon drive and dissipation, wherein its annihilation operator a and its resonant frequency f a Defining a target qubit characterized by, appropriately stabilizing the target qubit so that it has a phase determined conditionally on the state of the control qubit, and a longitudinally coupled Hamiltonian between the control qubit and the target cat qubit, the general formula

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[0006] In the following, the terms "longitudinal Hamiltonian," "longitudinal coupled," or CNOT or CX Hamiltonian are interchangeable, and this longitudinal coupled Hamiltonian is shown between the control qubit and the target cat qubit. [Prior art documents] [Patent Documents]

[0007] [Patent Document 1] European Patent Application No. EP22306815.6 Specification [Patent Document 2] European Patent Application No. EP22306816.4 [Non-patent literature]

[0008]

Non-licensed literature 1

Non-licensed Document 2

Non-licensed Document 4

Non-licensed Document 5

Non-licensed Document 6

[0009] This general proposal raises two fundamental challenges. The first challenge is that very few experimental implementations have been proposed. The second challenge is that the few experimental implementations that have been proposed are considered impractical because they require too many nonlinear components and / or too many parametric drives, and / or fail to address all the engineering challenges. These experimental proposals include Non-Patent Documents 1-3.

[0010] The objective of this invention is to improve this situation. [Means for solving the problem]

[0011] For this purpose, the applicant proposes a quantum system for executing CNOT gates. The above quantum system comprises a command circuit, a target cat qubit device, and a control superconducting qubit device. The above command circuit provides microwave radiation. The above-described target cat qubit device comprises a superconducting quantum interferometer having an asymmetric thread, the superconducting quantum interferometer having an asymmetric thread having flux lines through which radiation propagates that can be transmitted to modulate common flux and / or differential flux, and is connected to at least one resonant section. The above-described target cat qubit device has a first mode having a first resonant frequency and a second mode having a second resonant frequency. The superconducting quantum interferometer having the asymmetric thread described above is configured such that the command circuit drives the second mode by transmitting radiation having the second resonant frequency to the at least one resonant section, and modulates the common flux at a frequency equal to twice the first resonant frequency and the absolute difference between the second resonant frequency by transmitting radiation in the flux lines, thereby stabilizing the target cat qubit having the first resonant frequency. The above-described controlled superconducting qubit device has a third mode having a third resonant frequency for hosting a control qubit, and is configured such that Rabi oscillation of the third mode is induced when exposed to radiation having the third resonant frequency at an intensity lower than the control qubit confinement rate. The above-described controlled superconducting qubit device is linearly coupled to the target cat qubit device such that the third mode is linearly coupled to the superconducting quantum interferometer having the asymmetric thread. The command circuit described above is configured to operate a CNOT gate between the target cat qubit and the control qubit by transmitting only radiation with a frequency equal to the third resonant frequency in the flux line during the CNOT gate time window, and to drive the second mode by transmitting radiation having the second resonant frequency to at least one resonant part outside the CNOT gate time window, and to modulate the common flux at a frequency equal to twice the first resonant frequency and the absolute value of the difference between the second resonant frequency by transmitting radiation in the flux line.

[0012] This system is advantageous because it makes the minimum selection from a list of components of a general proposal and proposes a practical implementation with minimal hardware and controlled or parametric pumps. This reduces implementation complexity and therefore reduces potential points of failure. This system also addresses the effects of some unideal properties that arise from practical implementations.

[0013] In various embodiments, the method may present one or more of the following features.

[0014] The target cat qubit device and the control superconducting device are linearly coupled such that the phase difference across the superconducting quantum interference device having the asymmetric thread is

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[0015] The command circuit is configured to pump the magnetic flux line only by radiation at a frequency equal to the third resonance frequency with an amplitude ε CX during the CNOT gate time window. Thereby, a Hamiltonian having the equation

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[0016] The above command circuit transmits only radiation with a frequency equal to the third resonant frequency in the above magnetic flux line, and the equation

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[0017] The command circuit transmits radiation at a frequency equal to the third resonant frequency to the control qubit device, thereby during the CNOT gate time window, the equation

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[0018] The above-described controlled superconducting qubit device hosts transmon qubits, flux qubits, flaxonium qubits, cat qubits, or any bosonic qubit encoded in a resonator.

[0019] The superconducting quantum interferometer having the asymmetric thread described above is further configured such that the command circuit drives the second mode by transmitting radiation having the second resonant frequency, and modulates the common flux at a frequency equal to the sum of the second resonant frequency and twice the first resonant frequency and twice the first resonant frequency by transmitting radiation in the flux line, wherein the target cat qubit device is further configured to stabilize the target squeezed cat qubit having the first resonant frequency.

[0020] The target cat qubit device and the control superconducting qubit device are configured such that the absolute value of the difference between the first resonant frequency and the third resonant frequency exceeds 10 MHz.

[0021] The above command circuit modulates the common magnetic flux by transmitting radiation in the above magnetic flux line, using only radiation at a frequency equal to the third resonant frequency, and maintaining a constant intensity g CX It is configured to produce a CNOT Hamiltonian having the following characteristics.

[0022] The above command circuit modulates the common magnetic flux by transmitting radiation in the above magnetic flux line, using only radiation with a frequency equal to the third resonant frequency.

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[0023] The present invention also relates to a quantum system for performing iterative coding. The above quantum system comprises a command circuit, d data cat qubit devices (2 or more), and d-1 ancilla qubit devices. The above command circuit selectively applies microwave radiation. Each data cat qubit device has its own resonant frequency and is coupled to the command circuit described above to stabilize each data cat qubit. Each anscira qubit device has an anscira resonant frequency coupled to the command circuit described above for hosting an anscira qubit. Each of the d-1 ancilla qubit devices is linearly coupled to each of the d data cat qubit devices, Each of the d data cat qubit devices is connected to at most two ancilla qubit devices out of the d-1 ancilla qubit devices. The d data cat qubit devices and the d-1 ancilla qubit devices are such that each coupling of an ancilla qubit device with a data cat qubit device is controllable by the command circuit to realize the quantum system described in any of the preceding claims. The above command circuit is, 1) For each of the d-1 ancilla qubit devices mentioned above, each ancilla qubit in the d-1 ancilla qubit devices is prepared by the state of the X operator "|+>" or "|->", 2) For each of the d-1 ancilla qubit devices, a CNOT gate is executed by the first of the two data cat qubit devices to which it is coupled, and then a CNOT gate is executed by the second of the two data cat qubit devices to which it is coupled, 3) Applying the measurement to the operator X in each of the above ancilla qubit devices and This is further configured to perform d-1 quantum operations.

[0024] In this quantum system, every ancilla qubit device hosts cat qubits, such that every data cat qubit device and every ancilla qubit device are cat qubit devices. The command circuit described above defines which cat qubit devices are data cat qubit devices and which cat qubit devices are ansila qubit devices. [Brief explanation of the drawing]

[0025] [Figure 1] A diagram of a quantum circuit for implementing a phase-inverting iterative code is shown. [Figure 2] A schematic diagram of a conventional CNOT gate between a control qubit and a target cat qubit is shown. [Figure 3] A schematic diagram of a quantum system for executing the CNOT gate according to the present invention is shown. [Figure 4] Various embodiments relating to the target cat qubit device, the control qubit device, and the linear combination between the control qubit device and the target cat qubit device shown in Figure 3 are presented. [Figure 5] Various embodiments relating to the target cat qubit device, the control qubit device, and the linear combination between the control qubit device and the target cat qubit device shown in Figure 3 are presented. [Figure 6] Various embodiments relating to the target cat qubit device, the control qubit device, and the linear combination between the control qubit device and the target cat qubit device shown in Figure 3 are presented. [Figure 7] Various embodiments relating to the target cat qubit device, the control qubit device, and the linear combination between the control qubit device and the target cat qubit device shown in Figure 3 are presented. [Figure 8] Figure 3 shows a diagram illustrating how the quantum gate is executed. [Figure 9] Figure 8 shows two vertical connection establishment methods for the calculation. [Figure 10] A schematic diagram of a quantum system for executing an iterative code comprising a CNOT gate according to the present invention is shown. [Figure 11] The circuit diagram of the repeating code according to the present invention is shown. [Figure 12] A photograph of an experimental quantum circuit using the CNOT gate according to the present invention is shown. [Figure 13] A magnified view of the area shown in Figure 12 is provided. [Figure 14] This image shows a photograph of an experimental quantum circuit implementing a CNOT gate. [Figure 15] This shows an enlarged view of the section in Figure 14. [Modes for carrying out the invention]

[0026] Other features and advantages of the present invention will be readily apparent from the description of the following drawings illustrating exemplary embodiments of the present invention.

[0027] The drawings and the following description consist, for the most part, of explicitly and appropriately defined features. As a result, they are not only useful for understanding the invention, but can also be used, where necessary, to contribute to its definition.

[0028] This invention relates to the realization of a high-performance quantum gate in the context of cat qubits.

[0029] Stabilized cat qubits are known to benefit from noise bias. More precisely, the effective error channel (e.g., bit error or "bit inversion") is exponentially suppressed with respect to the "size" of the Schrödinger cat state of the cat qubit, i.e., the average number of photons.

[0030] According to current knowledge, this suppression should apply to handling a large class of physical noises that have local effects on the phase space of a harmonic oscillator. This includes, but is not limited to, photon losses, thermal excitations, photon phase relaxations, and various nonlinearities caused by connecting to Josephson junctions.

[0031] Recent experiments in the context of quantum superconducting circuits have observed this exponential suppression of the bit-flip error with respect to the average number of photons in the cat state.

[0032] Due to this noise structure, the use of a single repeating code is considered sufficient to correct the residual error channel. In fact, if only phase inversion needs to be corrected, it is sufficient to use only a phase inversion error correction code. This may be, for example, a repeating code defined on a dual basis, or any other prior art error correction code.

[0033] In the following, the convention for states and Pauli operator bases is such that the drive at the resonant frequency of a qubit (or cat qubit), also known as a Rabi drive, corresponds to a Z rotation. This is a common choice of bases in the field of cat qubits, but it is rare for the more standard two-level system qubits. In the following, this choice is made so that cat qubits and two-level system qubits have the same general notation. In particular, in the following, the states |+> and |-> correspond to the ground and excited states of a two-level system qubit, respectively. For cat qubits, the |+> and |-> states represent even and odd Schrödinger cat states proportional to |α>±|-α>, where the state |α> is a coherent state with complex amplitude α.

[0034] Figure 1 shows a diagram of a quantum circuit for executing a phase-inverted iterative code. The cat qubit iterative code is constructed using d cat qubits 4 (called data cat qubits) containing encoded logical information. The code is generated by repeatedly measuring a quantum operator that determines whether or not some errors have occurred in the data cat qubits. This is done using d-1 additional qubits 6 (called ancilla qubits or auxiliary qubits). The quantum circuit for the iterative code requires the preparation of an ancilla qubit in either state |+> or state |->, two CNOT gates between the ancilla qubit and the data cat qubit 10, and the measurement of the Pauli X operator 12 (which is equivalent to photon or excitation number parity and is a valid definition for both cat qubits and two-level system qubits; in the case of two-level system qubits, this also corresponds to a measurement in a standard-based ground / excited state). In the following, depending on whether the context is an iterative code (anscira / data) or a single CNOT gate (control / target), an anscira qubit is also called a control qubit, and a data CNOT qubit is also called a target CNOT qubit.

[0035] The challenge in implementing this iterative code is to execute the code on hardware that operates below a fault tolerance error threshold. This means that the fidelity of the quantum operations in this circuit must be extremely high for the code to have a positive effect.

[0036] More precisely, if the iterative code is operated above a threshold, i.e., if the fidelity of the physical operations constituting the iterative code is insufficient, the lifetime of the logical information decreases as the number of physical data qubits d increases. New errors introduced by the addition of quantum systems are not compensated for by error correction methods.

[0037] On the other hand, if the iterative code is operated below an error correction threshold, i.e., if the fidelity of the physical operation is sufficient, the lifetime of the logical information increases exponentially with respect to the number of physical data qubits d (which is also called the distance d of the iterative code).

[0038] Figure 1 shows only three physical data qubits (four data cat qubits) and two ancilla qubits. However, this figure illustrates how various qubits need to be connected in a quantum circuit to enable the execution of phase-inverting iterative codes.

[0039] The field of quantum computing is very young. This is all the more true in the cat-qubit domain. In many ways, it behaves like a research area. Consequently, the preferred way of development is through modification, which may appear very gradual at first glance, but in reality requires significant physical research to be validated and industrialized. In other words, whatever is considered cutting-edge technology today generally remains unchanged until a major obstacle is discovered. This means that solutions that are known to work are not easily replaced.

[0040] In this invention, the implementation of any superconducting circuit in a two-level system or an effective two-level system, such as a bosonic qubit encoded in a resonator, is intended by the qubit. These bosonic qubits include cat qubits.

[0041] In this invention, unless otherwise stated, the term "cat qubit" refers to any implementation of a cat qubit, and in particular, a two-photon dissipative Schrödinger cat qubit. Alternatively, other cat qubits may be used.

[0042] Such cat qubits may be stabilized or confined by the exemplary methods described below.

[0043] a) Jump operator

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[0044] b) Kerr Hamiltonian

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[0045] c) Detuned Kerr Hamiltonian

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[0046] d) Two-photon exchange (TPE) Hamiltonian

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[0047] e) Jump Operator

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[0048] f) Jump operator

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[0049] If the ancilla qubit is a cat qubit, it may be stabilized by one of the six stabilizations described above. With respect to the target cat qubit, the description below is limited to embodiments of stabilization mechanisms a) and e). In fact, as will become clear below, the command circuit should be able to turn the target cat qubit stabilization on or off. In the case of stabilization mechanism a), this can be done by turning off the parametric pump responsible for the 2:1 photon conversion. In the case of stabilization mechanism e), all parametric pumps must be turned off.

[0050] In previous proposals, the theoretical realization of a CNOT gate between a controlled qubit with an annihilation operator q and a stabilized cat qubit with an annihilation operator a typically relies on the use of the following three components. 1) Confinement of a control qubit such that microwave driving at its resonant frequency results in Rabi oscillation. This is typically native to two-level system qubits and is handled appropriately via parametric interactions for cat qubits. 2) Formula

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[0051] Figure 2 shows a schematic diagram of the CNOT gate. On the left, the quantum circuit representation of the CNOT gate is shown again. On the right, the target cat qubit 4 and control qubit 6, controlled via command circuit 8, are linked to execute the CNOT gate 10.

[0052] Generally speaking, a cat qubit is defined as an effective two-level system stabilized within a resonator embedded in a specific superconducting circuit that receives specific radiation controlled by a command circuit. As a result, a cat qubit may generally be specified as a resonator having a specific resonant frequency (which is the frequency of the cat qubit) controlled by a command circuit. Each cat qubit may be controlled by a specific command circuit, or a single circuit may be configured to control all of the cat qubits in a given circuit. In the embodiments described herein, a single command circuit 8 controls the target cat qubit 4 and the control qubit 6. Below, the term cat qubit mode may be used instead of cat qubit resonator to emphasize the fact that the cat qubit is effectively in the normal mode of the superconducting circuit, which may typically require several strongly coupled bare resonators.

[0053] As explained earlier, the dissipative stabilization (method a) or e)) of a cat qubit requires the proper realization of a nonlinear conversion and inverse between two photons in a first mode hosting a stabilized quantum manifold, also known as cat qubit mode a, and one photon in a second mode known as buffer mode b. Such a stabilization scheme makes it possible to suppress bit inversion exponentially with respect to the number of photons in the two coherent states. However, it will only be effective if the confinement rate of the two coherent states is greater than the escape rate caused by an external noise source. The confinement rate is positively related to the 2:1 photon conversion rate.

[0054] The initial implementations of this stabilization scheme (Non-Patent Documents 4 and 5) failed to observe exponential suppression of bit inversion. This is because the superconducting circuit elements used to properly realize the 2:1 photon conversion (so-called transmon) have a spurious cross-Kerr term, which introduces additional noise processing at the escape rate given by the very large transmon cat qubit dispersion shift. Non-Patent Document 6 disclosed an improved cat qubit implementation by properly handling the 2:1 photon conversion using a superconducting quantum interferometer (also called "ATS") with asymmetric threads. The ATS design has a much lower cross-Kerr term than the transmon, which allowed for the observation of exponential suppression of bit inversion. However, transmons were also used to measure the cat qubit state.

[0055] To date, this dissipative stabilization remains the most commonly used method for cat qubits, and the use of one or more ATSs is the de facto basis for best-performing experimental designs.

[0056] To improve the experimental realization of the CNOT gate, the applicant has strived to limit as much as possible the amount of theoretical CNOT components that need to be handled appropriately, as well as the number of parametric pumps and drives and the number of nonlinear components. The applicant has also carefully considered the spurious dynamics arising from this engineering technique and the provided solution, thereby limiting their negative effects.

[0057] This invention presents two extremely powerful circuit design options.

[0058] The first design choice is that the existing ATS, which adequately handles two-photon stabilization during the idling period, is the only nonlinear source required to execute the CNOT gate. This choice helps limit the complexity of the design.

[0059] The second design choice is that while component 3) improves the fidelity of CNOT, this is not decisive for its implementation. As a result, the implementation of component 3) was intentionally avoided. This eliminates the need for conditional stabilization and the engineering challenges it requires (direct exchange Hamiltonian between the control qubit and target cat qubit buffer and the two parametric pumps). Even if the data qubit is not stabilized during the CNOT gate, this does not impair bit-inversion suppression, and if the gate is fast compared to noise or defects, the stabilization is due to the Kerr effect resulting from the intrinsic nonlinearity of the resonator.

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[0060] These two design choices make CNOT feasible solely by properly handling the longitudinal Hamiltonian using the cat qubit device ATS. Since stabilization is turned off when longitudinal coupling is turned on, the ATS can be used to handle both terms (i.e., stabilization and longitudinal coupling) sequentially and appropriately without introducing the risk of parametric instability, which is more likely to occur when a single nonlinear component is parametrically driven simultaneously at several frequencies.

[0061] It should be noted that, when biased at its operating point, the Hamiltonian of the ATS embedded in the cat qubit device has the form "sin sin".

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[0062] To properly handle the CNOT Hamiltonian between the control qubit and the target cat qubit, the following is required: - Phase difference across ATS,

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[0063] In a rotating coordinate system, the parametric part of the Hamiltonian is expressed by the following equation:

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[0064] This Hamiltonian may be expressed in a way that emphasizes the desired dynamics.

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[0065] Since the buffer is a lossy mode coupled to a low-temperature environment, b is set such that a properly processed Hamiltonian satisfies the following equation. † We may also assume that b=0.

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[0066] This Hamiltonian is close to the CNOT Hamiltonian, with the exception of two additional terms. The first is the intensity

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[0067] The second clause cannot be fully compensated by simple means, given the design choices made in relation to the present invention. Instead, the applicant chose to design the system such that the amplitude of this term is much smaller than the amplitude of the CNOT Hamiltonian. In other words,

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[0068] If these two conditions (compensation for the second term and smaller amplitude) are met, the ATS Hamiltonian H ATS This may be appropriately processed to closely resemble a complete CNOT Hamiltonian.

[0069] The design choices described above resulted in the circuit design shown in Figure 3.

[0070] The quantum system 30 comprises a target cat qubit device 300 and a control qubit device 302, connected by a linear electromagnetic coupler 304.

[0071] In the embodiment described herein, the target cat qubit device 300 comprises a nonlinear superconducting circuit 306 to which several microwave sources 310, 311, 316 and a load 314 are connected.

[0072] When the nonlinear superconducting circuit 306 is coupled to the ATS 308, which acts as an inductive element, via a linear coupler 309, with a linear microwave network 320-322, the nonlinear superconducting circuit is involved with the ATS at a frequency f a and f b It comprises at least two normal modes (or eigenmodes) a and b. This involvement means that part or all of the mode magnetic energy is stored in the ATS. This involvement is in the case of mode a φ a It is expressed as follows, and in the case of mode b, φ b This can be quantified by the zero-point fluctuation of the superconducting phase across the ATS, represented by . Figures 4 to 7 further illustrate examples of embodiments for linear microwave networks. In these embodiments, the nonlinear superconducting circuit 306 may be a two-mode hybrid system (also known as a "galvanic cat," in which the applicant has filed Patent Documents 1 and 2) relating to the case where two lumped-parameter modes are strongly coupled via the ATS.

[0073] If an external DC magnetic field is set up such that a 0 mod 2π flux forms a loop and / or a thread in either a loop or a π mod 2π flux (or vice versa), we ensure that the ATS Hamiltonian has its "sin sin" form. For clarity, the configuration for applying the external DC magnetic field is not shown in Figure 3, but it may be applied via the two bottom mutual inductances of the ATS. A typical implementation involves interleaving a bias T-junction connected to a DC current source to input a DC current to the system while passing microwave radiation through it. Microwave sources 310 and 311 are configured to modulate the common and differential fluxes in the ATS, respectively. This is necessary to activate the parametric interaction. To clearly identify the roles of the two sources in the Hamiltonian, the schematic diagram provides a microwave network 324 that applies the correct phase offset. Alternatively, each microwave source may simply be coupled to a single node of the ATS, and their relative phases and amplitudes may be set to obtain the desired flux modulation. In that case, to modulate common flux, the two sources must engage circuits with different phases, and to modulate differential flux, the two sources must engage circuits with the same phase.

[0074] The microwave source 310 has a frequency f p =|2f a -f b When set to |, the nonlinear superconducting circuit 306 performs a 2:1 photon conversion between a first mode a having symbol 320 and a second mode b having symbol 322. To convert this 2:1 photon conversion into a 2-photon dissipation, mode b is connected to a linear coupler 312 and the frequency f b The load 314 is selectively coupled via a microwave filter 318 configured as a band-pass filter having a frequency f a It may be configured as a band-stop filter in and placed between the environment and the two modes to isolate the first mode and thus prevent the first mode from being affected by additional losses resulting from unwanted coupling to the load 314. Alternatively, fa >f b (or f b >f a In this case, it may be configured as a low-pass (or high-pass) filter. In other embodiments, the microwave filter 318 may be omitted if coupling between the load 314 and substantially only the second mode can be established. Thus, those skilled in the art will understand that the first mode has a high Q value and the second mode b has a low Q value.

[0075] In order to perform cat qubit stabilization in mode a, which requires two-photon drive and dissipation, mode b ultimately has its resonant frequency f b It is driven at a frequency f. This drive is typically performed at a frequency f b This is performed by the microwave source 316, which is set to f. Alternatively, the drive is performed at frequency f. p and f b If the microwave source 310 or 311 is configured to supply both, then this may be carried out by the microwave source 310 or 311. In the following, the first mode a hosts the cat qubit and is also known as the cat qubit mode, while the second mode b is used as a buffer between the cat qubit and the environment.

[0076] In the above, components 310-318 and 324 may be considered as part of the command circuit 8 in Figure 1. When the CNOT gate is not executed, i.e., in the so-called "idle mode," the command circuit 8 is configured to exclusively perform data cat qubit 4 stabilization. Herein, the control qubits and configuration required to execute the CNOT gate are described.

[0077] In the example shown in Figure 3, the control qubit device 302 hosts the control qubit at a resonant frequency f qThe control qubit device 302 comprises a mode q 305 having the following characteristics. In various embodiments, the control qubit device 302 may be any superconducting qubit such as a transmon qubit, a flux qubit, or a fluxonium qubit, or any bosonic qubit encoded in a resonator such as a Kerr-cat qubit (with or without detuning), another stabilized cat qubit device (with or without squeeze), or a cat qubit confined via a two-photon exchange Hamiltonian.

[0078] As described above, the control qubit 305 is coupled to the target cat qubit device 300 via a small linear coupler 304. The linear coupler 304 is configured such that the control qubit 305 is slightly hybridized with the cat qubit device 300, which results in a small involvement of the control qubit in the cat qubit device ATS308. This involvement is φ q This indicates that, as previously mentioned with respect to the second spurious element of the uncompensated, properly handled Hamiltonian, this involvement is the involvement of the target cat qubit mode φ a The fact that it remains small in comparison is critically important for accurately implementing the CNOT Hamiltonian. In various embodiments, the coupler 304 may be capacitive, inductive, or galvanic, or may be mediated via a resonant bus coupler or an additional linear microwave network. In the embodiment shown in Figure 3, the control qubit device 302 is coupled to the first mode a 320. In other embodiments, the control qubit device 302 may be coupled to the second mode b 322. Since modes a and b are typically moved from their usual locations in the microwave network a / b 320-322, this coupling location is not deterministic, and coupling to a particular location does not necessarily mean coupling to a particular mode unless the linear microwave network and ATS are specifically designed to do so.

[0079] The control qubit device 302 also includes a microwave source 303, which is coupled to the control qubit 305 in order to have the ability to drive it. The microwave source 303 may be used to compensate for the linear drive resulting from the CNOT Hamiltonian engineering. The microwave source 303 may be considered as part of the command circuit 8 in Figure 1.

[0080] As explained above, a further limitation on the control qubit device 302 is that the nonlinear superconducting circuit 306, cat qubit mode a, is more strongly involved in the ATS 308 than the control qubit mode 305. When the control qubit mode q joins the ATS via a small linear coupling with cat qubit mode a to perform a CNOT gate, typically via a capacitive coupling with a small capacitance value compared to the respective mode capacitances, a mode inductance mediated by a detuned bus resonator, and an inductive coupling with a small inductance value compared to the coupling, this coupling is generally moderate. More precisely, detuning the two coupled modes (here a and q, alternatively b and q) (i.e., frequency difference Δ=|f a -f q |) also plays a role. In fact, if the two modes have the same resonant frequency, even a slight linear combination will result in complete hybridization, φ a ≒φ q This would result in the following. However, for typical detuning Δ / 2π greater than several tens of MHz, the asymmetry of involvement is achieved by standard linear combinations.

[0081] Figures 4, 5, 6, and 7 show several possible circuit implementations of the invention in Figure 3. The microwave source is omitted for simplification but is configured in the same manner as in Figure 3 to operate the circuit. The ATS308 acts as a central reference point to which the rest of the components are connected. For this purpose, buffer-mode filtering and environment are also present.

[0082] As explained earlier, the key lies in the weaker coupling of the control qubit q to the ATS308 (compared to the target cat mode a coupling to the ATS308). In these diagrams, this weak coupling is represented by a small linear dipole. However, as explained earlier, this is typically not sufficient to guarantee the weak coupling described above. As is known in this art, if a linear coupling is characterized by its intensity g, it is also necessary to guarantee that the detuning Δ between the control qubit mode q and the mode it is coupled to (a, 320 or b, 322) satisfies g < Δ. This provides a design rule, but φ a , φ b , φ q To accurately calculate the value of , a complete microwave simulation or circuit diagonalization of the circuit layout is required. This feature helps to guarantee that the CNOT gate is precisely executable, that the circuit uses a single ATS among its components, and that parametric pumping always has a single purpose, as illustrated in the quantum gate diagram in Figure 8.

[0083] In Figure 4, the linear microwave network 320-322 consists of a capacitor galvanically coupled to the ATS308 to form buffer mode 322, and a parallel LC resonator 320 capacitively strongly coupled to the ATS308 to form cat qubit mode 320. The coupling to the ATS308 forms a linear coupler 309. Both modes a and b are strongly coupled to the ATS308, as indicated by the zero-point phase fluctuation of the two modes in the central inductance of the ATS308. The buffer is coupled to the outside world by capacitor 312. Finally, the transmon qubit 305, comprising a capacitor and a Josephson junction, is weakly coupled to cat qubit mode 320 by capacitor 304. The weak coupling should be understood as the involvement of the transmon qubit mode 305 in the ATS308 being smaller than that of the cat qubit 320. An experimental implementation of the illustrated circuit can be seen in Figure 14, where the control qubit is another dissipative cat qubit.

[0084] In Figure 5, the linear microwave network 320-322 consists of two series LC resonators galvanically coupled to the ATS 308 to form buffer mode 322 and cat qubit mode 320. Both modes are strongly coupled to the ATS 308, as indicated by the zero-point phase fluctuation of the two modes in the central inductance of the ATS 308. The buffer is coupled to the outside world by capacitance 312. Finally, the transmon qubit 305, comprising a capacitor and a Josephson junction, is weakly coupled to cat qubit mode 320 by capacitance 304. The weak coupling should be understood as the involvement of the transmon qubit mode 305 in the ATS 308 being smaller than that of cat qubit 320. An experimental implementation of the illustrated circuit can be seen in Figure 12, where the control qubit is another dissipative cat qubit.

[0085] In Figure 6, the linear microwave networks 320-322 are identical to those described in Figure 5. However, the control qubit 305 and its coupling are different. In this case, the control qubit 305 consists of a fraxonium qubit comprising a capacitor, a Josephson junction, and an inductance (which is typically made from a chain of Josephson junctions), configured as a loop through which a thread of magnetic flux passes. This control qubit 305 is inductively weakly coupled to the inductance of a series LC resonator defining the cat qubit mode 320 via a shared inductive portion 304. The weak coupling should be understood as the involvement of the transmon qubit mode 305 in ATS 308 being smaller than that of the cat qubit 320.

[0086] In Figure 7, the linear microwave networks 320-322 are identical to those described in Figure 4. However, the control qubit 305 has a different coupling. In this case, the control qubit 305 consists of a transmon qubit with a capacitor and a Josephson junction, and is weakly coupled to the ATS 308 by capacitance 304, which in its configuration is also the buffer mode 322. In this circuit, the cat qubit mode 320 and the control qubit mode 305 have very similar configurations, mainly differing in the strength of the linear coupling. This figure is provided to highlight that the weak coupling 304 in Figure 3 is a coupling to the linear microwave networks 320-322, and not necessarily to the cat qubit mode 320 only.

[0087] Figure 8 shows a diagram illustrating how the quantum gate in Figure 3 is executed.

[0088] In the first operation 800, which corresponds to the circuit's idling operation (idle mode) when only the cat qubit is stabilized, the quantum system 30 does not perform a CNOT operation. In this operation, the target cat qubit device 300 emits two types of radiation, namely modulating the common flux of the ATS to activate a 2:1 photon nonlinear transformation between the cat qubit mode and the buffer mode, at frequency f p =2f a -f b The parametric pump from the microwave source 310 (in the case of squeezed cat stabilization, as described above, two other parametric pumps are also active) and the second resonant frequency f b In this configuration, the second mode b 322 receives linear drive from the microwave source 316.

[0089] In operation 810, the quantum system 30 is modified to perform a CNOT operation over a selected duration. For that purpose, the radiation from both microwave sources 316 and 310 is temporarily turned off so that data-cat qubit stabilization ceases. As explained previously, it is critically important that the dynamics of the cat qubit mode be brought as close as possible to zero in the frame rotating at frequency f a To activate the CNOT Hamiltonian, microwave source 310 is configured to emit radiation at the control qubit frequency f q .

[0090] As explained previously, this has a first-order effect of strongly driving the control qubit q, which results in leakage or unwanted decoherence of the control qubit q. To cancel this effect, compensating radiation may be turned on with the correct phase and amplitude (experimentally fine-tuned while calibrating the CNOT gate in-situ). Since the goal of counterdrive is to avoid displacement of the control qubit, the natural approach is to send it to the control qubit via microwave source 303. While this approach may also work, it is better first to address the root cause of this displacement by directly counterdriving the ATS responsible for this displacement. This can be accomplished by modulating the differential flux of the ATS with the correct phase and amplitude, pre-calculated analytically and experimentally fine-tunable in-situ, by sending radiation at frequency f q using microwave source 311.

[0091] In operation 820, after the CNOT Hamiltonian has operated on the system for a duration T CX = π / (4αg CX ), the parametric pump at frequency f q from microwave source 310 and its compensation (303 or 311) are turned off. Then, at frequency f pTurn on the microwave source 310 again (and the other two pumps in the case of the squeezed cat), and the frequency f b By turning on the microwave source 316 again at, the two-photon stabilization of the data-cat qubit can be resumed.

[0092] As illustrated in FIG. 8, at any given time, the ATS 308 is under the influence of a single parametric pump, i.e., under the influence of the two-photon conversion pump during the idling / stabilization time and under the influence of the longitudinal pump during the CNOT gate. This minimal configuration ensures that the ATS non-linearity operates as close as possible to its theoretical ideal behavior, resulting in high gate speeds and fidelity. When the squeezed data-cat qubit is stabilized, operations 800 and 820 actually require three parametric pumps, all of which are required for a single purpose, namely, the stabilization of the squeezed cat state. This is what is referred to in the present application as "parametric pumping with a single purpose".

[0093] FIG. 9 shows exemplary signal timing for operations 800, 810, and 820. During the idle mode, the data-cat qubit is stabilized by two-photon dissipation κ2. When the gate starts, this dissipation is turned off so that the CNOT Hamiltonian can become effective. Finally, after time T CX = π / (4αg CX ), the CNOT Hamiltonian is turned off and two-photon dissipation is turned on again. At any given time, the ATS does not operate for two purposes at once, which ensures experimental robustness. Here, the limitations regarding the intensity and waveform (dotted or dashed line) of the CNOT pulse are described.

[0094] As explained above, the CNOT Hamiltonian effectively operates as a linear drive on the control qubit q with an intensity that depends on the photon number in the target cat qubit a. The nature of the control qubit q sets an upper bound on the maximum effective drive intensity g CX with respect to α. There are two cases. In the first case, the control qubit q is defined by a real number or a Hamiltonian gap, and in the case of a two-level system, this gap is an anharmonic |ω|. 12 -ω q | is the case here ω q ω is the qubit frequency, or the frequency of the transition from the ground state to the first excited state. 12 is the frequency of the transition from the first excitation to the second excited state. In that case, the adiabatic theorem applies, and g CX α<|ω 12 -ω q If the | condition is met, the qubit is considered to remain exponentially confined regardless of the effective driving operation. The amplitude α of the ancilla qubit is confined by the Kerr-Miltonian (b). c If it is a cat qubit having, the condition is,

number

number

number

number

number

[0095] In addition to the gate strength requirement, the properties of controlled qubit confinement impose constraints on how the gate should be applied. For dissipative ancilla qubits, as soon as the data cat qubit confinement is turned off (solid line κ2(t)), the CNOT Hamiltonian may be instantaneously turned on, as shown by the dashed line in Figure 9. For Hamiltonian ancilla qubits, pulse g CX The CNOT Hamiltonian should be smoothly turned on so that the spectral content of (t) (dotted line) does not contain frequency components on the gap. Typically, a Gaussian pulse can be used. In Figure 9, a cosine shape is used for its finite-time envelope. Time-dependent g CX In this case, the pulse amplitude is

number

[0096] Figure 10 shows a schematic diagram of a quantum system for executing iterative coding using the quantum gate of the present invention.

[0097] The quantum system in Figure 10 presents a classical iterative code architecture for cat qubits, namely, d data cat qubits 300 (3 or more) and d-1 ancilla qubits 302. In realizing a given mode, such as a circular iterative code, it is also possible to have d ancilla qubits 302 for d data cat qubits 300. Each ancilla qubit 302 is linked to two data cat qubits 300 by a CNOT gate 10. The links between the data cat qubits 4 and the ancilla qubits 302 are such that one data cat qubit 300 is connected to at most two ancilla qubits 302. Each ancilla qubit 302 is also connected to an instrument that measures an X Pauli operator of a number of photons 12, which is used to detect a phase error once a cycle of the error iterative code has been performed.

[0098] As explained in the introduction, the implementation of the repetition code is as follows: - For each Ancillane cat qubit 302, prepare in the |+> or |-> state, which is an eigenstate of the Pauli X operator. - Each CNOT gate 10 is applied such that the state of the Ancilla cat qubit 302 is changed according to the error syndrome of the data cat qubit 300 to which it is connected by each of the two CNOT gates 10. - For each of the 302 ancila qubits, the operator of X is measured.

[0099] Figures 11, 12, and 13 illustrate a practical implementation of the present invention. In this implementation, four dissipative cat qubits 1101, 1102, 1103, and 1104, having similar parameters (except that their frequencies are slightly detuned from each other to selectively engage each mode by emission, and that they do not have modes moved from their usual positions throughout the circuit), are arranged on a circular pattern superconducting chip with their nearest neighbors connected. When the chip is operated, some cat qubits are selected to operate as data cat qubits "a" and some are selected to operate as an ancilla cat qubits "q". This chip can operate an error correction experiment with a distance d=2 (strictly speaking, at d=2 only error detection is possible, and its location cannot be determined, so only error detection is possible). This error detection experiment requires d data qubits and d-1 ancilla qubits, thus 3 qubits. In this chip, there are four cat qubits so that the best 3 cat qubits for performing the error detection experiment can be inductively selected. For example, if qubits 1101, 1102, and 1103 have the best coherence time on the chip, then 1101 and 1103 are selected as data qubits and 1102 is selected as the ancilla qubit. Alternatively, if 1102, 1103, and 1104 are the best qubits on the chip, then 1102 and 1104 are selected as data qubits and 1103 is selected as the ancilla qubit. The fact that all qubits have the same properties makes this flexibility possible. If the data and ancilla qubits are different, then only the selection of ancilla qubits would be possible.

[0100] Figure 11 shows a partial schematic of the actual implementation. In this schematic, for readability, 1101 and 1103 are chosen as two data cat qubits, and 1102 or 1104 as an ancilla cat qubit, hence the labels "a" and "q". Both the data and ancilla cat qubits are dissipative cat qubits. The circuit representation is similar to Figure 5 (except for the control qubit), with a capacitive bus coupling 304 between the data 1101 and ancilla 1102 qubit modes. The microwave radiation source is not shown, but CPW (coplanar waveguide) transmission lines connecting the circuit to the outside, other than the quantum system, are represented as 1120, 1122, and 1124. Lines 1120 and 1122 are responsible for the DC current bias and parametric flux modulation of the ATS308, with the first one mainly working on the right loop and the second one mainly working on the left loop. By pumping any linear combination of the two, the control circuit becomes φ Σ or φ Δ It may be pumped. Line 1124 connects the buffer to a 50Ω environment, thereby enabling loss and drive. The ancilla qubit 1102 has the same input transmission line.

[0101] Figure 12 shows an optical microscope image of a chip with four cat qubits, as described in Figure 11. This chip consists of a superconducting layer 1200 (gray area) deposited on a dielectric (sapphire) substrate 1210, with white areas representing the substrate exposed where the superconducting layer has been removed. Brighter gray areas also correspond to the superconducting layer 1210, where the regularly spaced pores in the superconducting layer confine vortices caused by stray magnetic fields. The lithography pattern corresponds to the circuit representation described in Figure 11. In particular, the bus is coupled by a capacitance 304 corresponding to the CPW line portion. Buffer mode filtering is performed via a λ / 4 stub filter 318, which is capacitively coupled to the buffer and galvanically coupled to the input line (e.g., 1124), i.e., a passband filter at the buffer mode frequency. Sub-region 1220 is not shown in the circuit representation of Figure 11. These correspond to a standard configuration known in this art, comprising two transmon qubits capacitively coupled to two readout resonators, which are filtered by two Parcel filters capacitively coupled to a single input line. This configuration is coupled to a cat qubit mode to perform a Wigner tomography of the cat qubit, as is known in this art.

[0102] Figure 13 is an enlarged view of the dotted region 1300 in Figure 12, corresponding to the data qubit 1101. This corresponds to a typical stabilized data qubit device (in the case of the implementation in Figure 12, it is also used as an ancilla qubit in Figure 13). The components of the circuit representation in Figure 11 are readily recognizable, in particular the lumped-parameter inductances 1110 and 1112, which are understood as a Josephson junction array. The capacitively coupled bus 1310 couples cat qubit modes to the transmon qubit used for its tomography.

[0103] To give a typical value, in this device, the zero-point phase variation across the ATS308 is φ with respect to the cat qubit mode of this cell. a= 0.136, and regarding the buffer mode of this cell φ b = 0.186, and for the 1102 cat qubit mode acting as an ancilla qubit in this configuration, φ q The simulation is performed so that =0.019. The data cat qubit 1101 is also coupled to the ancilla qubit 1104 to perform other CNOT gates, so the zero-point fluctuation of the phase of the cat qubit mode of 1104 is also φ q ' = 0.018. These values ​​are constrained by φ q ≦φ a Verifying / 2 ensures a small parasitic term in CNOT Hamiltonian engineering.

[0104] Figure 14 shows an optical microscope image of a chip corresponding to another embodiment of the present invention. This chip is configured similarly to the equivalent circuit shown in Figure 7, except that the control qubit is another stabilized cat qubit. This chip consists of a superconducting layer 1200 (gray area) deposited on a dielectric (sapphire) substrate 1210, with darker gray areas representing areas where the superconducting layer has been removed and the substrate is visible. Wire bonding 1415 can be seen, which is used to balance the potential across the chip regardless of the circuit pattern and to connect the chip to the rest of the circuit. The chip comprises a target cat qubit device a / b, where the b-mode consists of an ATS (1500, see Figure 15) with a capacitive shunt to ground, which is capacitively coupled to a λ / 2 CPW resonator operating as cat qubit mode a. The buffer mode is dissipative by capacitive coupling to the input drive line 1124 via a λ / 4 stub filter 318, i.e., a passband filter at the buffer mode frequency, similar to Figure 12. The control qubit consists of another stabilized cat qubit device, where the cat qubit mode q consists of another λ / 2 CPW resonator coupled in a similar manner to its own buffer mode. The two cat qubits are coupled to a tomography system 1420, which includes a transmon qubit and a readout resonator, as in the case of Figure 12. In this chip, the bias T-junction 1430, mentioned with reference to Figure 3, is also represented to input a DC current in the RF flux lines (e.g., 1122 and 1120) and to bias the ATS at its DC operating point.

[0105] Figure 15 is an enlarged view of the dotted region 1500 in Figure 14, corresponding to the buffer mode of the data cat qubit device. The components of the circuit representation in Figure 4 are readily recognizable (except for the control qubit), and in particular, the ATS 308 is recognizable, which is directly coupled to a capacitance that is grounded to form buffer mode b 322, and to which cat qubit mode a 320 is capacitively coupled 309. The buffer mode is capacitively coupled to the environment via a drive line 1124 having a capacitor 312. The control qubit q 305 is finally capacitively coupled to the buffer mode via a capacitor 304 to participate in the ATS. Capacitor 304 is only slightly smaller than capacitor 309 in both dimensions and value. Finally, the smaller participation of the control qubit q in the ATS compared to one of the cat qubit modes a is ensured by greater detuning between q and b than between a and b. For illustrative purposes, the phase frequency and zero-point fluctuations in this device are given by the following equation: f a =4.82 GHz,f b = 5.38 GHz, f q =4.46 GHz;φ a =0.062,φ b =0.23,φ q =0.032. In this device, φ q ≒φ a If it's / 2, then it's sufficient to implement the CNOT Hamiltonian.

Claims

1. A quantum system for executing a CNOT gate, The above quantum system comprises a command circuit (8), a target cat qubit device (4), and a control superconducting qubit device (6). The above command circuit (8) provides microwave radiation, The above target cat qubit device (4) comprises a superconducting quantum interferometer (308) having asymmetric threads, the superconducting quantum interferometer (308) having asymmetric threads has flux lines through which radiation propagates that can be transmitted to modulate common flux and / or differential flux, and is connected to at least one resonant section (320-322), The above target cat qubit device (300) has a first mode (a) having a first resonant frequency and a second mode (b) having a second resonant frequency. The superconducting quantum interferometer (308) having the asymmetric thread described above is configured such that when the command circuit (8) drives the second mode (b) by transmitting radiation having the second resonant frequency to the at least one resonant section (320-322), and modulates the common magnetic flux at a frequency equal to twice the first resonant frequency and the absolute value of the difference between the second resonant frequency by transmitting radiation in the magnetic flux line, the target cat qubit device (4) is configured to stabilize the target cat qubit having the first resonant frequency. The above-described controlled superconducting qubit device (6) has a third mode (q) having a third resonant frequency for hosting a control qubit (305), and is configured such that Rabi oscillation of the third mode (q) is induced when exposed to radiation having the third resonant frequency at an intensity lower than the control qubit confinement rate. The above-mentioned controlled superconducting qubit device (6) is linearly coupled to the target cat qubit device (4) such that the third mode (q) is linearly coupled to the superconducting quantum interferometer (308) having the asymmetric thread. The command circuit (8) described above is configured to operate the CNOT gate (10) between the target cat qubit and the control qubit by transmitting only radiation with a frequency equal to the third resonant frequency in the magnetic flux line during the CNOT gate time window, and to drive the second mode (b) by transmitting radiation having the second resonant frequency to at least one resonant section (320-322) outside the CNOT gate time window, and is configured to modulate the common magnetic flux at a frequency equal to twice the first resonant frequency and the absolute value of the difference between the first and second resonant frequencies by transmitting radiation in the magnetic flux line. Quantum systems.

2. The above target cat qubit device (4) and the above control superconducting device (6) have a phase difference across the superconducting quantum interferometer (308) having the above asymmetric thread. [Math 1] As shown, they are linearly combined, Here, a is the photon annihilation operator of the first mode (a) described above, φ ａ This is the zero-point fluctuation of the phase of the first mode (a) across the superconducting quantum interferometer (308) having the asymmetric threads described above. b is the photon annihilation operator for the second mode (b) described above, φ ｂ This is the zero-point variation of the phase of the second mode (b) across the superconducting quantum interferometer (308) having the asymmetric thread described above. q is the photon annihilation operator of the third mode (q) described above, φ ｑ This is the zero-point variation of the phase of the third mode (q) across the superconducting quantum interferometer (308) having the asymmetric thread described above. φ ｑ ≦φ ａ Satisfying / 2, The quantum system according to claim 1.

3. The above command circuit (8) controls the amplitude ε during the CNOT gate time window. ＣＸ The system is configured to pump the magnetic flux lines by radiation only at a frequency equal to the third resonant frequency, thereby, [Math 2] A Hamiltonian having the following characteristics is obtained: Here, H ＣＸ is, formula [Math 3] and [Math 4] It is a CNOT Hamiltonian having the following characteristics: a is the photon annihilation operator of the first mode (a) described above, φ ａ This is the zero-point fluctuation of the phase of the first mode (a) across the superconducting quantum interferometer (308) having the asymmetric threads described above. q is the photon annihilation operator of the third mode (q) described above, φ ｑ is the zero-point fluctuation of the phase of the third mode (q) across the superconducting quantum interference device (308) having the above asymmetric thread, E Ｊ This is the Josephson energy of the transverse junction of the superconducting quantum interferometer (308) having the above-mentioned asymmetric thread, α ２ This is the number of photons in the first mode (a) described above, The above command circuit (8) is, during the CNOT gate time window, the formula [Math 5] Further configured to produce a compensated Hamiltonian having The quantum system according to claim 1 or 2.

4. The command circuit (8) above transmits only radiation with a frequency equal to the third resonant frequency in the magnetic flux line, thus the equation [Math 6] The differential magnetic flux φ substantially comprises Δ By modulating the above CNOT gate time window, the formula [Number 7] It is configured to produce a compensated Hamiltonian having, Here, φ Σ This is the common magnetic flux, E Ｌ This is the inductive energy of the central inductance of the superconducting quantum interferometer (308) having the above-mentioned asymmetric thread. The quantum system according to claim 3.

5. The command circuit (8) transmits radiation at a frequency equal to the third resonant frequency to the control qubit device (305), thereby during the CNOT gate time window, the equation [Number 8] Configured to produce a compensated Hamiltonian having The quantum system according to claim 3.

6. The above-mentioned controlled superconducting qubit device (6) hosts a transmon qubit, a flux qubit, a flaxonium qubit, a cat qubit, or any bosonic qubit encoded in a resonator. A quantum system according to one of claims 1 to 5.

7. The superconducting quantum interferometer (308) having the asymmetric thread described above is further configured such that the command circuit (8) drives the second mode (b) by transmitting radiation having the second resonant frequency, and modulates the common magnetic flux at a frequency equal to the sum of the second resonant frequency and twice the first resonant frequency and twice the first resonant frequency by transmitting radiation in the magnetic flux line, the target cat qubit device (4) is further configured to stabilize the target squeezed cat qubit having the first resonant frequency. A quantum system according to one of claims 1 to 6.

8. The target cat qubit device (4) and the control superconducting qubit device (6) are configured such that the absolute value of the difference between the first resonant frequency and the third resonant frequency exceeds 10 MHz. A quantum system according to any one of claims 1 to 7.

9. The command circuit (8) above modulates the common magnetic flux by transmitting radiation in the magnetic flux line, using only radiation at a frequency equal to the third resonant frequency, and maintaining a constant intensity g ＣＸ Configured to produce a CNOT Hamiltonian having A quantum system according to any one of claims 1 to 8.

10. The command circuit (8) above modulates the common magnetic flux by transmitting radiation in the magnetic flux line, using only radiation with a frequency equal to the third resonant frequency. [Number 9] The time-dependent intensity g satisfies this condition. ＣＸ Configured to produce a CNOT Hamiltonian having (t), A quantum system according to any one of claims 1 to 8.

11. A quantum system for executing iterative codes, The above quantum system comprises a command circuit (8), d data cat qubit devices (4) which are 2 or more, and d-1 ancilla qubit devices (6), The above command circuit (8) selectively applies microwave radiation, Each data cat qubit device (4) has a resonant frequency and is coupled to the command circuit (8) to stabilize each data cat qubit. Each ancilla qubit device has an ancilla resonant frequency coupled to the command circuit (8) for hosting an ancilla qubit, Each of the d-1 ancilla qubit devices is linearly coupled to each of the d data cat qubit devices, Each of the d data cat qubit devices is connected to at most two ancila qubit devices out of the d-1 ancila qubit devices. The d data cat qubit devices (4) and the d-1 ancilla qubit devices (6) are configured such that each connection between a certain ancilla qubit device (6) and a certain data cat qubit device (4) is controllable by the command circuit (8) to realize the quantum system described in any of claims 1 to 10. The above command circuit (8) is, 1) For each of the d-1 ancilla qubit devices, prepare each ancilla qubit in the d-1 ancilla qubit devices by the state of the X operator "|+>" or "|->", 2) For each of the d-1 ancilla qubit devices, a CNOT gate is executed by the first of the two data cat qubit devices to which it is coupled, and then a CNOT gate is executed by the second of the two data cat qubit devices to which it is coupled. 3) Applying the measurement to the operator X in each of the above ancilla qubit devices It is further configured to perform d-1 quantum operations. Quantum systems.

12. All ancilla qubit devices host cat qubits, just as all data cat qubit devices and all ancilla qubit devices are cat qubit devices. The above command circuit (8) defines which cat qubit device is a data cat qubit device (4) and which cat qubit device is an ancilla qubit device (6). The quantum system according to claim 11.

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