Quantum system for executing CNOT gates and quantum system for executing duplex codes using

Abstract

A quantum system for executing a CNOT gate includes a control circuit that provides radiation, a target cat qubit device and a control qubit device that are linearly coupled. The target cat quantum bit comprises a non-linear element, namely ATS, and has two purposes: the first purpose is to design two-photon conversion Hamiltonian for cat quantum bit stabilization, and the second purpose is to design CNOT Hamiltonian for executing a CNOT gate together with a quantum bit control device. At any time point, the ATS plays a role of cat quantum bit stabilization or plays a role of a CNOT gate.

Description

[0001] This invention relates to a system for executing quantum gates, and more particularly, to the application of such gates in cat-state qubit environments.

[0002] To extract joint information from multiple data qubits, conventional quantum circuits perform a series of two-qubit gate operations (typically CNOT or CZ gates) between the auxiliary qubit and the multiple data qubits before measuring the auxiliary qubit. This operation is often referred to as companion measurement.

[0003] Implementing these quantum gates is crucial for detecting errors and executing quantum error-correcting codes (or "QECC"), which are currently considered the only way to build reliable and usable quantum processors. The basic idea of ​​quantum error-correcting codes is to use multiple (at least two) physical data qubits to encode a logical qubit, and to employ an error detection scheme to verify that the information of these physical data qubits does not change over time. If an error is detected, correction is performed, either by changing the state of the physical data qubits themselves or by post-processing the results of quantum algorithms involving these qubits.

[0004] The quality of a quantum gate is assessed based on its execution time (i.e., the time required for gate operation) and the associated error probability. For a cat-state qubit, this error probability depends on the ratio κ1 / κ2, where κ1 is the single-photon loss rate (i.e., error rate) of the qubit used to execute the gate, and κ is the two-photon loss rate (i.e., correction or stabilization rate) of the qubit used to execute the gate.

[0005] One of the current goals of quantum hardware experiments is to realize quantum gates so that repetition codes become more efficient as the distance between error-correcting codes used to implement error-correction schemes (which is related to the number of physical qubits used to encode logic qubits) increases. Currently, it is necessary to control the ratio κ1 / κ2 within 5*10^5. -3 The following is a summary of previous references on CNOT dynamics for cat qubit quantum error-correcting codes (QECC), which rely on the following steps: defining a control qubit characterized by its annihilation operator q and resonance frequency f. q Define a target qubit, which consists of a two-photon driven and dissipatively stable cat qubit, characterized by its annihilation operator a and resonant frequency f. a Design a stabilization scheme for the target qubit, ensuring its phase is controlled by the state of the control qubit; and design the longitudinal coupling Hamiltonian between the control qubit and the target qubit, the general formula of which is: Where α is a complex number that defines the amplitude of the target cat qubit.

[0006] In the following text, the terms "longitudinal Hamiltonian", "longitudinal coupling", or CNOT or CX Hamiltonian are used interchangeably and all refer to this longitudinal coupling Hamiltonian between the control qubit and the target cat qubit.

[0007] This overall approach raises two fundamental problems. The first problem is that very few experimental implementations have been proposed. The second problem 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 drivers, and / or fail to address all the engineering challenges. These experimental proposals include Guillaud et al.'s article "Repeating cat qubits for fault-tolerant quantum computing," published in Physical Review X, Volume 9, page 041053, at https: / / doi.org / 10.1103 / PhysRevX.9.041053; Chamberland et al.'s article "Building fault-tolerant quantum computers using cascaded cat codes," published in Physical Review Letters Quantum, Volume 3, page 010329, at https: / / doi.org / 10.1103 / PRXQuantum.3.010329; and Gautier et al.'s article "Combining dissipation and Hamiltonian constraints in cat qubits," published in Physical Review Letters Quantum, Volume 3, page 020339, at https: / / doi.org / 10.1103 / PRXQuantum.3.020339.

[0008] This invention aims to improve this situation. To this end, the applicant proposes a quantum system for executing CNOT gates, the system comprising:

[0009] - Control circuitry used to provide microwave radiation

[0010] A target cat qubit device includes an asymmetric magnetically penetrating superconducting quantum interference device (MPU) having flux lines through which radiation can be transmitted to modulate a common magnetic flux and / or a differential magnetic flux. The device is connected to at least one resonant section. The target cat qubit device has a first mode and a second mode. The first mode has a first resonant frequency, and the second mode has a second resonant frequency. The asymmetric MPU is configured to stabilize the target cat qubit having the first resonant frequency when the control circuit transmits radiation with the second resonant frequency to the at least one resonant section to drive the second mode and transmits radiation in the flux lines to modulate the common magnetic flux at a frequency equal to twice the absolute value of the difference between the first and second resonant frequencies.

[0011] A control superconducting quantum bit device, having a third mode and a third resonant frequency, for carrying control qubits, is configured such that Rabi oscillations are induced when the third mode is subjected to radiation of the third resonant frequency with an intensity less than the control qubit confinement rate. The control superconducting quantum bit device is linearly coupled to the target cat quantum bit device, such that the third mode is linearly coupled to the asymmetric magnetically penetrating superconducting quantum interference device. The control circuit is configured to transmit radiation with a frequency equal to the third resonant frequency only on the flux line within a CNOT gate time window to operate the CNOT gate between the target cat quantum bit and the control qubit; outside the CNOT gate time window, it is configured to transmit radiation with a second resonant frequency to the at least one resonant portion to drive the second mode and to transmit radiation on the flux line to modulate the common flux at a frequency equal to the absolute value of the difference between twice the first resonant frequency and the second resonant frequency.

[0012] This system has advantages because it selects the fewest options from a general list of components and proposes a practical implementation using minimal hardware and control drivers or parameter pumps. This reduces implementation complexity, thereby reducing potential points of failure. The system also addresses the impact of some non-ideal factors in practical implementation.

[0013] In various embodiments, the method may have 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 on the asymmetric transmagnetic superconducting quantum interference device satisfies Where 'a' is the photon annihilation operator of the first mode. Let be the phase zero-point fluctuation of the first mode on the asymmetric transmagnetic superconducting quantum interference device, and let b be the photon annihilation operator of the second mode. Let q represent the phase zero-point fluctuation of the second mode on the asymmetric transmagnetic superconducting quantum interference device, and let q be the photon annihilation operator of the third mode. The third mode represents the phase zero-point fluctuation on the asymmetric transmagnetic superconducting quantum interference device, and

[0015] - The control circuit is configured to inject an amplitude of ∈ into the magnetic flux lines only at a frequency equal to the third resonant frequency within the CNOT gate time window. cx The radiation from this induces a Hamiltonian, whose formula is: Where H CX It is the CNOT Hamiltonian, and its formula is: 'a' is the photon annihilation operator for the first mode. q represents the phase zero-point fluctuation of the first mode on the asymmetric transconducting superconducting quantum interference device, and q is the photon annihilation operator of the third mode. It is the phase zero-point fluctuation of the third mode on the asymmetric transconducting superconducting quantum interference device, F J It is the Josephson energy of the side junction of an asymmetric transconducting superconducting quantum interference device, α. 2 This is the number of photons in the first mode. The control circuit is also configured to induce a compensated Hamiltonian within the CNOT gate time window, the formula of which is:

[0016] The control circuit is configured to modulate the differential magnetic flux during the CNOT gate time window by transmitting radiation with a frequency equal to the third resonant frequency only in the flux lines. This leads to the following formula: The compensated Hamiltonian, the differential flux The formula is in It is the common magnetic flux, E L It is the induced energy of the central inductor of an asymmetric transconducting superconducting quantum interference device.

[0017] - The control circuit is configured to induce the following formula during the CNOT gate time window by transmitting radiation with a frequency equal to the third resonant frequency to the control qubit device: The compensation Hamiltonian.

[0018] The controlled superconducting quantum bit device carries an anti-quantum bit, flux quantum bit, flux-sub quantum bit, cat-state quantum bit, or any boson quantum bit encoded in a resonator.

[0019] The asymmetric transmagnetic superconducting quantum interference device is further configured such that when the control circuit transmits radiation with the second resonant frequency to drive the second mode and transmits radiation in the flux lines to modulate the common magnetic flux, and the modulation frequency is equal to the absolute value of the difference between twice the first resonant frequency and the second resonant frequency, the second resonant frequency, and the sum of twice the first resonant frequency and the second resonant frequency, the target cat qubit device stabilizes the target compressed cat qubit with the first resonant frequency.

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

[0021] The control circuit is configured to transmit radiation in the magnetic flux lines to modulate the common magnetic flux using only radiation with a frequency equal to the third resonant frequency, thereby inducing a magnetic flux with a constant intensity g.cx The CNOT Hamiltonian, and

[0022] The control circuit is configured to transmit radiation in the magnetic flux lines to modulate the common magnetic flux using only radiation with a frequency equal to the third resonant frequency, thereby inducing a CNOT Hamiltonian g whose intensity varies with time. CX (t), such that ∫4αg CX (t)dt=π.

[0023] The present invention also relates to a quantum system for executing repeating codes, the system comprising a control circuit for selectively applying microwave radiation, d data cat qubit devices, and d-1 auxiliary qubit devices, wherein d is equal to or greater than 2. Each data cat qubit device has a respective resonant frequency and is coupled to the control circuit to stabilize its respective data cat qubit. Each auxiliary qubit device has an auxiliary resonant frequency coupled to the control circuit for carrying one auxiliary qubit. Each of the d-1 auxiliary qubit devices is linearly coupled to two of the respective data cat qubit devices, and each of the d data cat qubit devices is connected to at most two of the d-1 auxiliary qubit devices. The d data cat qubit devices and the d-1 auxiliary qubit devices enable each coupling between the data cat qubit devices and the auxiliary qubit devices to be controlled by the control circuit to realize the quantum system according to any one of the preceding claims. The control circuit is further configured to perform d-1 quantum operations in such a way that:

[0024] 1) For each of the d-1 auxiliary qubit devices, prepare an auxiliary qubit with the X operator state "|+>" or "|->" in each of the d-1 auxiliary qubit devices.

[0025] 2) For each of the d-1 auxiliary qubit devices, firstly, the first of the two data cat qubit devices coupled to it performs a CNOT gate operation, and then the second of the two data cat qubit devices coupled to it performs a CNOT gate operation.

[0026] 3) Measure the operator X on each of the auxiliary qubit devices.

[0027] In the quantum system described above, all auxiliary qubit devices carry cat qubits, such that all data cat qubit devices and all auxiliary qubit devices are cat qubit devices, and the control circuit defines which cat qubit device is a data cat qubit device and which cat qubit device is an auxiliary qubit device.

[0028] Other features and advantages of the present invention will become apparent from the following description of the accompanying drawings, which illustrate exemplary embodiments of the invention, in which:

[0029] - Figure 1 A quantum circuit diagram for implementing phase-flipping repeating codes is shown.

[0030] - Figure 2 A general schematic diagram of a conventional CNOT gate is shown, which controls the connection between the qubit and the target cat qubit.

[0031] - Figure 3 A general schematic diagram of a quantum system that performs CNOT gates according to the present invention is shown.

[0032] - Figures 4 to 7 The target cat qubit device, the control qubit device, and... are shown. Figure 3 Various embodiments of linear coupling between the central control qubit device and the target cat qubit device.

[0033] - Figure 8 It explains how to execute. Figure 3 A schematic diagram of quantum gates in the image.

[0034] - Figure 9 It shows Figure 8 The two vertical coupling establishment schemes shown in the operation are as follows:

[0035] - Figure 10 A general schematic diagram of a quantum system for executing repeating codes is shown, the system comprising the CNOT gate according to the present invention.

[0036] - Figure 11 A circuit diagram of the repeat code according to the present invention is shown.

[0037] - Figure 12 A photograph of an experimental quantum circuit using CNOT gates according to the present invention is shown, and...

[0038] - Figure 13 It shows Figure 12 A close-up of a certain part of the film.

[0039] - Figure 14 A photograph of an experimental quantum circuit implementing a CNOT gate is shown, and...

[0040] - Figure 15 It shows Figure 14 A close-up of a certain part of the film.

[0041] The accompanying drawings and the following description consist largely of clear and distinct features. Therefore, they not only aid in understanding the invention but can also be used to refine the definition of the invention if necessary.

[0042] This invention relates to the realization of high-performance quantum gates in the context of cat qubits.

[0043] It is known that stable cat-state qubits can benefit from noise bias. More specifically, as the "size" (i.e., the average number of photons) of the Schrödinger cat state of the cat-state qubit increases, effective error channels (e.g., bit errors or "bit flips") are suppressed exponentially.

[0044] Based on current knowledge, this suppression should apply to a large class of physical noise processes that have a local effect on the phase space of the harmonic oscillator. These include, but are not limited to, photon loss, thermal excitation, photon decoherence, and various nonlinear effects caused by coupling with the Josephson junction.

[0045] Recent experiments in the context of quantum superconducting circuits have shown that bit-flipping errors are exponentially suppressed as the average number of photons in the cat state increases.

[0046] Because of this noisy structure, it is assumed that using a single repetition code is sufficient to correct the remaining errors in the channel. In fact, if only phase flipping correction is desired, then a phase flipping error-correcting code alone is sufficient. This could be, for example, a repetition code defined in a bibasic code, or any other state-of-the-art error-correcting code.

[0047] In the following, the basis conventions for states and the Pauli operator are as follows: the driving at the resonant frequency of the qubit (or cat qubit), also known as the Rabi driving, corresponds to the Z-rotation. This is a common basis choice in the cat qubit field, but less common for the more standard two-level system qubits. This choice is made in the following to give cat qubits and two-level system qubits the same general notation. Specifically, in the following, the states |+> and |-> correspond to the ground state and excited state of a two-level system qubit, respectively. For a cat qubit, the states |+> and ||-> represent the even and odd Schrödinger cat states ∝ |α>±|-α>, where the state |α> is a coherent state with a complex amplitude α.

[0048] Figure 1A quantum circuit diagram for executing a phase-flipping repeating code is shown. The cat qubit repeating code is constructed using d cat qubits 4 (called data cat qubits), where logical information is encoded. The code operates by repeatedly measuring quantum operators that reveal whether some error has occurred on the data cat qubits. This is achieved by using d-1 additional qubits 6 (called auxiliary qubits or auxiliary quantum bits). The quantum circuit for the repeating code requires preparing the auxiliary qubits to be in either the |+> or |-> state, two CNOT gates between the auxiliary qubits and the data cat qubit 10, and a measurement of the Pauli X operator 12 (this is equivalent to the parity of photon or excitation number, a definition valid for both cat qubits and two-level system qubits; for two-level system qubits, this also corresponds to the measurement of the standard ground state / excited state). In the following text, the auxiliary qubits are also referred to as control qubits, and the data cat qubits are also referred to as target cat qubits, depending on whether the context is a repeating code (auxiliary / data) or a single CNOT gate (control / target).

[0049] The challenge in implementing this repeating code lies in running it on hardware with a fault tolerance threshold. This means that the fidelity of quantum operations in the circuit must be extremely high for the code to have a positive impact.

[0050] More precisely, when the repetition code operates above the threshold, that is, when the fidelity of the physical operations constituting the repetition code is insufficient, the lifetime of the logical information will be shortened as the number of physical data qubits d increases: new errors introduced by increasing the quantum system cannot be compensated by error correction strategies.

[0051] On the other hand, when the repetition code is running below the error correction threshold, that is, when the fidelity of the physical operation is sufficient, the lifetime of the logical information increases exponentially with the number of physical data qubits d (also known as the distance d of the repetition code).

[0052] exist Figure 1 The diagram only shows 3 physical data qubits (data cat qubit 4) and 2 auxiliary qubits. However, it illustrates how various qubits can be connected on a quantum circuit to enable the execution of phase-flipping repeating codes.

[0053] The field of quantum computing is still quite young, especially in the area of ​​cat qubits. In many ways, it resembles a research field. Therefore, the preferred way to make progress is to make changes that may seem minor at first glance, but in reality, these changes require a great deal of physical work to verify and industrialize. In other words, nothing considered state-of-the-art technology is typically changed until a major hurdle is discovered. This means that known effective solutions are not easily replaced.

[0054] In this invention, a qubit refers to any two-level system or effective two-level system implemented in the form of a superconducting circuit, such as a boson qubit encoded in a resonator. These boson qubits include cat-state qubits.

[0055] In this invention, unless otherwise stated, the term "cat qubit" refers to any form of cat qubit implementation, particularly the two-photon dissipative Schrödinger cat qubit. Alternatively, other forms of cat qubits may also be used.

[0056] Such cat-state qubits can be stabilized or constrained using the following example schemes:

[0057] a) Dissipative stabilization, its jump operator is Where κ² is the two-photon dissipation rate, a is the photon annihilation operator, and α is the complex number defining the cat qubit. This hopping operator can be used to switch between qubits with a dissipation rate κ². b The lossy buffered mode b is coupled to the cat qubit mode by a four-wave mixer (usually a Josephson junction or an ATS—an asymmetric transmagnetic superconducting quantum interference device), and the Hamiltonian is designed. This is achieved by, where b is a buffered-mode photon annihilation operator with a frequency of 2f. a -f b The pump, and the frequency f b The buffered mode is driven under the condition that g2 < k b .

[0058] b) A Kerr Hamiltonian Where K is the complex amplitude of the Kerr Hamiltonian, a is the photon annihilation operator, and α 2 It is the average number of photons.

[0059] c) A detuned Kerr Hamiltonian Where K is the complex amplitude of the Kerr Hamiltonian, a is the photon annihilation operator, α is the complex number defining the cat qubit, and Δ is the detuning factor.

[0060] d) Two-photon exchange (TPE) Hamiltonian Where g2 is the complex amplitude of the TPE Hamiltonian, a is the photon annihilation operator, α is the complex number defining the cat qubit, and σ ± It is the dimensionality reduction and dimensionality increase operator for a two-level system. This Hamiltonian can be designed in the same way as dissipative stabilization a).

[0061] e) A dissipative compression stabilization method, wherein the jump operator of the dissipative compression stabilization method is: Where κ² is the two-photon dissipation rate, a is the photon annihilation operator, α is the complex number defining the cat qubit, and ξ = reiθ For complex compression parameters. This method stabilizes "compressed cat qubits," also known as shift-compressed vacuum states. This hopping operator can be used to convert qubits with a dissipation rate κ. b The lossy buffered mode b is coupled to the cat qubit mode by a four-wave mixer (typically a Josephson junction or an automatic tuning system (ATS)) and the Hamiltonian is designed. To achieve this, where b is the photon annihilation operator of mode b, having multiple frequencies of 2f. a -f b ,f b f b and 2f a +f b The pump, and at frequency f b The condition for driving the buffered mode is g2 < κ. b .

[0062] f) A resonant dissipative stabilization method, whose jump operator is: Where κ² is the two-photon dissipation rate, a is the photon annihilation operator, and α is the complex number defining the cat qubit. This hopping operator can be used to switch between qubits with a dissipation rate κ². b This is achieved by coupling a lossy buffered mode b and a three-wave mixer with the cat qubit mode a. This coupling operation is designed with a Hamiltonian. Where b is the photon annihilation operator for the buffered mode, assuming the mode frequency satisfies 2f. a =f b And g2 < κ b At this time, at frequency f b Apply a drive to the buffered mode.

[0063] If the auxiliary qubit is a cat qubit, it can be stabilized using one of the six stabilization methods described above. For the target cat qubit, only stabilization mechanisms a) and e) will be described below. In fact, as explained below, the control circuit should be able to turn the stabilization of the target cat qubit on or off. For stabilization mechanism a), this can be achieved by turning off the parametric pumps responsible for the 2:1 photon conversion. For stabilization mechanism e), all parametric pumps need to be turned off.

[0064] In previous schemes, the theoretical approach to implementing a CNOT gate between a control qubit with the annihilation operator q and a stable cat qubit with the annihilation operator a typically relies on the following three elements:

[0065] 1) Controlling the qubit's limitations allows microwave drive at its resonant frequency to produce Rabi oscillations. This is typically natural in two-level qubit systems, while for cat-state qubits, it is designed through parameter interactions.

[0066] 2) Add a 'CNOT' Hamiltonian or a 'vertical' Hamiltonian to the circuit, the formula of which is: Where g CX This is the amplitude of the Hamiltonian (chosen as a real number for lack of generality). This longitudinal coupling can be viewed as a driver of the control qubit (the first factor), whose amplitude depends on the number of photons in the target cat qubit (the second factor). For this Hamiltonian to be effective for the target cat qubit, the constraints on the target cat qubit also need to be turned off; therefore, the constraint strength of the target cat qubit should be controllable.

[0067] 3) The performance of conditionally constraining the target cat qubit, which depends on the state of the control qubit. For example, in the case of a two-photon dissipative target cat qubit and a control cat qubit (with the annihilation operator q and the same amplitude α as the target cat qubit), such conditional stability can be achieved through a hopping operator. To achieve this, where κ2 is the two-photon dissipation rate required for the target cat qubit.

[0068] Figure 2 A general schematic diagram of a CNOT gate is shown. The left side shows the quantum circuit representation of the CNOT gate. On the right side, a target cat qubit 4 and a control qubit 6 are both controlled by a control circuit 8. The target cat qubit 4 and the control qubit 6 are connected to execute the CNOT gate 10.

[0069] Generally, a cat qubit is defined as an efficient two-level system that exists stably within a resonator embedded in a specific superconducting circuit that receives specific radiation controlled by a control circuit. Therefore, a cat qubit can typically be specified as a resonator with a specific resonant frequency (i.e., the frequency of the cat qubit), which is controlled by the control circuit. Each cat qubit can be controlled by a specific control circuit, or a single circuit can be configured to control all cat qubits in a given circuit. In the example described here, a single control circuit 8 controls the target cat qubit 4 and the control qubit 6. In the following text, to emphasize that the cat qubit actually exists in a normal mode of the superconducting circuit (typically involving multiple bare resonators in strong coupling), the term "cat qubit mode" may be used instead of "cat qubit resonator."

[0070] As previously mentioned, dissipative stabilization of cat qubits (scheme a or e) requires designing a nonlinear transition between two photons of the first mode carrying the stable quantum manifold, also known as cat qubit mode a, and one photon of the second mode, called buffer mode b, and vice versa. This stabilization scheme can exponentially suppress bit flipping as the number of photons in the two coherent states increases. However, it is only effective when the confinement rate of the two coherent states is greater than the escape rate caused by external noise sources. The confinement rate is directly related to a 2:1 photon conversion rate.

[0071] The first realizations of this stable scheme (Leghtas et al., Science 347,853 (2015) "Confining the state of light to a quantum manifold by engineered two-photon loss" and Touzard et al., Physical Review X 8,021005 (2018) "Coherent oscillations within a dissipative stable quantum manifold") failed to observe exponential suppression of bit flips because the superconducting circuit elements used for engineered 2-to-1 photon conversion (i.e., so-called transoms) have spurious cross-Kerr terms that introduce additional noise processes whose escape rate is determined by the very large transom-cat quantum ratio dispersion shift. Lescanne R. et al.'s 2020 article in Nature Physics, "Exponential suppression of bit flips in qubits encoded in an oscillator" (hereinafter referred to as Lescanne 2020), disclosed an improved method for realizing cat qubits, using an asymmetric magnetically penetrating superconducting quantum interference device (also known as an "ATS") to design a 2:1 photon conversion. The ATS design has a much lower cross-Kerr term than the transect, thus enabling the observation of exponential suppression of bit flips. However, the transect is also used to measure the cat qubit state.

[0072] To date, this dissipative stabilization remains the most common approach for cat qubits, and the use of one or more auxiliary transmission systems (ATS) is the basis for the best-performing experimental designs.

[0073] To improve the experimental implementation of the CNOT gate, the applicant has minimized the number of theoretical CNOT components, parametric pumps and actuators, and nonlinear elements that need to be designed. The applicant has also carefully considered the parasitic dynamics resulting from this engineering design and provided solutions to limit its negative impacts.

[0074] This invention demonstrates two excellent circuit design options.

[0075] The first design choice was that the Automated Test System (ATS) used to design two-photon stability during idle periods was the only source of nonlinearity required to perform the CNOT gate. This choice helped reduce design complexity.

[0076] The second design choice was that, although component 3) improves the fidelity of the CNOT (Controlled Inverse Gate), it is not critical to its implementation. Therefore, the implementation of component 3) was deliberately avoided. This eliminates the need for conditional stabilization and the associated engineering challenges (direct exchange of Hamiltonians between the control qubit and the target cat qubit buffer, as well as two parametric pumps). Although the data qubit is not stabilized during CNOT gate operation, this is acceptable as long as the gate operation speed is much faster than the noise or defects protected by stabilization (such as the Kerr effect due to the inherent nonlinearity of the resonator). Or decoherence due to mode frequency fluctuations This will not affect bit flip suppression. If these conditions are met, with the interleaving operation of the CNOT gates, two-photon stabilization will be effectively enabled in a stroboscopic manner while maintaining the noise bias of the datacat qubits.

[0077] With these two design options, CNOT (controlled NOT gate) operations can be performed by manipulating only the longitudinal Hamiltonian of the cat qubit device's ATS. Since stabilization is off when longitudinal coupling is on, the ATS can be used to manipulate two terms (i.e., stabilization and longitudinal coupling) sequentially without incurring the risk of parameter instability, which is more likely to occur when a single nonlinear element is driven by multiple frequencies simultaneously.

[0078] As a reminder, when the automatic tuning system (ATS) embedded in a cat qubit device shifts its operating point, its Hamiltonian has a "sin sin" form.

[0079]

[0080] in, This represents the total superconducting phase difference through the auxiliary transmission line (ATS). This represents the phase zero-point fluctuation of the cat qubit mode via the auxiliary transmission line. E represents the phase zero ripple in the buffered mode of the auxiliary transmission line. J E represents the Josephson energy of the lateral junction. L Inductance energy represents the inductance of the center inductor. This corresponds to the common flux modulation of the two loops of the auxiliary transmission line, while This corresponds to differential flux modulation of the two loops of the auxiliary transmission line. The former can be achieved by sending out-of-phase microwave radiation to the two flux lines of the auxiliary transmission line, while the latter can be achieved by sending in-phase microwave radiation to the two flux lines of the auxiliary transmission line. Parametric pumping of the auxiliary transmission line is usually achieved by pumping the common flux, because pumping the differential flux only shifts the modes coupled to the auxiliary transmission line. For example, when at a frequency f... p=2f a -f b When pumping the common magnetic flux, The nonlinear part of the Hamiltonian in the rotating coordinate system is

[0081]

[0082] This is typically the 2:1 photon-swapping Hamiltonian required for designing two-photon stability.

[0083] To construct a CNOT Hamiltonian between the control qubit and the target cat qubit, it is necessary to: - Couple the control qubit to an automatic tuning system (ATS) such that the phase difference through the ATS is...

[0084] - In controlling the frequency f of the quantum bits q Down-pump common flux, i.e.

[0085] In a rotating coordinate system, the parametric part of the Hamiltonian is:

[0086]

[0087] This Hamiltonian can be written in a form that highlights the desired dynamic characteristics.

[0088]

[0089] in,

[0090] Since the buffer is a lossy mode coupled to a cold environment, it is possible to further assume... Thus, the Hamiltonian of this project is

[0091]

[0092] This Hamiltonian is very close to the CNOT Hamiltonian, differing by only two terms. The first term corresponds to an intensity of... Linear drive. This linear drive can be easily compensated by directly sending a reverse drive with the correct relative phase and amplitude on the control qubit, both of which can be experimentally tuned. The applicant also found that the accuracy and range of compensation can be greatly improved by pumping the differential flux of the assisted transmission system (ATS) with the correct phase and amplitude. CX Under the first-order approximation, the differential magnetic flux can be expressed as: While the amplitude and phase of the drive can be calculated analytically, fine-tuning is required experimentally to ensure that the control qubit is not driven under the correct conditions. Experimentally, this compensation is more appropriate than reverse driving on the control qubit because it directly compensates for the parasitic linear drive at its source (i.e., at the ATS), rather than merely attempting to compensate for its primary consequence (i.e., the shift of the control qubit).

[0093] Given the design choices of this invention, the second item cannot be fully compensated for in a simple manner. Instead, the applicant has chosen to design the system such that the amplitude of this item is much smaller than the amplitude of the CNOT Hamiltonian. In other words, Simplified to Although further experimental studies are needed to fully characterize the adverse effects of this second item, the applicant found that when At that time, these effects are negligible.

[0094] Under the premise of satisfying these two conditions (compensation and small amplitude of the second term), the ATS Hamiltonian H can be adjusted. ATS Design it to be close enough to the perfect CNOT Hamiltonian.

[0095] The above design choices led to Figure 3 The circuit design shown.

[0096] The quantum system 30 includes a target cat qubit device 300 and a control qubit device 302, which are connected by a linear electromagnetic coupler 304.

[0097] In the example described here, the target cat qubit device 300 includes a nonlinear superconducting circuit 306 that connects multiple microwave sources 310, 311, 316 and a load 314.

[0098] The nonlinear superconducting circuit 306 includes a linear microwave network 320-322. When this network is coupled to an automated test system (ATS) 308, which acts as an inductor, via a linear coupler 309, the nonlinear superconducting circuit contains at least two normal modes (or eigenmodes) a and b with frequencies fa and fb, respectively. These two modes participate in the ATS. This participation means that some or all of the mode's magnetic energy is stored in the ATS. The degree of this participation can be quantified by the zero-point fluctuation of the superconducting phase on the ATS, where the zero-point fluctuation of mode a is denoted as fb. The zero-point fluctuation of pattern b is denoted as Examples of implementation methods for linear microwave networks are shown in Figures 4 to 7Further description is provided below. In these embodiments, the nonlinear superconducting circuit 306 can be a dual-mode hybrid system (also known as an “electrically coupled cat,” for which the applicant has filed patent applications EP22306815.6 and EP22306816.4), in which the two lumped modes are strongly coupled via an ATS.

[0099] When the external DC magnetic field is set to cause a 0-mode 2π magnetic flux through one of the loops, and a π-mode 2π flux (or vice versa), we ensure that the Hamiltonian of the Automated Test System (ATS) has a “sin sin” form. For clarity, Figure 3 The setup for applying an external DC magnetic field is not shown, but it can be achieved using two bottom current transformers in the automated test system. A typical implementation involves inserting a bias tee connected to a DC current source into the system to input DC current, while allowing microwave radiation to pass through. Microwave sources 310 and 311 are configured to modulate the common and differential magnetic flux in the automated test system, respectively, which is necessary to activate the parameter interactions. To clearly distinguish the roles of the two sources in the Hamiltonian, a microwave network 324 with the correct phase offset is shown in the schematic. Alternatively, each microwave source can be simply coupled to a single node in the automated test system, and their relative phase and amplitude can be set to achieve the desired flux modulation. In this case, to modulate the common magnetic flux, the two sources need to act out of phase with the circuit; to modulate the differential flux, the two sources need to act in phase with the circuit.

[0100] When the frequency of microwave source 310 is set to f p =|2f a -f b In this configuration, the nonlinear superconducting circuit 306 performs a 2:1 photon conversion between the first mode a (reference numeral 320) and the second mode b (reference numeral 322). To convert this 2:1 photon conversion into two-photon dissipation, mode b is selectively coupled to the load 314 via a linear coupler 312 and a microwave filter 318 configured as a bandpass filter at a frequency of fb. Alternatively, the filter 318 may be configured as a bandstop filter at a frequency of fa and may be placed between the environment and the two modes to isolate the first mode, thereby preventing the first mode from suffering additional losses due to unnecessary coupling with the load 314. Alternatively, if fa > fb (or fb > fa), the filter may be configured as a low-pass (or high-pass) filter. In other embodiments, the microwave filter 318 may be omitted when coupling can be established between the load 314 and essentially only the second mode. Thus, those skilled in the art will understand that the first mode a has a high quality factor, while the second mode b has a low quality factor.

[0101] Stabilizing the cat qubit in mode a requires two-photon drive and dissipation, ultimately driving mode b at the resonant frequency fb. This drive is typically performed by microwave source 316 with a frequency set to fb. Alternatively, microwave source 310 or 311 can also be used for drive if it is configured to provide both frequencies fp and fb simultaneously. In the following text, the first mode a, carrying the cat qubit, is also referred to as the cat qubit mode, while the second mode b acts as a buffer between the cat qubit and the environment.

[0102] In the above text, components 310 to 318 and 324 can be considered as Figure 1 This is part of the control circuit 8. When the CNOT gate is not executed, i.e., in the so-called "idle mode," the control circuit 8 is configured to only perform stabilization of the data cat qubit 4. The control qubits and configuration required to execute the CNOT gate will now be described.

[0103] exist Figure 3 In the example, the control qubit device 302 includes a mode q 305 with a resonant frequency fq, which carries the control qubit. In various embodiments, the control qubit device 302 can be any superconducting qubit, such as a transonic qubit, a fluxonic qubit, or a fluxonic qubit, or any boson qubit encoded in a resonator, such as a Kerr cat qubit (whether detuned or not), another stabilized cat qubit device (whether compressed or not), or a cat qubit confined by two-photon exchange Hamiltonian.

[0104] 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 to allow slight mixing of the control qubit 305 with the cat qubit device 300, resulting in minimal involvement of the control qubit in the cat qubit device auxiliary test system (ATS) 308. This involvement is used... This is indicated by the second parasitic term above regarding the uncompensated engineering Hamiltonian, related to the participation of the target cat qubit mode. In contrast, this relatively small involvement is crucial for the accurate realization of the CNOT Hamiltonian. In various embodiments, coupler 304 can be capacitive, inductive, current-coupled, or mediated via a resonant bus coupler or an additional linear microwave network. Figure 3In the illustrated embodiment, the control qubit device 302 is coupled to a first mode a 320. In other embodiments, the control qubit device 302 may be coupled to a second mode b 322. This coupling location is not critical because modes a and b are typically delocalized within the microwave network a / b 320-322, and coupling to a specific location does not necessarily imply coupling to a specific mode unless the linear microwave network and auxiliary test system are specifically designed for this purpose.

[0105] The control qubit device 302 also includes a microwave source 303 coupled to the control qubit 305 to enable it to be driven. The microwave source 303 can be used to compensate for linear drive caused by CNOT Hamiltonian engineering. The microwave source 303 can be considered as... Figure 1 Part of the control circuit 8.

[0106] As mentioned above, another limitation of the control qubit device 302 is that the cat qubit mode a in the nonlinear superconducting circuit 306 has a higher participation in the auxiliary test system (ATS) 308 compared to the control qubit mode 305. This coupling is typically moderate when the control qubit mode q participates in the ATS to execute CNOT gates via a small linear coupling with the cat qubit mode a (typically capacitive coupling, with a capacitance value smaller than the capacitance of either mode; inductive coupling, with an inductance value smaller than the inductance of either mode; or coupling mediated by a detuned bus resonator). More specifically, the detuning (i.e., the frequency difference Δ = |f|) of the two coupled modes (here, a and q, or b and q) a -f q |) also plays a role. In fact, if the two modes have the same resonant frequency, any slight linear coupling will lead to complete hybridization and make However, when the typical detuning Δ / 2π is greater than tens of megahertz, the asymmetry of participation can be achieved through standard linear coupling.

[0107] Figure 4 , 5 6 and 7 show Figure 3 Several possible circuit implementations of the invention are shown. For simplicity, the microwave source is omitted, but its arrangement is similar to... Figure 3 The components are identical to each other to ensure circuit operation. The Automatic Test System (ATS) 308 serves as the central reference point, to which all other components are connected. For this purpose, buffered mode filtering and environmental conditions are also provided.

[0108] As previously mentioned, the weak coupling between the control qubit q and the auxiliary transmission system (ATS) 308 (compared to the target cat state a coupled to the ATS 308) is crucial. In these diagrams, this weak coupling will be represented by a small linear dipole. However, as previously stated, this is generally insufficient to guarantee the weak coupling explained above. If the strength of the linear coupling is represented by g, as is known in the art, then it is also necessary to ensure that the detuning Δ between the control qubit mode q and its coupled modes (a, 320 or b, 322) satisfies g < Δ. While this provides some design rules, precise calculations are still necessary. The value still requires a complete microwave simulation or circuit diagonalization of the circuit layout. This characteristic helps ensure that CNOT gates execute accurately, and that only a single auxiliary transfer system (ATS) is used in the circuit elements, while parameter pumping always has a single purpose, specifically in... Figure 8 The quantum gate diagram will be used for illustration.

[0109] exist Figure 4 In the diagram, the linear microwave network 320-322 consists of a capacitor electrically coupled to an automatic tuning system (ATS) 308 to form a buffer mode 322, and a parallel LC resonator 320 strongly capacitively coupled to the ATS 308 to form a cat qubit mode 320. The coupling with the ATS 308 forms a linear coupler 309. Modes a and b are both strongly coupled to the ATS 308, as can be seen from the phase zero-point fluctuations of the two modes in the central inductor of the ATS 308. The buffer is coupled to the outside world via capacitor 312. Finally, the transnic qubit 305, consisting of a capacitor and a Josephson junction, is weakly coupled to the cat qubit mode 320 via capacitor 304. This weak coupling should be understood as the transnic qubit mode 305 having less involvement in the ATS 308 than the cat qubit 320. An experimental implementation of the circuit is shown below. Figure 14 As shown, the control qubit is another dissipative cat qubit.

[0110] exist Figure 5 In the diagram, the linear microwave network 320-322 consists of two LC resonators connected in series, electrically coupled to an automatic tuning system (ATS) 308 to form a buffer mode 322 and a cat qubit mode 320. These two modes are strongly coupled to the ATS 308, as indicated by the phase zero-point oscillations of the two modes in the center inductor of the ATS 308. The buffer is coupled to the outside via capacitor 312. Finally, a transnic qubit 305, comprising a capacitor and a Josephson junction, is weakly coupled to the cat qubit mode 320 via capacitor 304. Weak coupling should be understood as the transnic qubit mode 305 having a lesser involvement in the ATS 308 than the cat qubit 320. An experimental implementation of the circuit is shown below. Figure 12 As shown, the control qubit is another dissipative cat qubit.

[0111] exist Figure 6 In the middle, the linear microwave network 320-322 and Figure 5 The network described herein is the same. However, the control qubit 305 and its coupling method differ. In this case, the control qubit 305 consists of a flux-transient qubit comprising a capacitor, a Josephson junction, and an inductor (typically composed of a series of Josephson structures) arranged in a loop through which magnetic flux passes. This control qubit 305 is weakly inductively coupled to the inductor of the series LC resonator defining the cat qubit mode 320 via a shared inductance portion 304. This weak coupling should be understood as the transconductance qubit mode 305 having a smaller involvement in the automatic tuning system (ATS) 308 compared to the cat qubit 320.

[0112] exist Figure 7 In the middle, the linear microwave network 320-322 and Figure 4 The same as described in the figure. However, the coupling method of the control qubit 305 is different. In this case, the control qubit 305 consists of a transconductance qubit containing a capacitor and a Josephson junction, weakly coupled to the ATS 308 through capacitor 304, which is also buffer mode 322 in this configuration. In this circuit, the cat qubit mode 320 and the control qubit mode 305 have very similar configurations, mainly the difference in the strength of the linear coupling. This figure is provided to emphasize that... Figure 3 The weak coupling 304 in the diagram is coupled with the linear microwave network 320-322, and not necessarily only with the cat qubit mode 320.

[0113] Figure 8 A diagram is shown that explains Figure 3 How do quantum gates in [the context of quantum mechanics] work?

[0114] In the first operation 800, which corresponds to the idling operation (idling mode) of the circuit when the cat qubit is only stabilizing, the quantum system 30 does not perform the CNOT operation. In this operation, the target cat qubit device 300 receives two types of radiation: radiation from the microwave source 310 at frequency f. p =2f a -f b The parametric pump modulates the common flux of the Auxiliary Transmission System (ATS) to activate the 2:1 photonic nonlinear conversion between the cat qubit mode and the buffer mode (with two other parametric pumps also active as described above in the case of compressed cat stabilization), and a linear drive from microwave source 316 drives the second mode b322 at the second resonant frequency fb.

[0115] In operation 810, the quantum system 30 is modified to perform a CNOT operation for a selected duration. For this purpose, radiation from microwave sources 316 and 310 is temporarily shut off to allow the data cat qubit to come to a stable stop. As previously mentioned, it is crucial that the dynamics of the cat qubit mode be as close to 0 as possible during frames rotating at frequency fa to avoid bit flips during the gate operation time. To activate the CNOT Hamiltonian, microwave source 310 is configured to emit radiation at the control qubit frequency fq.

[0116] As previously mentioned, this produces a first-order effect that strongly drives the control qubit q, leading to leakage or unwanted decoherence in q. To counteract this effect, compensating radiation with precise phase and amplitude can be turned on (factually fine-tuned experimentally while calibrating the CNOT gate in situ). Since the goal of reverse driving is to avoid displacement of the control qubit, a natural approach is to send it to the control qubit via microwave source 303. While this approach is feasible, a better practice is to directly reverse drive the auxiliary transmission system (ATS) that initially caused this displacement, thus addressing the root cause of the displacement. This can be achieved by sending radiation at a frequency of fq using microwave source 311 to modulate the ATS differential flux with the correct phase and amplitude, which have been calculated in advance through analysis and can be fine-tuned experimentally in situ.

[0117] In operation 820, the duration T of the effect of the CNOT Hamiltonian on the system is... cx =π / (4αg cx After that, the parametric pump at frequency fq from microwave source 310 and its compensation (303 or 311) are turned off. Then, by turning on microwave source 310 at frequency fp (and, in the case of a compressed cat, turning on two other pumps) and microwave source 316 at frequency fb, the two-photon stability of the data cat qubits can be restored.

[0118] like Figure 8 As shown, at any given time point, the Automatic Tuning System (ATS) 308 is acted upon by a single parametric pump: a two-photon conversion pump during idle / steady-state time and a longitudinal pump during CNOT gate operation. This minimized setup ensures that the nonlinear performance of the ATS is as close as possible to its theoretically ideal behavior, resulting in high gate speed and high fidelity. In the case of stabilizing the squeezed cat qubits, operating 800 and 820 actually requires three parametric pumps, but all of these pumps serve a single purpose: stabilizing the squeezed cat state. This is what this paper refers to as a “single-purpose parametric pump.”

[0119] Figure 9Exemplary signal timings for operations 800, 810, and 820 are shown. In idle mode, the datacat qubits are stabilized via a two-photon dissipation κ2. This dissipation is turned off when gating begins so that the CNOT Hamiltonian can function. Finally, at time T... cx =π / (4αg cx After this, the CNOT Hamiltonian is turned off, and the two-photon dissipation is turned on again. At any given time, the Auxiliary Test System (ATS) does not serve two purposes simultaneously, ensuring the robustness of the experiment. The limitations on the CNOT pulse intensity and shape (dashed or dotted lines) will now be explained.

[0120] As mentioned above, the CNOT Hamiltonian effectively acts as a linear actuator for the control qubit q, and its strength depends on the number of photons in the target cat qubit a. The property of the control qubit q is the maximum effective actuator strength g. cx α sets an upper limit. There are two cases. In the first case, the control qubit q is defined by a real number or Hamiltonian gap—for a two-level system, this gap is anharmonic |ω. 12 -ω q | where ωq is the frequency of the qubit or the frequency of the transition from the ground state to the first excited state, and ω12 is the frequency of the transition from the first excited state to the second excited state. In this case, the adiabatic theorem applies, and it states that as long as g cx α<|ω 12 -ω q |, then the qubit will remain exponentially confined under effective driving. If the auxiliary qubit is confined by the Kerr Hamiltonian (b) with an amplitude of α c For a cat qubit, the condition is: If constrained by the detuned Kerr Hamiltonian (c), then the condition is: Finally, if restricted by the TPE Hamiltonian (d), the condition is g. cx α<α c g2. In the second case, the control qubit q is defined by the imaginary gap, or equivalently stabilized by dissipation. In this case, the adiabatic theorem does not apply, and the gate strength must be much smaller than the imaginary gap to avoid undesirable decoherence of the auxiliary qubit during gate operations. In practice, for the auxiliary cat qubit (a) stabilized by two-photon dissipation (rate κ2(a)), the condition is... For the compressed auxiliary cat qubit (e), the condition is: Where r is the compression parameter.

[0121] Besides the requirement for gate strength, the constraint properties of the control qubit also impose restrictions on how the gate is applied. In the case of dissipative auxiliary qubits, once the constraint of the data cat qubit is closed (solid line κ2(t)), the CNOT Hamiltonian can be expressed as follows: Figure 9 The dashed line in the diagram indicates immediate activation. In the case of Hamiltonian-assisted qubits, the activation of the CNOT Hamiltonian should be smooth to ensure that the pulse g... CX The spectral content of (t)(dashed line) does not include frequency components above the bandgap. Typically, a Gaussian pulse can be used. Figure 9 In order to obtain a finite time envelope, a cosine shape was used. In g CX When the pulse amplitude varies with time, it should satisfy ∫4ag CX (t)dt=π.

[0122] Figure 10 A schematic diagram of a quantum system that uses the quantum gates of this invention to execute repeating codes is shown.

[0123] Figure 10 The quantum system shown illustrates a classical repeating code architecture for cat qubits: d (d > 3) data cat qubits 300 and d-1 auxiliary qubits 302. In some implementations, such as cyclic repeating codes, there may also be d auxiliary qubits 302 for the d data cat qubits 300. Each auxiliary qubit 302 is connected to two data cat qubits 300 via a CNOT gate 10. The connection between the data cat qubit 4 and the auxiliary qubits 302 is such that one data cat qubit 300 is connected to at most two auxiliary qubits 302. Each auxiliary qubit 302 is also connected to a measurement device for measuring the X Pauli operator with 12 photons, which can be used to detect phase errors once the erroneous repeating code cycle has been executed.

[0124] As mentioned in the introduction, the implementation of the repeating code is as follows:

[0125] - For each auxiliary cat qubit 302, preparation is made in the state |+> or |->, which are eigenstates of the Pauli X operator.

[0126] - Apply each CNOT gate 10 to modify the state of the auxiliary cat qubit 302 according to the error corrector of the data cat qubit 300 connected to it via two corresponding CNOT gates 10, and

[0127] -Measure the X operator for each of the 302 auxiliary qubits.

[0128] Figure 11 , Figure 12 and Figure 13 A practical implementation of the invention is illustrated. In this implementation, four dissipative cat qubits 1101, 1102, 1103, and 1104 with similar parameters (except that their frequencies are slightly detuned to each other to selectively address each mode radiatively without delocalizing the modes across the entire circuit) are arranged in a circular pattern on a superconducting chip, with connections between adjacent qubits. When operating the chip, some cat qubits are selected as data cat qubits "a", and others as auxiliary cat qubits "q". The chip can perform error correction experiments at a distance d = 2 (strictly speaking, only error detection is possible, because at d = 2, errors can only be detected, not located). This error detection experiment requires d data qubits and d-1 auxiliary qubits, thus requiring 3 qubits. On this chip, there are 4 cat qubits so that the optimal 3 cat qubits can be selected afterward to run the error detection experiment. For example, if qubits 1101, 1102, and 1103 have the optimal coherence time for the chip, then qubits 1101 and 1103 are chosen as data qubits, and qubit 1102 is chosen as an auxiliary qubit. Alternatively, if qubits 1102, 1103, and 1104 are the optimal qubits for the chip, then qubits 1102 and 1104 are chosen as data qubits, and qubit 1103 is chosen as an auxiliary qubit. This flexibility is made possible by the fact that all qubits have the same properties. If the data and auxiliary qubits are different, then only the auxiliary qubit can be chosen.

[0129] Figure 11 A partial schematic diagram of the actual implementation is depicted. In this diagram, for improved readability, qubits 1101 and 1103 are chosen as the two data cat qubits, and qubit 1102 or qubit 1104 is chosen as the auxiliary cat qubit, thus labeled "a" and "q" respectively. Both the data cat qubits and the auxiliary cat qubits are dissipative cat qubits. The circuit representation is as follows... Figure 5 Similar to the control qubits, the data 1101 and auxiliary 1102 qubit modes are coupled via a capacitive bus 304. The microwave radiation source is not shown in the figure, but the coplanar waveguide (CPW) transmission lines connecting the circuitry to the rest of the quantum system are shown as 1120, 1122, and 1124. Lines 1120 and 1122 are responsible for DC current bias and parametric flux modulation of the automatic tuning system (ATS) 308; the first line primarily acts on the right loop, and the second line primarily acts on the left loop. The control circuit can pump any linear combination of these two lines. or Line 1124 connects the buffer to a 50Ω environment for loss and drive. The auxiliary qubit 1102 has the same input transmission line.

[0130] Figure 12 An optical micrograph of a chip containing four cat-state qubits is shown, such as... Figure 11 As shown. The chip consists of a superconducting layer 1200 (gray area) deposited on a dielectric (sapphire) substrate 1210. The white areas indicate that the superconducting layer has been removed, revealing the substrate. The lighter gray areas also correspond to the superconducting layer 1210, in which small holes are regularly spaced to trap eddy currents induced by stray magnetic fields. The photolithographic pattern corresponds to... Figure 11 The circuit representation is as described in the diagram. Specifically, the bus is coupled via capacitor 304, which corresponds to a portion of the coplanar waveguide (CPW) line. Buffer mode filtering is performed using a λ / 4 stub filter 318, which is a passband filter at the buffer mode frequency, coupled to the buffer capacitor and electrically coupled to the input line (e.g., 1124). Sub-region 1220 is... Figure 11 The circuit representation is not shown. These correspond to a standard arrangement known in the art, comprising two transmon qubits coupled to two readout resonator capacitors, filtered by two Purcell filters coupled to a single input line capacitor. This arrangement is coupled to a cat-state qubit mode to perform Wigner tomography of cat-state qubits known in the art.

[0131] Figure 13 yes Figure 12 The magnified view of the dashed area 1300 in the image corresponds to the data qubit 1101. This corresponds to a typical stable data qubit device (in... Figure 12 In its implementation, the device is also used as Figure 13 (Auxiliary qubits in the process). Figure 11 The components in the circuit representation are easily identifiable, particularly the lumped inductors 1110 and 1112, which are composed of an array of Josephson junctions. The capacitively coupled bus 1310 couples the cat-state qubit modes to the transconductance qubits used for their tomographic imaging.

[0132] To provide typical values, phase zero-point fluctuations were simulated in this device via an automated test system (ATS) 308. For the cat qubit mode of this unit, the simulated values ​​are... For the buffer mode of this unit, the simulated value is For the 1102 cat qubit mode, which serves as an auxiliary qubit in this configuration, the simulated value is... Since the data cat qubit 1101 is also coupled to the auxiliary qubit 1104 to execute another CNOT gate, the phase zero-point fluctuation of the cat qubit mode of the auxiliary qubit 1104 is also... These values ​​validate the constraints. This condition ensures that parasitic terms are small in CNOT Hamiltonian engineering.

[0133] Figure 14 An optical micrograph of a chip according to another embodiment of the present invention is shown. The chip is arranged in a manner similar to... Figure 7 The equivalent circuit shown is similar, except that the control qubit is a different type of stable cat qubit. The chip consists of a superconducting layer 1200 (gray area) deposited on a dielectric (sapphire) substrate 1210; darker gray areas indicate portions of the substrate where the superconducting layer has been removed. Wire bonds 1415 are visible, used to balance the potential across the entire chip, despite the presence of circuit patterns, and to connect the chip to other parts of the circuit. The chip includes target cat qubit devices a / b, where mode b is capacitively tapped to ground by an ATS (1500, see...). Figure 15 The ATS is capacitively coupled to a λ / 2 coplanar waveguide (CPW) resonator that acts as the cat qubit mode a. The buffer mode is filtered through a λ / 4 stub filter 318 (a passband filter at the buffer mode frequency), and... Figure 12 (Similarly) capacitively coupled to the input drive line 1124, exhibiting dissipative characteristics. The control qubit consists of another stable cat qubit device, where the cat qubit mode q is composed of another λ / 2CPW resonator coupled to its own buffer mode in a similar manner. The two cat qubits are coupled to a tomographic imaging system 1420 containing transconductance qubits and a readout resonator, as shown... Figure 12 As shown. A reference is also shown on this chip. Figure 3 The bias tee 1430 is used to input a direct current (DC) (e.g., 1122 and 1120) on the radio frequency (RF) flux line and bias the ATS to its DC operating point.

[0134] Figure 15 yes Figure 14 The magnified view of the dashed area 1500 in the image corresponds to the buffer mode of the Datacat qubit device. Figure 4 The components represented in the circuit (except for the control qubits) are easily identifiable, particularly ATS 308, which is directly coupled to the ground capacitor to form buffer mode b 322, and cat qubit mode a 320, which is capacitively coupled to it via capacitor 309. The buffer mode is coupled to the ambient capacitor via drive line 112 and capacitor 312. The control qubit q 305 is ultimately capacitively coupled to the buffer mode via capacitor 304 to participate in the ATS. Capacitor 304 is slightly smaller than capacitor 309 in both size and value. Ultimately, by making the detuning between q and b greater than that between a and b, the participation of control qubit q in the ATS is ensured to be less than that of cat qubit mode a. As an example, the following are the values ​​for the frequency and phase zero-point fluctuations in this device: f a=4.82GHz, f b =5.38GHz, f q =4.46GHz; In this device, This is sufficient to realize the CNOT Hamiltonian.

Claims

1. A quantum system for executing CNOT gates, including: Control circuit (8) for providing microwave radiation, The target cat qubit device (4) includes an asymmetric magnetically penetrating superconducting quantum interference device (308) having flux lines that modulate common and / or differential magnetic flux by emitting radiation, and the device is connected to at least one resonant section (320-322). The target cat qubit device (300) has a first mode (a) and a second mode (b), wherein the first mode (a) has a first resonant frequency and the second mode (b) has a second resonant frequency. Furthermore, the asymmetric magnetically penetrating superconducting quantum interference device (308) is configured such that when the control circuit (8) sends radiation with the second resonant frequency to the at least one resonant part (320-322) to drive the second mode (b), and sends radiation on the flux line to modulate the common magnetic flux, and the modulation frequency is equal to the absolute value of the difference between twice the first resonant frequency and the second resonant frequency, the target cat qubit device (4) stabilizes the target cat qubit with the first resonant frequency, and A control superconducting quantum bit device (6) having a third mode (q) and a third resonant frequency, for carrying control qubits (305), wherein the control qubits (305) are configured such that when the third mode (q) is subjected to radiation having the third resonant frequency and an intensity less than the control qubit confinement rate, Rabi oscillations of the third mode (q) are induced; the control superconducting quantum bit device (6) is linearly coupled to the target cat quantum bit device (4), such that the third mode (q) is linearly coupled to the asymmetric magnetically penetrating superconducting quantum interference device (308). The control circuit (8) is configured to transmit radiation at a frequency equal to the third resonant frequency only on the flux line within the CNOT gate time window to operate the CNOT gate (10) between the target cat qubit and the control qubit; while outside the CNOT gate time window, it transmits radiation with the second resonant frequency to the at least one resonant portion (320-322) to drive the second mode (b) and transmits radiation on the flux line to modulate the common flux when the frequency is equal to the absolute value of the difference between twice the first resonant frequency and the second resonant frequency.

2. The quantum system according to claim 1, wherein the target cat qubit device (4) is linearly coupled to the control superconducting device (6), such that the phase difference between the two ends of the asymmetric punch-through superconducting quantum interference device (308) is Where 'a' is the photon annihilation operator of the first mode (a). Let b be the phase zero-point fluctuation of the first mode (a) at both ends of the asymmetric punch-through superconducting quantum interference device (308), and let b be the photon annihilation operator of the second mode (b). Let q be the phase zero-point fluctuation of the second mode (b) at both ends of the asymmetric punch-through superconducting quantum interference device (308), and let q be the photon annihilation operator of the third mode (q). The phase zero-point fluctuation of the third mode (q) at both ends of the asymmetric punch-through superconducting quantum interference device (308) is given.

3. The quantum system according to claim 1 or claim 2, wherein the control circuit (8) is configured to operate only at a frequency and amplitude equal to the third resonant frequency within the CNOT gate time window. CX The radiation pumps the magnetic flux lines, which induces a Hamiltonian, the formula of which is: Where H CX The formula is CNOT Hamiltonian, a is the photon annihilation operator of the first mode (a). q is the phase zero-point fluctuation of the first mode (a) on the asymmetric transmagnetic superconducting quantum interference device (308), and q is the photon annihilation operator of the third mode (q). It is the phase zero-point fluctuation of the third mode (q) on the asymmetric transmagnetic superconducting quantum interference device (308), E J It is the Josephson energy of the side junction of the asymmetric magnetically penetrating superconducting quantum interference device (308), α 2 It is the number of photons in the first mode (a), and the control circuit (8) is also configured to induce the formula within the CNOT gate time window. The compensation Hamiltonian.

4. The quantum system according to claim 3, wherein the control circuit (8) is configured to modulate the differential magnetic flux within the CNOT gate time window by transmitting radiation with a frequency equal to the third resonant frequency only on the magnetic flux lines. This leads to the following formula: The compensation Hamiltonian, of which For the common magnetic flux, E L The induced energy is the central inductance of the asymmetric transmagnetic superconducting quantum interference device (308).

5. The quantum system according to claim 3, wherein the control circuit (8) is configured to induce the generation of the formula during the CNOT gate time window by transmitting radiation at a frequency equal to the third resonant frequency to the control qubit device (305). The compensation Hamiltonian.

6. The quantum system according to any one of the preceding claims, wherein the control superconducting quantum bit device (6) carries an anti-qubit, flux qubit, pass qubit, cat qubit or any boson qubit encoded in a resonator.

7. The quantum system according to any one of the preceding claims, wherein, The asymmetric transmagnetic superconducting quantum interference device (308) is further configured such that when the control circuit (8) transmits radiation having the second resonant frequency to drive the second mode (b) and transmits radiation on the flux line to modulate the common magnetic flux at a frequency equal to the absolute value of the difference between the first resonant frequency and the second resonant frequency, the second resonant frequency, and the sum of the first resonant frequency and the second resonant frequency, the target cat qubit device (4) stabilizes the target compressed cat qubit having the first resonant frequency.

8. The quantum system according to any of the preceding claims, wherein the target cat qubit device (4) and the control superconducting qubit device (6) are configured such that the absolute difference between the first resonance frequency and the third resonance frequency exceeds 10 MHz.

9. The quantum system according to any of the preceding claims, wherein the control circuit (8) is configured to transmit radiation in the magnetic flux lines to modulate the common magnetic flux using only radiation with a frequency equal to the third resonant frequency, thereby inducing a constant intensity g. CX The CNOT Hamiltonian.

10. The quantum system according to any one of claims 1 to 8, wherein the control circuit (8) is configured to transmit radiation in the magnetic flux lines to modulate the common magnetic flux only with radiation having a frequency equal to the third resonant frequency, thereby inducing a CNOT Hamiltonian g whose intensity varies with time. CX (t), such that ∫4αg CX (t)dt=π.

11. A quantum system for executing repeating codes, comprising a control circuit (8) for selectively applying microwave radiation; d data cat qubit devices (4), d equal to or greater than 2, each data cat qubit device (4) having its own resonant frequency and coupled to the control circuit (8) to stabilize its respective data cat qubit; and d-1 auxiliary qubit devices (6), each auxiliary qubit device having an auxiliary resonant frequency coupled to the control circuit (8) for carrying auxiliary qubits, each of the d-1 auxiliary qubit devices being linearly coupled to two of the respective data cat qubit devices, and each of the d data cat qubit devices being connected to at most two of the d-1 auxiliary qubit devices. The d data cat qubit devices (4) and the d-1 auxiliary qubit devices (6) enable each coupling between the data cat qubit devices (4) and the auxiliary qubit devices (6) to be controlled by the control circuit (8) to realize the quantum system according to any one of the preceding claims, wherein the control circuit (8) is further configured to perform d-1 quantum operations in the following manner: 1) For each of the d-1 auxiliary qubit devices, an auxiliary qubit with the X operator state "|+>" or "|->" is prepared in each of the d-1 auxiliary qubit devices. 2) For each of the d-1 auxiliary qubit devices, the first of the two data cat qubit devices coupled to it performs a CNOT gate operation, and then the second of the two data cat qubit devices coupled to it performs a CNOT gate operation. 3) Measure the operator X on each of the auxiliary qubit devices.

12. The quantum system of claim 11, wherein all auxiliary qubit devices carry cat qubits such that all data cat qubit devices and all auxiliary qubit devices are cat qubit devices, and wherein the control circuit (8) defines which cat qubit device is a data cat qubit device (4) and which cat qubit device is an auxiliary qubit device (6).

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