A method for calibrating parameters of a sequence for performing quantum operations.

Abstract

A method for calibrating at least one parameter of an interaction sequence used to perform quantum operations defined on a single- or multi-qubit system in a cat-qubit-based superconducting circuit, the cat-qubit having a reduced bit-flip error rate as a noise bias, comprising: for a list of qubit systems affected by a change in at least one parameter and at least one of their states, for a given parameter value and for each state in the list, performing: preparing (S20) a state; applying (S30) an interaction sequence to the prepared state using the parameter value and measuring (S40) the bit-flip population and non-bit-flip population of the affected qubit at the end of the application; iterating (S50, S60, S10, S20) the preparation, application, and measurement steps using different parameter values, scanning a range of candidate parameter values; determining (S70) an optimal value for maintaining the noise bias from the measured population, and implementing the associated derived parameter value as a calibration value (S80).

Description

Technical Field

[0001] This disclosure relates to the field of quantum technology, and more specifically to the calibration of quantum circuits involved in quantum information processing and quantum computing. Background Technology

[0002] Quantum information processing, which enables quantum computing, consists of encoding and manipulating quantum information, typically within a two-level quantum system called a qubit. The state of a qubit is represented on a sphere called a Bloch sphere, with the north and south poles corresponding to the standard basis vectors |0〉 and |1〉, respectively. Any pure state in a two-level quantum system can be written as a superposition of the basis vectors |0〉 and |1〉, where the contribution from each of these two basis vectors is a complex number.

[0003] In this context, the operations involved (including logic gates) are implemented through sequences of interactions comprising wave pulses (particularly microwave pulses). Some of these operations and gates operate on single-qubit systems, while others operate on multi-qubit systems. Each operation or gate has a name and may have one or more parameters (particularly numerical parameters).

[0004] Typical examples of single-qubit gates are the X(θ) and Z(θ) gates, which rotate by an angle θ around the X-axis and Z-axis of the Bloch sphere, respectively.

[0005] An example of a multi-qubit gate is the two-qubit controlled-NOT gate (CNOT or CX). The effect of a CNOT gate on its input depends on the state of the other qubit designated as the control qubit, applying a NOT operation to the qubit designated as the target qubit, or leaving it unchanged. The control qubit remains unchanged in all operations of the CNOT gate.

[0006] Different parameters of the physical implementation of operations and gates need to be calibrated in order to implement these operations on quantum hardware. This calibration can be performed all at once, or more likely, it can be performed and adjusted periodically or at critical moments in hardware use.

[0007] In terms of hardware design, encoding quantum information in the states of a harmonic oscillator, i.e., boson modes (the most well-known boson being the photon), is a solution adopted by several players in this field. Superconducting microwave circuits, including LC resonators and Josephson junctions, are used in cryogenic cryostats to implement qubits and quantum gates. Specifically, in such superconducting circuits, quantum gates can be implemented using a single microwave pulse or a combination of pulses, and the parameters to be optimized include pulse amplitude, phase, and duration.

[0008] However, when considering operations and gates, quantum noise must be taken into account because isolated qubits are susceptible to two fundamental errors (i.e., bit flips and phase flips), and the error rate increases with the size of the qubit. In general, noise is a major concern in quantum circuits, and various strategies have been devised to control its effects with limited resources.

[0009] One of these strategies is to design noisy bias qubits, and specifically, so-called cat qubits. In these qubits, one of two types of errors is exponentially suppressed as the size of the cat qubit increases—the size of the cat qubit is equal to the average number of photons in the harmonic oscillator—while the other type of error increases only linearly with size. The protection or stabilization of the qubit is implemented by driving two-photon dissipation, or by using Kerr-type Hamiltonians and two-photon drive.

[0010] Based on existing knowledge, this suppression is applicable to physical noise processes such as photon loss, thermal excitation, photon phase shift, and various nonlinearities caused by coupling to Josephson junctions.

[0011] Furthermore, due to this noisy structure, error correction becomes easier in order to realize logical qubits.

[0012] Marquet et al. (arXiv:2307.06761 (2023)) demonstrated self-parametric resonance that extends the bit-flipping time of a cat qubit to 0.3 seconds. Reglade et al. (arXiv:2307.06617 (2023)) demonstrated control of a cat qubit with a bit-flipping time exceeding 10 seconds.

[0013] The requirement is for noise-preserving quantum gates and operations to act on noise-biased qubits in order to enable coherent implementations of quantum circuits. Guillaud and Mirrahimi (Phys. Rev. X 9, 041053 (2019)) list the following basic bias-preserving operations at the cat qubit level: preparation of |+> or |-> cat states, X measurement, X gate, CNOT gate, Toffoli gate, rotation around the Z-axis by a given angle, and two-qubit entanglement gate.

[0014] Marquet et al. envisioned calibrating the driving phase of a Z-gate by observing the rate of movement of Wigner function fringes over time and extracting the oscillation rate. Reglade et al. (arXiv:2307.06617 (2023)) published essentially the same principle on this subject. While this was an achievement at the time, the applicants recognized that further research was needed to improve the precision, accuracy, and reliability of quantum gate and operational calibration methods. Summary of the Invention

[0015] In this context, a method for calibrating at least one parameter of an interaction sequence for performing quantum operations is provided, and in particular, when the operation is a gate, a method for calibrating the parameters of a physical quantum gate is provided. The interaction sequence typically consists of a pulse sequence, but is not limited to pulse sequences.

[0016] Generally, this operation is defined on single-qubit or multi-qubit systems in cat-qubit-based superconducting circuits, where the cat qubit has a reduced bit-flip error rate as a noise bias. The method includes: for a qubit in the system affected by a change in at least one parameter, and for a list of at least one state of the qubit system, performing the following steps for the value of the at least one parameter and for each state in the list: -Preparation state; - Apply an interaction sequence to the prepared state using the parameter values; -At the end of the application, measure the bit-flipped or non-bit-flipped population of the affected qubit; - Iterate the preparation, application, and measurement steps using the variation value of at least one parameter to scan the candidate value range of at least one parameter, then determine the optimal value of the maintenance noise bias by operation from the population measured during the iteration steps, and implement the inferred parameter value associated with the determined optimal value as the calibration value of at least one parameter.

[0017] The present invention features highly accurate calibration of the detected parameters by measuring the bit-flipped and / or non-bit-flipped populations. In fact, the dependence of the bit-flipped rate on the gate parameters is very high, and the results are highly accurate with a lower uncertainty range and better final results. Robustness is ensured by noise bias—the bit-flipped rate can be virtually arbitrarily small.

[0018] In various non-limiting embodiments, the following features are advantageous:

[0019] At least one parameter may include the phase angle of the microwave pulse sent to the nonlinear coupling element, which couples the memory of the qubits in the system to the buffer of the qubits (at a frequency). The pump can be a nonlinear coupling element, such as an ATS or an asymmetric cross-flow SQUID. and This refers to the frequency of the memory and buffer. In this case, the following characteristics can be used: the system has two qubits, including a control cat qubit and a target cat qubit; the gate can be a CNOT gate; the phase angle of the microwave pulse is the phase of the pulse relative to the stabilization angle of the control cat qubit; the microwave pulse has a frequency of Send to a nonlinear coupling element that couples the memory of the target cat qubit to the buffer of the target cat qubit. The interaction sequence may include: preparing a first state from a list, wherein the list includes: (i) + α C +α T and -α C +α T ; or (ii) + α C -α T and -α C -α T (i.e., α of the list) T (To maintain consistency); applying the interaction sequence may include sending microwave pulses, where the parameter value is the phase angle; the measurement includes measuring the number of bit flips in the initial preparation state of the cat qubit (therefore, if +α was initially chosen). C +α T Then, the bit flips from +αC are measured, and finally, the sequence is repeated for the other states in the list; when the number of bit flips of the control qubits from the first state in the list is equal to the number of bit flips of the control qubits from the other state in the list, the optimal value for preserving the noise bias of the gate is determined.

[0020] At least one parameter may include the phase angle of the microwave pulse transmitted at the frequency of the system's qubit memory. In this case, the following features may be used: Microwave pulses can be sent from anywhere on the circuit, not necessarily from the memory itself (although the effect will indeed reach the memory). Pulses can be sent via transmission lines directly coupled to the memory, but this is optional. The system can be a single-cat qubit system, the gate can be a Z(θ) gate, and the phase angle of the microwave pulse is the phase of the pulse relative to the stabilization angle of the cat qubit. The interaction sequence can be: preparing a first state from a list, where the list includes +α and -α. Applying the interaction sequence includes sending a microwave pulse, where the parameter value is the phase angle. Measurement includes measuring the number of bit flips from the prepared state. This process is repeated for other states in the list. When the number of bit flips from the first state in the list is equal to the number of bit flips from another state in the list, the optimal value for the gate to maintain the noise bias is determined. However, the system can also be a multi-cat qubit system (possibly having two qubits, including a control cat qubit and a target cat qubit), the gate can be a CNOT gate, and the phase angle of the microwave pulse is the phase of a pulse sent at the frequency of the controlled memory as a spurious effect elimination pump; the phase angle of this elimination microwave pulse can be explicitly defined as the phase of the pulse relative to the stabilization angle of the control cat qubit. The interaction sequence can be: preparing a first state from a list, wherein the list includes: no photons in the target qubit, and +α, iα, -α, and –iα in the control qubit. Applying the interaction sequence includes sending a microwave pulse, wherein the parameter value is the phase angle. Measurement includes measuring the number of bit flips in the control cat qubit from the prepared state. This process is repeated for the other states in the list. When the number of bit flips from the states in the list is equal to each other, the optimal value for the gate to maintain the noise bias is determined.

[0021] At least one parameter includes the duration of destabilization of the driving state of the qubits in the system. The system can be a single-cat qubit, and the gate can be an X-gate. The interaction sequence can begin with preparation of +α or -α. Applying the interaction sequence can include the duration of destabilization of the single cat qubit. Measurements can include measuring the number of bit flips from the preparation state (or equivalently, measuring the number of bit flips from another state). It is not necessary to repeat for other states in the list. The list can include only one state. The optimal value for the gate's preservation of the noise bias is determined when: if +α is prepared, it is determined when the number of bit flips from -α is at its minimum (i.e., the minimum population in +α); or if -α is prepared, it is determined when the number of bit flips from +α is at its minimum (i.e., the minimum population in -α).

[0022] At least one parameter may include the integral of the amplitude of the microwave pulse sent during operation to a nonlinear coupling element that couples the memory of the qubit in the system to the buffer of that qubit. In this case, the following feature may be used: the system has two qubits, including a control cat qubit and a target cat qubit. The CNOT microwave pulse is at a frequency... The sequence involves sending a nonlinear coupling element to the buffer that couples the target cat qubit's memory to the target cat qubit. This sequence includes preparing a first state from a list, where the list includes: (i) + α C +α T and -α C +α T ; or (ii) + α C -α T and -α C -α T(i.e., α of the list) T (To maintain consistency). The interaction sequence involves sending a CNOT microwave pulse, where the parameter value is the time integral of the amplitude. The measurement involves measuring the number of bit flips of the target cat qubit. The interaction sequence may also include destabilizing the target cat qubit. These steps are repeated for the other states in the list. The optimal value for preserving the noise bias of the gate is determined when the number of bit flips of the target from the first state in the list simultaneously satisfies the following conditions: (i) equal to the number of bit flips of the target from another state in the list; and (ii) is the minimum value.

[0023] At least one parameter may include the integral of the amplitude of a microwave pulse transmitted at the frequency of the memory of the qubits in the system during the implementation of the operation, over time. In this case, the following feature may be used: the microwave pulse can be transmitted anywhere on the circuit, not necessarily on the memory itself (although its effect does reach the memory). The pulse can be transmitted via a transmission line directly coupled to the memory, but this is optional. The system may also be a multi-cat qubit system, the gate may be a CNOT gate, and the integral of the amplitude of the microwave pulse over time is the integral of a pulse transmitted at the frequency of the control memory as a spurious effect elimination pump; the interaction sequence may be: - preparing a first state from a list, wherein the list includes +α, iα, -α, and -iα on the control, or other state pairs with a π phase difference between the two states of that pair, and no excitation in the target. Applying the interaction sequence includes transmitting a microwave pulse, and wherein the parameter value includes the amplitude of the pulse, and may include the phase of the pulse. Measurement includes measuring the number of bit flips on the control qubit from the prepared state. This process is repeated for the other states in the list. When the number of bit flips from the states in the list is equal to each other, the optimal value for preserving the noise bias of the gate is determined.

[0024] At least one parameter may include the phase angle difference of two microwaves transmitted as a driving stabilization after the operation is performed. These two microwaves are respectively transmitted to a buffer associated with a qubit in the system, and a nonlinear coupling element of a memory coupling the associated buffer to the qubit, at a frequency associated with both the buffer and the nonlinear coupling element. In this case, the following features may be used: the system may be a multi-cat qubit system, the gate may be a CNOT gate, the stabilization is the stabilization of the target cat qubit; the interaction sequence may be: preparing a first state from a list, wherein the list includes +α C and + / -α T and -α C and + / -α TApplying the interaction sequence involves sending microwave pulses, where the parameter value is a phase angle difference. Measurements include measuring the number of bit flips from the prepared state. This process is repeated for the other states in the list. When the number of bit flips from the states in the list tends to be equal to each other, the optimal value for preserving the gate's noise bias is determined.

[0025] In some embodiments, the application of the interaction sequence may be performed several times prior to the measurement, wherein the number N of the applied interaction sequence is further from N... min Change to N max , and perform the measurement for each value of N used.

[0026] At least one parameter may include two real numbers, and the applied interaction sequence may be executed several times while independently changing these two real numbers.

[0027] The state list may include one or more pairs of coherent states, or cat states, in which there is a π phase difference between two states in a given pair, or it may be a list of only one coherent state.

[0028] Determining the optimal value may include: determining the minimum bit flip rate or number of bit flips from the population measured during the iteration step, or the minimum distance (absolute value of the difference) between the bit flip rates or number of bit flips associated with different states of the list, or simultaneously finding the minimum bit flip rate or number of bit flips and the minimum distance between the bit flip rates or number of bit flips associated with different states of the list.

[0029] This operation can be one or a combination of Z-gate, X-gate, CNOT gate, Tofoli gate, state preparation operation, and parity check measurement operation.

[0030] Cat qubits can be stabilized by driving two-photon dissipation.

[0031] The measurement is performed via Wigner tomography or any parity measurement procedure, particularly direct or indirect parity measurement, and especially by mapping different parities to different populations in memory. The parity of a state can be measured using a transmon.

[0032] Measurements can be performed not with any parity checks, but by measuring the photon / population ratio in the memory. A Z(π / 2) gate is executed using a dynamical Zeno effect, an X(π / 2) gate using a full gate, and a Pauli operator is measured using longitudinal readout. The first two steps are based solely on a two-photon dissipation mechanism. To achieve the third step, terms of the ATS Hamiltonian are pumped to design the required longitudinal coupling.

[0033] Therefore, measurements are performed by Wigner tomography or any parity measurement procedure, particularly direct or indirect parity measurement, and especially by mapping different parities to different populations in the memory of the affected qubit; or by determining the number of photons in the memory of the affected qubit.

[0034] A computer program including instructions is also provided, which, when executed by a computer, cause the computer to perform the methods described above. Attached Figure Description

[0035] Figure 1 The mapping between resonator states and qubit states, and the corresponding complex plane representation, are shown.

[0036] Figure 2 A lumped element model of the circuit that achieves stabilization in one embodiment is shown.

[0037] Figure 3 A pulse for an X-gate is shown according to an embodiment of the present invention.

[0038] Figure 4 A graphical representation of the effect of the Hamiltonian in executing the Z-gate is shown.

[0039] Figure 5 A pulse for a Z-gate according to an embodiment of the present invention is shown.

[0040] Figure 6 The effect of the CNOT gate is shown when the target qubit is in the initial state of |+α〉.

[0041] Figure 7 A simplified graphical representation of the effect of the Hamiltonian in executing the CNOT gate is shown.

[0042] Figure 8 A pulse for a CNOT gate is shown according to an embodiment of the present invention.

[0043] Figure 9 A schematic representation of the calibration method according to the present invention is shown.

[0044] Figure 10 The calibration of the X gate according to an embodiment of the present invention is shown.

[0045] Figure 11 The calibration results of the phase angle of a CNOT gate according to an embodiment of the present invention are shown.

[0046] Figure 12The calibration results of the parameters of the CNOT gate (the integral of the CNOT pulse amplitude over time and the target stabilization angle) according to an embodiment of the present invention are shown. Detailed Implementation Cat qubit

[0047] [ Figure 1 Stabilized cat qubits encode quantum information in the quantum states of a linear resonator, typically represented by functions defined in the complex plane. Coherent states (i.e., specific states of the quantum harmonic oscillator) |+α〉 and |-α〉 are represented as 2D Gaussian distributions centered at opposite points in the complex plane.

[0048] Coherent states |+α〉 and |-α〉 can be used to encode quantum information in qubits, where |+α〉 is identified as |0〉 and |-α〉 as |1〉. A universal qubit state can be represented as a superposition of |+α〉 and |-α〉.

[0049] Of particular note is the superposition state known as the cat state, that is, given N + and N - Normalization constant, where:

[0050] ;

[0051] |C+〉= N + x (|+α〉+ |-α〉), denoted as |+〉;

[0052] as well as,

[0053] |C-〉= N - x(|+α〉- |-α〉), denoted as |-〉;

[0054] These are even cat states and odd cat states, and other cat states can be obtained by using different relative phases between |+α〉 and |-α〉.

[0055] The cat states, |+〉 and |-〉, exhibit interference patterns between the Gaussian distributions of |+α〉 and |-α〉, which is a typical characteristic of cat states.

[0056] A cat qubit is defined as a two-dimensional manifold spanned by two cat states.

[0057] Any physical realization of these systems is affected by defects that cause quantum states to deviate from the desired |+α>, |-α>, and superposition states. One type of error is called a local error. These errors affect the system in such a way that the complex plane representation of the state is deformed and drifts towards the origin of the complex plane over time.

[0058] Stabilization (also known as confinement) protects the system from these errors, effectively fixing the coherent states |+α> and |-α>, as well as their superposition states, in their proper positions in the complex plane. In particular, R. Lescanne et al. (2020) demonstrated that cat qubits can be stabilized by a nonlinear transition between two photons in a first mode a (memory mode or cat qubit mode) and one photon in a second mode b (called buffer mode).

[0059] Despite stabilization, the stabilized cat qubit is not error-free. A bit flip error is an unwanted transition between two qubit states. Therefore, a bit flip error occurs when a constrained coherent state transitions to another constrained coherent state: |+α> coherent state transitions to |-α> coherent state, and vice versa.

[0060] [ Figure 2 At the hardware level, the unrestricted model of the circuit consists of a linear resonator called a memory called a qubit (where quantum information is stored as |+α〉, |-α〉, and their superposition states) and an additional resonator called a buffer for the qubit. The buffer comprises a circuit element called an asymmetric through-current (ATS) SQUID (superconducting quantum interference device), which enables the stabilization of the memory states. The ATS is a SQUID loop shunted by a linear inductor, forming two loops through which magnetic flux can pass to achieve different operations. Figure 2 The diagram shows a simplified circuit, illustrating a lumped-element model of a circuit that includes components capable of achieving stabilization.

[0061] Stabilization or limiting is activated by two microwave drives: one applied to the buffer resonator and the other to the ATS, such that the sum of these two frequencies equals twice the memory mode frequency. These drives are introduced via external drives.

[0062] More generally, cat qubits can be stabilized or restricted through one or more of the following exemplary schemes: Stabilization scheme

[0063] a) Parameter dissipation stabilization, with jump operators ,in, It is the two-photon dissipation rate. It is the photon annihilation operator, and It is a complex number that defines a cat qubit. This jump operator can be used to jump qubits with dissipation rate... The lossy buffered mode b and a four-wave mixer (typically a Josephson junction, ATS, or a DC bias coupled to one or more Josephson junctions (for which the applicant has filed European patent application EP 23306839.4)) are coupled to the cat qubit mode a, and the Hamiltonian is designed... To achieve this, in which, It is the photon annihilation operator for buffered mode b, and It is the two-photon coupling rate, and the above design is in Under these conditions, by applying a frequency of [frequency value] to the four-wave mixer [process / method]... The pump, and apply a frequency of to the buffer mode. This is achieved through a driver.

[0064] b) Dissipative compression stabilization with jump operator ,in, It is to compress the two-photon dissipation rate. It is the photon annihilation operator. It is the complex number that defines a cat qubit. and These are complex compression parameters. The modulus and argument. This jump operator can be used by having a dissipation rate The lossy buffered mode b and a four-wave mixer (typically a Josephson junction or ATS) are coupled to the cat qubit mode a, and the Hamiltonian is designed... To achieve this, in which, It is the photon annihilation operator for mode b, and It is a compression of two-photon coupling ratio, in Under the condition of, the frequency of use is , and Several pumps, and frequency The buffer mode b is implemented by the driver.

[0065] c) A variation of the aforementioned stabilization scheme b), for which the applicant filed European patent application EP23175147.0.

[0066] d) Kerhammill ,in, It is the amplitude of the Kerhammeter. It is the photon annihilation operator, and It is the average number of photons.

[0067] e) Detuned Kelhamnton ,in, It is the amplitude of the Kerhammeter. It is the photon annihilation operator. It is the complex number that defines a cat qubit, and It is a detuning factor.

[0068] f) Two-photon exchange (TPE) Hamiltonian ,in, It is the complex two-photon coupling rate. It is the photon annihilation operator. It is the complex number that defines a cat qubit, and It is the decrement and elevation operator for a two-level system. The Hamiltonian can be designed in the same way as parameter dissipation stabilization a).

[0069] g) Resonant dissipation stabilization with jump operator ,in, It is the two-photon dissipation rate. It is the photon annihilation operator, and It is a complex number that defines a cat qubit. This jump operator can be used to jump qubits with dissipation rates. The lossy buffer mode b and the three-wave mixer are coupled to the design Hamiltonian. The cat qubit mode a is used to achieve this, where, It is the photon annihilation operator for buffered mode b. It essentially satisfies the following at the mode frequency. ,as well as Under the condition that, add a frequency of .

[0070] In particular, dissipative stabilization schemes, including resonant dissipative stabilization scheme g) (especially schemes a)-c), have been found to be particularly effective in realizing cat qubits with extremely low bit-flipping errors. Furthermore, combinations of various dissipative stabilization schemes with other stabilization schemes (resulting in the retention of some dissipative stabilization components) are also particularly effective. Bit Flip Error

[0071] Bit-flipping errors occur when one restricted state transitions to another. For example, the |+α〉 state transitions to the |-α〉 state, and vice versa. Bit-flipping errors are quantified by measuring the population of the stabilized coherent states |+α〉 and |-α〉.

[0072] For example, bit-flipping errors caused by an operation can be quantified by: a) preparing and stabilizing the state |+α〉; b) performing the operation—an interaction sequence that causes a quantum gate or any operation, for example, simply waiting for a period of time during the stabilization process; c) measuring the populations of the states |+α〉 and |-α〉.

[0073] Assuming the desired outcome of the operation is state |+α>, in the case of few bit-flip errors, the high population in |+α> and the low population in |-α> are then measured; however, if many bit flips occur during the operation, the populations of |+α> and |-α> are measured to be equal or nearly equal, as they tend to be balanced.

[0074] The populations of states |+α〉 and |-α〉 can be measured, for example, by measuring the parity at the corresponding +α and -α points in the complex plane, where even parity corresponds to a high population and zero parity corresponds to no population.

[0075] The population of a state can be measured by performing Wigner tomography and comparing the values ​​at two "blob" locations. The population P(±α) of states |+α〉 and |-α〉 is then given by the values ​​of the Wigner function at the two locations +α and -α, as follows:

[0076] P(±α)=W(±α) π / 2.

[0077] The Wigner function is proportional to parity, therefore measuring the Wigner function is equivalent to measuring parity.

[0078] Parity can be measured, for example, by encoding parity into the state of an auxiliary transport and measuring the state of that transport.

[0079] Parity can be measured by transportless readout. Parity measurement and photon number measurement are known. See, for example, Reglade et al. (arXiv:2307.06617 (2023)) for a description of the measurement process using transportless readout; and Lescanne et al. (Nature Physics, Vol. 16, pp. 509–513 (2020)) for a description of the measurement process using transports.

[0080] Since the two specific locations ±α are previously known, a full Wigner tomography is not required: Wigner tomography that only provides the Wigner function at these locations is useful for population measurements.

[0081] Using the population measure, the bit flip error rate Γ can be given / extracted by the following formula:

[0082] Γ=[P(+α)-P(-α)] / Δt,

[0083] Where Δt is the duration of the operation.

[0084] The above calculations can be performed starting from the |-α> state.

[0085] Lescanne et al. provide an example of measuring bit flip time using the above method in Nature Physics, Vol. 16, pp. 509–513, 2020. Dissipation circuit

[0086] In the dissipative stabilization scheme, it is represented by the design as Jump operators (or dissipators) To stabilize the cat qubit. Parameter dissipation stabilization depends on the generation term. and .item This corresponds to the nonlinear conversion between two photons in the first mode a (memory) and one photon in the second mode b (buffer). This term requires a frequency of... The pump, in which It is the resonant frequency of the first mode a, and This is the resonant frequency of mode b. Therefore, this pumping is two-photon pumping. Required frequency is The second mode b is driven. Therefore, this driving is two-photon driving. Resonant dissipation stabilization corresponds to Specific circumstances.

[0087] The quantum system comprises a nonlinear superconducting quantum circuit and an instruction circuit. The nonlinear superconducting quantum circuit is arranged such that three-wave mixing or four-wave mixing is possible between a first mode a and a second mode b. In the following text, the first mode a serves as a memory carrying the cat qubit, while the second mode b serves as a buffer between the cat qubit and the external environment.

[0088] The first mode a and the second mode b each correspond to the inherent resonant frequencies of the nonlinear superconducting quantum circuit. Therefore, the first mode a and the second mode b each have their own resonant frequencies. The first mode a has a resonant frequency. Furthermore, the second mode b has a resonant frequency. ,in, and These are the corresponding angular frequencies of the first mode a and the second mode b. Nonlinear superconducting quantum circuits therefore include components operating in the superconducting state, which carry modes independently or simultaneously. The first mode a and the second mode b can be carried on different subsets of the components of the superconducting circuit or on the same subset of the components.

[0089] A nonlinear superconducting quantum circuit is subjected to microwave radiation delivered by a command circuit, thereby inducing various nonlinear interactions between a first mode a and a second mode b. The frequency of each microwave radiation is tuned to select a specific term within the rotating wave approximation.

[0090] Nonlinear superconducting quantum circuits include nonlinear elements and at least one resonant component.

[0091] Nonlinear elements generate hopping operators for stabilizing cat qubits. Alternatively, the nonlinear element and at least one resonant component are arranged to otherwise confine the cat qubit by schemes d), e), or f).

[0092] When implementing parameter dissipation stabilization, the nonlinear element can be a four-wave mixing nonlinear element.

[0093] Four-wave mixing nonlinear elements are, for example, ATS. Those skilled in the art know that ATS can be used to design 2-to-1 photon conversion, i.e., hopping operators. The first item This is to facilitate parameter dissipation stabilization, as successfully demonstrated by R. Lescanne et al. (2020). More specifically, by using frequency as... The parameters are used to pump the ATS to obtain this 2-to-1 photon conversion between the first mode a and the second mode b. Advantageously, the pump frequency satisfies... To make the pump work as well as possible with this parameter.

[0094] In contrast to earlier implementations in which the superconducting circuit element used as a four-wave mixer was a stabilization scheme of a transporter with a single Josephson junction, the solution developed by R. Lescanne et al. (2020) utilizes an ATS design with a much lower cross-Kerr term than the transporter, and thus allows for exponential suppression of bit flips.

[0095] Alternatively, the nonlinear element performing four-wave mixing can be an element other than an ATS. For example, the nonlinear element can alternatively include only one or more Josephson junctions.

[0096] When resonant dissipation stabilization is implemented, the nonlinear element is a three-wave mixing nonlinear element. As mentioned above, this resonant dissipation case corresponds to... ( This can be particularly advantageous in specific cases. It utilizes three-wave mixing nonlinear elements and a DC bias that enables the system to achieve 2-to-1 photonic resonance, as described in EP4207005A1.

[0097] The resonant portion of the circuit is arranged as a nonlinear element connected to the circuit to provide the nonlinear superconducting quantum circuit with a corresponding resonant frequency. and The first mode a and the second mode b. More specifically, the first mode a and the second mode b "participate" in the nonlinear element, which means that some or all of the mode magnetic energy is stored in the nonlinear element.

[0098] This nonlinear superconducting quantum system includes at least one resonant section, and typically includes two or more resonant sections to form two electromagnetic modes, including a cat qubit mode and a buffer mode.

[0099] The command circuit is arranged to deliver microwave radiation and optional DC bias. It is arranged to transmit at least at a frequency equal to the second resonant frequency. The frequency of radiation is delivered to the resonant part to drive the second mode b, because both parameter dissipation stabilization and resonant dissipation stabilization require this driving of the second mode b. Bias-holding gate X Gate

[0100] [ Figure 3 The X(π) gate (hereinafter referred to as the X gate) transforms |+α〉=|0〉 into |-α〉=|1〉 and vice versa, which is equivalent to performing a π rotation around the X-axis. It can be implemented in software or by implementing a Hamiltonian of the following form:

[0101] Where Δ is the coefficient and a is the ladder operator for memory mode.

[0102] This gate can maintain the noise bias where bit flipping is suppressed.

[0103] The Hamiltonian in Equation 1 can be viewed as a rotation of the coherent Gaussian distribution around the center of the complex plane at a rate Δ. Therefore, by implementing the Hamiltonian in Equation 1 for a duration such that the rotation angle is π, the X-gate is achieved.

[0104] Since stabilization prevents this process from occurring, it needs to be turned off and on again at the correct time. A bit-flipping error will occur if the rotation angle is different from π.

[0105] Therefore, the stabilization time T when the gate remains closed is a key parameter of the X gate.

[0106] Other implementations of the X-gate may exist, but are not described here. Z Gate

[0107] [ Figure 4 The Z(θ) gate (hereinafter referred to as the Z gate) is a single-qubit gate. It performs a rotation of angle θ about the Z-axis. It can be implemented using a displacement Hamiltonian of the following form:

[0108] Where ε is the displacement amplitude. Let be the shift phase, and let a be the ladder operator of the memory mode of the qubit. Such a Hamiltonian causes a rotation about the Z-axis of the qubit's Bloch sphere.

[0109] This gate can maintain the noise bias where bit flipping is suppressed.

[0110] [ Figure 5 The Hamiltonian in Formula 2 can be obtained by microwave driving at the memory frequency.

[0111] When applied to a stabilized cat qubit, the gate creates a transition from |C| via an angle θ. + >=|+> to state|C - The conversion between >=|-> is also true, and vice versa. Stabilization prevents the coherent state from shifting in the complex plane, and the net effect of the Hamiltonian in Equation 2 is a change in the cat-state interference pattern. The amplitude ε and duration T of the pulse are chosen to obtain the interference fringe rotation at the desired angle θ.

[0112] For the optimal phase modulo π This achieves better gate performance. Otherwise, bit-flipping errors occur during the gate, and the duration or amplitude must be increased to achieve the angle θ.

[0113] Therefore, ε, T and These are the parameters of the Z-gate. CNOT gate (also known as CX gate, used for controlled X gate)

[0114] The operation of the CNOT gate is shown in the following truth table, where the first bit is the state of the control qubit and the second bit is the state of the target qubit. If the first bit is 0, the gate leaves both bits unchanged. If the first bit is 1, the gate leaves the first bit unchanged and performs an X gate on the second bit, that is, changes the second bit to its opposite bit.

[0115] This gate can maintain the noise bias where bit flipping is suppressed.

[0116] The natural expression for the CNOT gate is: when the control qubit is in a coherent state |+α> C The target qubit remains idle, and the control qubit is in a coherent state |-α>. C Rotate by π (in the complex plane).

[0117] [ Figure 6Otherwise, a symmetric picture is adopted by performing a reference frame rotation on the target qubit, such that it is always rotated, but by +π / 2 or -π / 2 around the origin in the complex plane depending on the state of the control qubit. The corresponding truth table is as follows:

[0118] The physical system on which CNOT is implemented consists of two coupled circuits. One circuit is the control qubit, and the other circuit is the target qubit.

[0119] [ Figure 7 The Hamiltonian used to implement this gate, controlling the longitudinal coupling between the qubit and the target qubit, can be written as:

[0120] Among them, g CNOT is the CNOT interaction strength, and a and b are the ladder operators used to control the qubit and the target qubit, respectively.

[0121] Figure 7 The simplified effect of the CNOT gate is shown. In addition to what is shown, controlling the cat state in the qubit also causes the target qubit to split into two clumps. Parameters of CNOT pulse

[0122] [ Figure 8 As can be seen from the Hamiltonian in Equation 3, the rotation of the target qubit in the complex plane is related to g. CNOT It occurs at a rate proportional to the speed.

[0123] T CNOT This is called the CNOT gate time. At the end of the CNOT gate time, the target qubit rotates as needed, so the parameter of the gate has the following form:

[0124] Where k is a real number.

[0125] The Hamiltonian in Equation 3 is activated by a microwave drive that is sent to the ATS of the target qubit at a frequency controlled by the memory of the qubit.

[0126] Starting with the stabilized control qubit and target qubit, the stabilization of the target qubit is turned off to allow conditional rotation of the target qubit. Conversely, the stabilization of the control qubit remains on.

[0127] Then, the conditional rotation of the target qubit is activated by a microwave drive sent to the target ATS at the frequency of the memory controlling the qubit. This drive lasts for a duration T. CNOT Stay active during this period.

[0128] Once the desired conditional rotation is completed (i.e., at T...) CNOT After a certain period of time, stabilization on the target qubit is restarted, while the CNOT pulse (i.e., the drive on the ATS of the target qubit) is turned off.

[0129] The parameters of the CNOT pulse itself (the ATS sent to the target cat qubit to implement the Hamiltonian of Equation 3) need to be calibrated, and one of these parameters is a) its phase angle, and the other is b) its duration T. CNOT The integral of the amplitude over time. a) CNOT pulse phase angle

[0130] Generally, there is a free phase degree of freedom between the stabilized control qubit cat state and the longitudinal Hamiltonian in Equation 3. Without loss of generality, this relative phase can be written as being carried by the longitudinal Hamiltonian in Equation 3, as follows:

[0131] Physically, phase difference This stems from the different optical paths traversed by the stabilization drive and the CNOT pulse.

[0132] For a given CNOT pulse amplitude g CNOT and duration T CNOT , Each value corresponds to the rotation of the target qubit cat state in the complex plane.

[0133] The effects on the target qubit and the control qubit are very different.

[0134] They are equal in the sense of modulo π. The optimal value corresponds to the maximum rotation angle that the coherent state of the target qubit spans around the origin in the complex plane.

[0135] And, more importantly, phase Non-optimal values ​​can cause bit-flipping errors in the control qubits during gate execution. b) The integral of the CNOT pulse amplitude over time during the pulse duration.

[0136] If the CNOT pulse amplitude is the integral of the pulse duration over time... Miscalibration can lead to undershoot or overshoot in the conditional rotation of the target qubit. When the control qubit is in a cat state and the target qubit is a single cluster, the target cluster will split into two clusters rotating in opposite directions. If undershoot or overshoot exists due to miscalibration, the final separation between these clusters will not be the maximum value (i.e., a separation of π angles). Therefore, a bit-flip error will occur when stabilization is reactivated. Thus, the time integral of the miscalibrated pulse amplitude over the pulse duration will cause a bit-flip error in the target qubit. Additional driver parameters

[0137] Stray terms emerge, and experimental defects need to be corrected. One of these stray terms corresponds to the direct drive on the control qubit and has a Hamiltonian of the following form:

[0138] The coefficient ξ depends on the amplitude and phase of the pulse transmitted on the ATS of the target cat qubit.

[0139] The spurious Hamiltonian in Equation 6 can be canceled out by an additional drive of the following form sent from the memory controlling the qubit:

[0140] Make:

[0141] In practice, this is implemented by an additional pulse (called a cancellation pulse) emitted simultaneously with the main gate pulse, i.e., a pulse sent on the ATS of the target cat qubit (as discussed above).

[0142] The cancellation pulse is used to control the frequency of transmission of the cat qubit memory.

[0143] The amplitude and phase of the tuning canceling pulse are effectively tuned to the parameter g in the Hamiltonian of Formula 7. cancel and This verifies the equation in Formula 8. Parameters for restarting the stabilization pulse on the target cat qubit

[0144] Finally, after the gate is completed, when the coherent state +π / 2 or +π / 2 rotation of the target qubit has been obtained in the complex plane with the gate parameters properly calibrated, the stabilization on the target qubit is switched back on.

[0145] As explained above, this stabilization involves sending waves to the buffers of the ATS and the target qubit, respectively. These two waves must be restarted with a defined phase angle.

[0146] Due to the detuning of the target qubit's memory and the influence of other stray terms, an overall rotation of the target qubit's coherent state is further superimposed on the target qubit conditional rotation provided by the gate.

[0147] Therefore, set the angle θ T The angle θ T It is defined as the phase angle difference between the stabilized waves sent to the buffers of the ATS and the target qubit, respectively, when stabilization is reactivated.

[0148] The miscalibrated target stabilization differential phase angle will actually cause bit flip errors in the target qubit due to the offset between the stabilization and the actual position of the coherent state. Summary of CNOT parameters

[0149] therefore , g cancel and θ T These are the parameters for the CNOT gate.

[0150] Incorrect calibration of certain gate parameters can lead to bit-flip errors. This disclosure relates to measuring bit-flip errors across various settings of the physical implementation parameters of a scan gate. This allows for sensitive and robust optimization of the gate using bit-flip errors as a metric. This disclosure also relates to a set of calibration procedures for pulse parameters. Using bias-hold gates, the reduction in bit-flip errors is maintained during gate execution. The calibration procedures utilize bit-flip errors, taking full advantage of stabilization and noise bias to tune the hardware parameters used for the gate implementation. Thanks to stabilization and noise bias, bit-flip-based calibration is robust. Detailed calibration method

[0151] [ Figure 9 The parameter calibration method according to the present invention includes, in a general manner, calibrating at least one parameter, defined on a single-qubit or multi-qubit system, for an interaction sequence used to perform physical quantum gates in a superconducting circuit based on a cat qubit having a reduced bit-flip error rate as a noise bias, for a cat qubit having a reduced bit-flip error rate as a noise bias.

[0152] For a qubit in the system that is affected by a change in at least one parameter, the qubit is identified as a tested qubit, and a list of at least one state of the qubit system is provided.

[0153] For the value of the at least one parameter selected in step S10 of selecting parameter values, and for each state in the list, the following steps are performed: -Prepared in step S20; - In step S30, an interaction sequence is applied to the prepared state using the parameter values; - At the end of the application, measure the unmodified population of the affected (tested) qubits and the bit-flipped population; - In step S50, the states in the list are changed, and the preparation, application, and measurement steps are iterated. - In step S60, the parameter value is then changed, and the preparation, application, and measurement steps are iteratively performed for different values ​​of at least one parameter, scanning the candidate value range of at least one parameter; - In step S70, the optimal value for maintaining the gate against the noise bias is then determined based on the population measured during the iteration step, and in step S80, the derived parameter value associated with the determined optimal value is implemented as a calibration value for at least one parameter.

[0154] The optimal value for gate-to-noise bias preservation can be: (i) the symmetry between the bit-flipped populations of the measurements of two or more states in the list; and / or (ii) the minimum value of the bit-flipped populations of the measurements of one or more states.

[0155] The following paragraphs provide a calibration procedure for bit-flipping errors using the gates described above. X Gate

[0156] [ Figure 10 The X-gate can be destabilized for an appropriate duration, resulting in a rotation angle of π, and then stabilization can be reactivated. Calibration can be performed as follows during the destabilization period:

[0157] Preparation state |+α〉;

[0158] Destabilize over a given duration;

[0159] Restore stabilization and immediately measure the number of bit-flip errors relative to the expected final state |-α〉. In practice, stabilization can be restored, and the process can wait between 10 ns and the inherent bit-flip time, depending on the parameters, before measuring only the population.

[0160] Then perform this operation multiple times at different durations of stabilization release.

[0161] Here you can use a list with only one state—then there's no need to iterate over another state.

[0162] This can also be done by preparing the state |-α〉 instead of the state |+α〉 and measuring the error with respect to the expected final state (i.e., |+α〉).

[0163] The optimal duration is the duration with the minimum bit-flipping error.

[0164] This figure illustrates a qualitative representation of the X-gate calibration assuming |+α〉 as the initial state. The stabilization release time is scanned along the x-axis, with the populations of |+α〉 and |-α〉 measured at each time step. The optimal release time minimizes the bit flips relative to the target state |-α〉 (indicated by the black vertical line), thereby maximizing the population contrast. Z(θ) gate

[0165] The Z(θ) gate can be implemented using a pulse at the memory frequency, characterized by its amplitude, duration, and phase angle. Once the duration is known (predetermined) and fixed, the phase angle of the pulse can be calibrated as follows:

[0166] Preparation state |+α〉;

[0167] Stabilization is enabled, and a Z-gate pulse with a given phase angle is applied to the qubit for a duration equal to an integer multiple of a fixed and predetermined duration (e.g., 1, 2, ... N).

[0168] At the end of the Z-gate pulse, the populations of |+α〉 and |-α〉 are measured, and thus the number of bit-flipping errors can be derived if necessary;

[0169] Repeat with different values ​​of a given phase angle;

[0170] Optionally, but advantageously, the duration may be repeated in other integer multiples;

[0171] Then the process is repeated with the prepared state |-α> (because a list of two coherent states with a phase difference of π is used in this embodiment—however, an alternative is given below).

[0172] The optimal phase angle of the pulse ensures that the number of bit-flipping errors is equal for both states of preparation (|+α〉 and |-α〉). Compared to existing methods, this approach finds the optimal phase from clear and sharp features in the data, reducing experimental uncertainty.

[0173] For each state of preparation, the data analysis is as follows:

[0174] for For each value of , the measured population p(N) of the qubit will be fitted with respect to N as an exponential decay in the following form:

[0175] Where A is the pre-factor, B is the offset, and Γ is the decay rate of the curve, which is related to the bit flip rate during gate execution. For each Obtain a value for Γ.

[0176] Then, the Γ pair is fitted as a linear function of the following form:

[0177] Where m and q are the fitting parameters.

[0178] The intersection of the two linear fits corresponds to the optimal point. .

[0179] As described above, the scenario begins with the preparation state |+α> and is repeated with the preparation state |-α>. However, this method can also utilize cat states for preparation, i.e., states |+> or |->. In this case, the procedure could be, for example: prepare state |->, apply a CNOT pulse, and measure bit flips. The optimal phase minimizes bit flips. Using this alternative, the list could be a list of states with only one state, but that state is a cat state. CNOT Gate

[0180] Bit flip errors can be used to calibrate different parameters of CNOT gates. 3.1) Additional driver H CANCEL

[0181] The given H CANCEL The amplitude and phase of the cancellation pulse can be tuned using a bit-flip error in the following way:

[0182] Preparation state |+α> C |0〉 T ;

[0183] The CNOT pulse and the canceling pulse are applied to the qubit with a given amplitude and phase of the canceling pulse;

[0184] Measuring the number of bit-flipping errors in a control qubit;

[0185] Repeated to cancel out different values ​​of pulse amplitude and phase;

[0186] In preparation state |-α> C |0〉 T 、|+iα〉 C |0〉 T 、|-iα〉 C |0〉 T The process is repeated, where i is the imaginary unit (in this embodiment, there are two pairs of coherent states in the list, with a phase difference of π between the two states in each pair). This is for one qubit; the other qubit remains unchanged in the list.

[0187] The optimal amplitude and phase of the cancellation pulse ensure that the number of bit-flip errors is as equal as possible for all four states.

[0188] Note that the alternative usage is |-α〉. C |0〉 T 、|-α〉 C |0〉 T 、|-α〉 C |0〉 T This method can also be performed using any two pairs of coherent states, where the phase difference between the states in each pair of coherent states is π. 3.2)

[0189] [ Figure 11 The phase of the CNOT pulse can be tuned using bit-flip errors in the following way:

[0190] Preparation state |+α> C |+α〉 T ;

[0191] Activate the stabilization on the control qubit and apply a CNOT pulse to the qubit with a given phase value and a given duration;

[0192] Measurement |+α〉 C The population number;

[0193] Similar to the Z-gate mentioned above, in practice, stabilization on the control qubit can be restored by waiting between 10 ns and the inherent bit-flipping time of the control qubit according to the parameters, and then only the population of the control qubit is measured.

[0194] exist Figure 11 In the diagram, sections a) and c) show the measured population of the control qubits: in a) for the preparation of |+α> C |±α〉 T State of time |+α〉 C The population number, in c), is the number of cells used to prepare |-α>. C |±α〉 T State of time |-α〉 C The population. The x-axis shows the duration of the applied pulse increment (represented by integers from 1 to 446 of the predetermined basic pulse length, where each N consists of a CNOT pulse followed by some stabilization time), and the y-axis shows the phase angle value used. Figure 11 In the experiment, the delay between the end of the CNOT pulse and the measurement of the control qubit (i.e., the stabilization time) was approximately 2 ms.

[0195] Optionally, but advantageously, the pulses can be repeated for other CNOT pulse durations (which can be integer multiples of the predetermined pulse duration N).

[0196] Repeat with other values ​​of the CNOT pulse phase angle.

[0197] For example, the CNOT pulse phase is scanned in the range of -0.3 rad to 0.1 rad, and the CNOT pulse number N is in the range of 1 to 446.

[0198] In preparation state |-α> C |+α〉 T Repeating the process, where the state of the target qubit remains unchanged across all fabrications, but the state of the control qubit changes to the opposite state—in this case, measuring |-α> C The population number.

[0199] The optimal phase of the CNOT pulse ensures that the number of control bit flip errors is equal for both states.

[0200] This can also be achieved by preparing the state |+α> C |-α〉 T And then |-α〉 C |-α〉 T To complete.

[0201] Calibration can be performed by applying multiple CNOT pulses (changing N), which is an optional step that can be repeated with different pulse durations. By increasing the number N, the bit flip can be "amplified," resulting in higher resolution bit flip measurements.

[0202] For each state of preparation, the data analysis is as follows:

[0203] for Each value of will control the measured population p(N) of the qubit to fit an exponential decay with respect to N as follows:

[0204] Where A is the pre-factor, B is the offset, and Γ is the decay rate of the curve, associated with the bit flip rate during gate execution. For each The value is obtained by taking a Γ value. It should be understood that for each... Obtaining a value for Γ is effective in the following experimental mechanism: the control qubit is stabilized for a short period of time after the CNOT pulse ends and before the measurement. This short period of time is between 2 ns and 5 times the intrinsic bit-flip time of the control qubit, and most preferably between 10 ns and the intrinsic bit-flip time of the control qubit. In other experimental mechanisms (e.g., when the measurement of the control qubit is performed almost immediately after the CNOT pulse ends), the pre-factor A or offset B can alternatively contain the required information, thus tailored to each... The value is obtained from A and / or B.

[0205] exist Figure 11 In the figure, parts b) and d) show the fitting of the data in parts a) and c) to Equation 11, respectively.

[0206] exist Figure 11 The coordinate axes are labeled in the same way in all sections a) through d).

[0207] Γ pair Fit to a linear function of the following form:

[0208] Where m and q are the fitting parameters.

[0209] exist Figure 11 In section e), the example shows the action for |+α>. C |±α〉 T Preparation (decreasing line) and targeting |-α> C |±α〉 T Preparation (incremental line) extraction of Γ and The comparison is performed, and Equation 12 is fitted. The optimal phase is found at the intersection of these fitted equations.

[0210] The intersection of the linear fits prepared from the two states corresponds to the optimal point. .

[0211] Determine the optimal CNOT phase as The associated exponential decay rate is approximately 0.008 ns. -1 .

[0212] As mentioned above, this scheme begins with the preparation state |+α> C |+α〉 T And through the preparation of state |-α> C |+α〉 T The process is repeated. In this embodiment, there are two pairs of coherent states in the list, with a phase difference of π between the two states in each pair. This applies to one qubit; the other qubit remains unchanged in the list.

[0213] However, this method can also be used to prepare the state |+> or |-> by controlling the cat state within the qubit. Using this alternative, the list can contain only one state, but that state is the cat state controlling the qubit. The other qubit remains unchanged in the list. In this case, the procedure could be, for example, to prepare the state |+>. C |+α〉 T A CNOT pulse is applied, and bit flips in the control qubits are measured. The optimal phase minimizes bit flips.

[0214] Typically, the states in the target qubit are independent of the phase calibration of the CNOT pulse. 3.3) CNOT pulse amplitude and stabilization angle

[0215] [ Figure 12 [The CNOT pulse amplitude g used to restore stability after the pulse] CNOT Difference angle θ with the target qubit T (Hereinafter referred to as the stabilization angle, or "buffer phase"), can be tuned together within the same calibration procedure using bit flip errors in the following manner:

[0216] Preparation state |+α> C |+α〉 T ;

[0217] Given a CNOT pulse amplitude value, activate the stabilization on the control qubit, apply the CNOT pulse, and simultaneously destabilize the target qubit.

[0218] While maintaining stability on the control qubit, the CNOT pulse is applied to the target qubit;

[0219] Given a stabilization angle value θ T In this case, stabilization is initiated on the target qubit;

[0220] Before measurement, the sequence of controlling qubit stabilization and subsequently stabilizing the target qubit CNOT pulse on the target qubit can be repeated more than once—N times, i.e., integer multiples; this increases the resolution.

[0221] Measurement |+α〉 T and |-α〉 T The population number is then used to infer the number of bit-flipping errors in the target qubit.

[0222] Figure 12 Section a) shows that: the preparation is controlled to be |+α> C The target qubit p+ is shown in the case. The x-axis shows the CNOT pulse amplitude with relative units ranging from approximately 0.9 to 1.1. The y-axis shows the target qubit buffer phase (stabilization angle) ranging from approximately -4 rad to 4 rad.

[0223] exist Figure 12 The coordinate axes are labeled in the same way in all sections a) to h).

[0224] Figure 12 Section b) shows that: the control of the quantum bit preparation is as |+α> C The target qubit p- in the case is shown in section c): the control qubit is prepared as |-α>C The target qubit p+ in the case of; and as shown in section d): the preparation of the control qubit as |-α> C The target qubit p- under the condition.

[0225] Repeatedly scan with various CNOT pulse amplitude values ​​and stabilization angle values—CNOT pulse amplitude scanned in any unit range from 0.9 to 1.1, for example, and target stabilization angle θ. T Over the entire range from 0 radians to 2π radians.

[0226] Repeated preparation state |-α> C |+α〉 T .

[0227] In this embodiment, a list of two coherent states with a phase difference of π is used for one qubit. The other qubit remains unchanged in the list.

[0228] The optimal CNOT pulse amplitude and stabilization angle minimize the number of bit flips for both preparation methods.

[0229] This can also be achieved by preparing the state |+α> C |-α〉 T And the subsequent |-α〉 C |-α〉 T To complete.

[0230] For each state of preparation, the data analysis is as follows:

[0231] For each amplitude of the CNOT pulse, it will be respectively in |+α> T and |-α〉 T The target populations p+ and p- in the model are fitted to the following functions:

[0232] Where A is the pre-factor, B is the population offset, and θ0 is the offset of the stabilization angle; for each amplitude of the CNOT pulse, the optimal stabilization angle θ T This maximizes the difference between p+ and p-.

[0233] exist Figure 12 In the diagram, parts e), f), g), and h) are the fits to the data in parts a), b), c), and d), respectively. The dashed line represents the optimal θ found by fitting the amplitude of each CNOT pulse. T .

[0234] The optimal stabilization angle θ T The CNOT pulse amplitude is fitted as a linear function.

[0235] exist Figure 12 In part i), the optimal θ T The CNOT pulse amplitude is fitted as a linear function.

[0236] |+α> were prepared in two states. C |+α〉 T and |-α〉 C |+α〉 T Find the optimal θ at the intersection of the linear fits. T and CNOT pulse amplitude.

[0237] The intersection of these fits corresponds to the optimal CNOT pulse amplitude and the target stabilization angle θ. T .

Claims

1. A method for calibrating at least one parameter of an interaction sequence for performing a quantum operation, the quantum operation being defined on a single-qubit or multi-qubit system in a cat-qubit-based superconducting circuit, the cat-qubit having a reduced bit-flip error rate as a noise bias, the method comprising: For a qubit in the system affected by a change in the at least one parameter, and for a list of at least one state of the qubit system, perform the following steps for the value of the at least one parameter and for each state in the list: The state described in (S20) is prepared; In the prepared state, the interaction sequence is applied (S30) using the parameter values; At the end of the application, the bit-flipped or non-bit-flipped population of the affected qubit is measured (S40); The steps of preparation, application, and measurement are iterated (S50, S60, S10, S20) using the variation value of the at least one parameter to scan the candidate value range of the at least one parameter, and then the optimal value maintained by the operation for the noise bias is determined (S70) by the population measured during the iterative steps, and the inferred parameter value associated with the determined optimal value is implemented (S80) as the calibration value of the at least one parameter.

2. The calibration method according to claim 1, wherein, The at least one parameter includes the phase angle of a microwave pulse sent to a nonlinear coupling element that couples a memory of the qubits in the system to a buffer of the qubits.

3. The calibration method according to claim 1, wherein, The at least one parameter includes the phase angle of a microwave pulse transmitted at the frequency of the memory of the qubits in the multi-qubit system.

4. The calibration method according to claim 1, wherein, The at least one parameter includes the duration of the de-stabilization of the drive stabilization of the qubits in the system.

5. The calibration method according to claim 1, wherein, The at least one parameter includes the integral of the amplitude of a microwave pulse sent to a nonlinear coupling element during the implementation of the operation, the nonlinear coupling element coupling a memory of the qubits in the system to a buffer of the qubits.

6. The calibration method according to claim 1, wherein, The at least one parameter includes the integral of the amplitude of a microwave pulse transmitted at the frequency of the memory of the qubits in the system over time during the implementation of the operation.

7. The calibration method according to claim 1, wherein, The at least one parameter includes the difference in phase angle between two microwaves, which, after the operation is performed, are respectively sent as drive stabilization to a buffer associated with the qubit of the system and a nonlinear coupling element of a memory coupling the associated buffer to the qubit.

8. The calibration method according to any one of claims 1-7, wherein, Prior to the measurement, the interaction sequence is applied several times, wherein the number N of times the interaction sequence is applied is further from N... min Change to N max , and perform the measurement for each value of N used.

9. The calibration method according to any one of claims 1-8, wherein, The at least one parameter comprises two real numbers, and the application of the interaction sequence is performed several times while the two real numbers are changed independently.

10. The calibration method according to any one of claims 1-9, wherein, The list of states includes one or more pairs of coherent states, wherein there is a phase difference π between the two states in a given pair, or the list of states includes cat states, or the list of states is a list consisting of only one coherent state.

11. The calibration method according to any one of claims 1-10, wherein, Determining the optimal value includes: By measuring the population during the iteration steps, determine the minimum bit flip rate or bit flip number associated with the different states of the list, or determine the minimum distance between the bit flip rates or bit flip numbers associated with the different states of the list, or find both the minimum bit flip rate or bit flip number associated with the different states of the list and the minimum distance between the bit flip rates or bit flip numbers associated with the different states of the list.

12. The calibration method according to any one of claims 1-11, wherein, The quantum operation is one or a combination of X-gate, CNOT gate, Toveley gate, state preparation, and parity check measurement.

13. The calibration method according to any one of claims 1-12, wherein, The cat qubit is stabilized by driven two-photon dissipation.

14. The calibration method according to any one of claims 1-13, wherein, Measurements are performed via Wigner tomography or any parity-checking measurement procedure, particularly direct or indirect parity-checking measurements, and especially mapping different parities to different populations in the memory of the affected qubits; or where... The measurement is performed by determining the number of photons in the memory of the affected qubit.

15. A computer program comprising instructions that, when executed by a computer, cause the computer to perform the method according to any one of claims 1-14.

Images