Method for calibrating sequence parameters to perform quantum operations
Patent Information
- Application Number
- KR1020267029310
- Authority / Receiving Office
- KR · KR
- Patent Type
- Applications
- Current Assignee / Owner
- Priority Date
- 2024-03-05
- Filing Date
- 2025-02-27
- Publication Date
- 2026-09-17
Smart Images
Figure PCT00109_ABST
Abstract
Description
Technology Field
[0001] background
[0002] The present disclosure relates to the field of quantum technologies, and more specifically to the calibration of quantum circuits associated with quantum information processing and quantum computing. Background Technology
[0003] Quantum information processing that enables quantum computing consists of encoding and manipulating quantum information in two-level quantum systems—named qubits. Qubit states are represented on a sphere called the Bloch sphere, where the north and south poles of the Bloch sphere correspond to the standard basis vectors |0> and |1>. Any pure state of a two-level quantum system can be described as a superposition of the basis vectors |0> and |1>, where the contribution from each of the two basis vectors is a complex number.
[0004] In this context, logic gates and related operations are implemented as sequences of interactions including wave pulses, particularly microwave pulses. Some of these operations and gates operate on a single qubit, while others operate on multi-qubit systems. Each operation or gate has a name and possibly one or several parameters, particularly numerical parameters.
[0005] Typical examples of single-qubit gates are X(θ) and Z(θ) gates, which are rotations of an angle θ centered on the X and Z axes of the Bloch sphere, respectively.
[0006] An example of a multi-qubit gate is a 2-qubit gate control NOT (abbreviated as CNOT or CX). The action of the CNOT gate on an input is to apply a NOT operation to one of the qubits—designated as the target qubit—or to leave it unchanged, conditioned on the state of the other qubit—designated as the control qubit. The control qubit remains unchanged throughout all operations of the CNOT gate.
[0007] Different parameters of the physical realization of operations and gates need to be calibrated to implement operations on quantum hardware. Calibration can be performed possibly only once, or most likely, at critical points in hardware usage or at any time.
[0008] Regarding hardware construction, encoding quantum information in the states of harmonic oscillators, which are boson modes (the most widely known bosons are photons), is a solution adopted by various entities in the field. Superconducting microwave circuits within cryostats, including LC resonators and Josephson junctions, are used to implement qubits and gates. Specifically, in such superconducting circuits, gates can be implemented as individual microwave pulses or combinations of pulses, and thus the parameters to be optimized are, in particular, the amplitudes, phases, and durations of the pulses.
[0009] However, when considering operations and gates, the problem of quantum noise must be taken into account because isolated qubits undergo two fundamental errors, namely bit flips and phase flips, and error rates increase with the size of the qubit. Noise is generally a major topic in quantum circuits, and strategies are devised to control its effects with hopefully limited resources.
[0010] One of these strategies is the design of noise-biased qubits, and in particular, so-called cat qubits are noise-biased qubits. In these qubits, one of two types of errors is suppressed exponentially with the cat qubit size—which is the average number of photons in the oscillator—while the other type of error increases only linearly with size. Protection or stabilization of the qubits is achieved through two-photon driven dissipation or through Kerr-type Hamiltonian and two-photon drives.
[0011] According to current knowledge, suppression is applied to physical noise processes such as photon loss, thermal excitations, photon phase shifts, and various nonlinearities induced by coupling to Josephson junctions.
[0012] And due to this noise structure, error correction becomes easier to implement logic qubits.
[0013] Autoparametric resonance extending the bit flip time of a cat qubit to 0.3 seconds was presented by Marquet et al. in arXiv:2307.06761 2023. Control of a cat qubit having bit flip times exceeding 10 seconds was presented by Reglade et al. in arXiv:2307.06617 2023.
[0014] Quantum gates and operations operating on noise-biased qubits need to be noise-preserving to enable coherent implementations of quantum circuits. Guillaud and Mirrahimi Phys. Rev. X 9, 041053 2019 listed the following basic operations at the cat qubit level as bias-preserving: preparation of the |+> cat state or |-> cat state, measurement of X, X gate, CNOT gate, Tofoli gate, rotation of a given angle around Z, and 2-qubit entanglement gate.
[0015] Marquet et al. consider the calibration of the drive phase of a Z gate by observing how quickly Wigner function fringes shift over time and extracting the oscillation rate. Reglade et al., arXiv:2307.06617 2023 disclosed essentially the same principles regarding this subject. Although this was an achievement at the time, the applicant recognized that further research is needed to improve the precision, accuracy, and reliability of gate and operation calibration methods.
[0016] In this context, a method for calibrating at least one parameter of an interaction sequence for performing a quantum operation, and in particular, a method for calibrating the parameter of a physical quantum gate when the operation is a gate, is provided. The interaction sequence is generally composed of a pulse sequence, but is not limited to a pulse sequence.
[0017] Generally, the operation is defined for a single or multiple qubit system in a superconducting circuit based on cat qubits having a bit flip error rate reduced as a noise bias, and the method is for a qubit of the system affected by a variation of at least one parameter and a list of at least one state of said qubit system, for a value of at least one parameter and for each state of the list,
[0018] - Step of preparing the above state,
[0019] - A step of applying an interaction sequence with parameter values to a prepared state,
[0020] - A step of measuring the bit-flipped or bit-unflipped populations of the affected qubits, or both, at the end of the above application,
[0021] The method includes the step of repeating the preparation, application, and measurement steps with varied values of at least one parameter to sweep a range of candidate values of at least one parameter, and then, the step of identifying an optimum of the preservation of the noise bias by calculation from the occupancy rates measured during the step of repeating the steps, and the step of implementing an inferred parameter value associated with the identified optimum as a calibrated value of at least one parameter.
[0022] The features of the present invention allow for highly accurate calibration of the parameters under review by utilizing measurements of bit-flip and / or bit-non-flip occupancy rates. In practice, the dependence of bit-flip rates on gate parameters is very high, and the results are highly accurate with lower uncertainty ranges and better final results. Robustness is ensured by noise bias—bit-flip rates can actually be arbitrarily small.
[0023] In various non-limiting embodiments, the following features are advantageous:
[0024] At least one parameter is the phase angle (frequency) of a microwave pulse transmitted to a non-linear coupling element that couples the memory of a qubit of the system to the buffer of said qubit. Pumping a non-linear coupling element, possibly an ATS or asymmetrically threaded SQUID, f a and f b It may also include (which is the frequency of the memory and buffer). In that case, the following features may be used: the system has two qubits including a control cat qubit and a target cat qubit; the gate may be a CNOT gate; the phase angle of the microwave pulse is the phase of the pulse with respect to the stabilization angle of the control cat qubit; the microwave pulse is frequency In this case, the memory of the target cat qubit is transmitted to a non-linear coupling element that couples the target cat qubit's buffer. The interaction sequence may include preparing a first state from a list, wherein the list includes: (i) +alpha_C + alpha_T and -alpha_C + alpha_T; or (ii) +alpha_C - alpha_T and -alpha_C - alpha_T (i.e., the alpha_Ts in the list are maintained consistently). The step of applying the interaction sequence may include transmitting a microwave pulse, wherein the parameter value is the phase angle. The measuring step includes measuring the number of bit flips of the initially prepared state of the control cat qubit (thus, if +alpha_C + alpha_T is initially selected, bit flips from +alpha_C are measured), and ultimately, the sequence is repeated for other states in the list. The optimal value for the preservation of the noise bias by the gate is identified when the number of bit flips of the control in the first state of the list is equal to the number of bit flips from the control in another state of the list.
[0025] At least one parameter may include the phase angle of a microwave pulse transmitted at the frequency of the system's qubit memory. In that case, the following features may be used:
[0026] - Microwave pulses can be transmitted to any part of the circuit and do not necessarily need to be transmitted to the memory itself (even if their effects reach the memory). The pulses can be transmitted through a transmission line directly coupled to the memory, but this is optional.
[0027] - The system may be a single cat qubit system, the gate may be a Z(θ) gate, and the phase angle of the microwave pulse is the phase of the pulse with respect to the stabilization angle of the cat qubit; the interaction sequence may be as follows - a first state ready from a list, where the list includes +alpha and -alpha. Applying the interaction sequence involves transmitting a microwave pulse, where the parameter value is the phase angle. Measuring involves measuring the number of bit flips from the ready state. This is repeated for other states from the list. The optimal value of the preservation of the noise bias by the gate is identified when the number of bit flips from the first state of the list is equal to the number of bit flips from other states of the list.
[0028] However, the system may also be a multi-cat qubit system (possibly having two qubits including a control cat qubit and a target cat qubit), the gate may be a CNOT gate, and the phase angle of the microwave pulse is the phase of the pulse transmitted at the frequency of the control's memory as a spurious effect cancellation pump; the phase angle of this cancellation microwave pulse may explicitly be the phase of the pulse with respect to the stabilization angle of the control cat. The interaction sequence may be as follows: a first state preparation from a list, wherein the list contains nothing for the target and includes: +alpha, i alpha, -alpha, and -i alpha for the control. Applying the interaction sequence involves transmitting a microwave pulse, wherein the parameter value is the phase angle. Measuring involves measuring the number of bit flips from the prepared state in the control cat qubit. This is repeated for other states from the list. The optimal value of the preservation of the noise bias by the gate is identified when the number of bit flips from the states of the list becomes equal.
[0029] At least one parameter may include the duration of the release of the driven stabilization of the system's qubit. The system may be a single CAT qubit, and the gate may be an X gate. The interaction sequence may begin by preparing +alpha or -alpha. Applying the interaction sequence may include the duration of the release of the stabilization of the single CAT qubit. The measurement may include measuring the number of bit flips from the prepared state (or equivalently, measuring the number of bit flips from other states). It is not necessary to repeat for other states in the list. The list may contain only a single state. The optimal value for the preservation of the noise bias by the gate is identified when: if +alpha is prepared, the number of bit flips from -alpha is minimal (i.e., minimum occupancy in +alpha); or if -alpha is prepared, the number of bit flips from +alpha is minimal (i.e., minimum occupancy in -alpha).
[0030] At least one parameter is a non-linear coupling element that couples the memory of the system's qubit to the buffer of said qubit, and may include the integral of the amplitude of the transmitted microwave pulse over time during the implementation of the operation. In that case, the following features may be used: the system has two qubits, including a control cat qubit and a target cat qubit. The CNOT microwave pulse is frequency In this case, the memory of the target cat qubit is transferred to a non-linear coupling element that couples the target cat qubit's buffer. The sequence includes preparing a first state from a list, wherein the list includes: (i) +alpha_C + alpha_T and -alpha_C + alpha_T; or (ii) +alpha_C - alpha_T and -alpha_C - alpha_T (i.e., the alpha_Ts in the list are maintained consistently). The step of applying the interaction sequence includes transmitting a CNOT microwave pulse, wherein the parameter value is the time integral of the amplitude. The measurement step includes measuring the number of bit flips of the target cat qubit. The interaction sequence may also include releasing the stabilization on the target cat qubit. The steps are repeated for other states in the list. The optimal value for the preservation of the noise bias by the gate is identified when the number of bit flips from the target in the first state of the list is both of the following: (i) equal to the number of bit flips from the target in other states of the list; and (ii) minimum value.
[0031] At least one parameter may include the integral of the amplitude of a microwave pulse transmitted over time during the implementation of the operation at the frequency of the memory of the system's qubit. In that case, the following features may be used: the microwave pulse may be transmitted anywhere on the circuit and does not necessarily need to be transmitted anywhere on the memory itself (even if its effect reaches the memory). The pulse may be transmitted through a transmission line directly coupled to the memory, but this is optional. Also, the system may 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 of the pulse transmitted at the frequency of the memory of the control as a spurious effect cancellation pump; the interaction sequence may be as follows: a first state preparation from a list, where the list includes +alpha, i-alpha, -alpha, and -i-alpha on the control, or other pairs of states where the phase difference between the two states of the pair is π, and nothing on the target. The step of applying the interaction sequence includes transmitting a microwave pulse, wherein the parameter value includes the amplitude of the pulse and may also include the phase of the pulse. The measuring step includes measuring the number of bit flips from a prepared state on the control qubit. This is repeated for other states from the list. The optimal value for the preservation of the noise bias by the gate is identified when the number of bit flips from the states of the list becomes equal to one another.
[0032] At least one parameter, as a drive stabilization after the operation is performed, may include the difference in phase angles of two microwaves transmitted at the respective frequencies to a buffer associated with the system's qubit and a non-linear coupling element coupling the memory of said associated buffer and qubit. In that case, the following features may be used: the system may be a multi-cat qubit system, the gate may be a CNOT gate, and the stabilization is the stabilization of a target cat qubit; the interaction sequence may be as follows: a first state preparation from a list, wherein the list includes +alphaC and + / -alphaT and -alphaC and + / -alphaT. The step of applying the interaction sequence includes transmitting a microwave pulse, wherein the parameter value is the phase angle difference. The measurement step includes measuring the number of bit flips from the prepared state. This is repeated for other states from the list. The optimal value of the preservation of the noise bias by the gate is identified when the number of bit flips from the states of the list tends to be equal to one another.
[0033] In certain embodiments, the step of applying the interaction sequence may be performed multiple times prior to the measuring step, and the number of times N the interaction sequence is applied also varies from Nmin to Nmax, and the measuring step is performed for each value of N used.
[0034] At least one parameter may include two numeric real numbers, and the step of applying the interaction sequence may be performed multiple times while varying the two numeric real numbers independently.
[0035] The above list of states may include one or more pairs of coherent states or cat states having a phase difference of π between two states of a given pair, or it may be a list of only one coherent state.
[0036] Identifying the optimal value may include identifying, from the shares measured during the step of repeating the above steps, a minimum value of the bit flip rate or the number of bit flips or a minimum value of the distance (absolute value of the difference) between the bit flip rates or the number of bit flips associated with distinct states of the list, or finding both the minimum value of the bit flip rate or the number of bit flips and the minimum value of the distance between the bit flip rates or the number of bit flips associated with distinct states of the list.
[0037] The operation may be one or a combination of Z gates, X gates, CNOT gates, Topolly gates, state preparation operations, and parity measurement operations.
[0038] Cat qubits may also be stabilized by driven two-photon dissipation.
[0039] The measurement step may be performed via Wigner tomography or any parity measurement process, particularly direct or indirect parity measurements, and particularly by mapping different parities to different occupancy rates in memory. The parity of the state can be measured using transmon.
[0040] The measurement step may be performed without any parity measurement, but by measuring the number of photons / occupancy rates in memory. The Z(pi / 2) gate is performed using the dynamic Zeno effect, the X(pi / 2) gate is performed using the holonomic gate, and the Pauli operator Z is measured using a longitudinal readout. The first two steps are based solely on the two-photon dissipation mechanism. To achieve the third step, terms of the ATS Hamiltonian are pumped to engineer the required longitudinal coupling.
[0041] Accordingly, the measuring step is performed by Wigner tomography or any parity measurement process, in particular direct or indirect parity measurements, and in particular by mapping different parities to different occupancy rates in the memory of the affected qubit, or the measuring step is performed by determining the number of photons in the memory of the affected qubit.
[0042] A computer program is also provided, comprising instructions that cause the computer to perform the aforementioned method when the program is executed by the computer. Brief explanation of the drawing
[0043] FIG. 1 illustrates the mapping between resonator states and qubit states, along with corresponding complex plane representations. FIG. 2 illustrates a lumped element model of a circuit that enables stabilization in one embodiment. FIG. 3 illustrates pulses for an X gate according to one embodiment of the present invention. Figure 4 illustrates a graph representation of the action of the Hamiltonian performing the Z gate. FIG. 5 illustrates pulses for a Z gate according to one embodiment of the present invention. Figure 6 illustrates the operation of a CNOT gate having a target qubit in the initial state of |+ α>. Figure 7 illustrates a simplified graph representation of the action of a Hamiltonian performing a CNOT gate. FIG. 8 illustrates pulses for a CNOT gate according to one embodiment of the present invention. FIG. 9 illustrates a schematic representation of a calibration method according to the present invention. FIG. 10 illustrates the calibration of an X gate according to one embodiment of the present invention. FIG. 11 illustrates the results of calibration of the phase angle of a CNOT gate according to one embodiment of the present invention. FIG. 12 illustrates the results of calibration of parameters of a CNOT gate (amplitude integrated over time of a CNOT pulse and target stabilization angle) according to one embodiment of the present invention. Specific details for implementing the invention
[0044] Examples
[0045] Cat Cubits
[0046] [Fig. 1] Stabilized cat qubits encode quantum information in the quantum states of a linear resonator, which is generally represented as a function whose domain is the complex plane. Coherent states (i.e., specific states of a quantum harmonic oscillator) |+α> and |-α> are each represented as 2D Gaussians centered on opposite points in the complex plane.
[0047] The coherent states |+α> and |-α> can be used to encode quantum information into a qubit by identifying |+α> as |0> and |-α> as |1>. A general qubit state can be represented as a superposition of |+α> and |-α>.
[0048] Of particular interest are the superpositions referred to as cat states, that is, Given N+ and N- normalization constants,
[0049] |C+> = N + x(|+α> + |-α>) is denoted as |+> and
[0050] and
[0051] |C-> = N - x(|+α> - |-α>) is denoted as |->.
[0052] These are even and odd cat states, and other cat states can be obtained when different relative phases are used between |+α> and |-α>.
[0053] The |+> and |-> cat states have interference patterns between the |+α> and |-α> Gaussians, which are typical signatures of cat states.
[0054] A cat qubit is defined as a two-dimensional manifold consisting of two cat states.
[0055] Any physical implementation of these systems is affected by incompletenesses leading to quantum state deviations from desired |+α>, |-α>, and their superpositions. One class of errors is referred to as local errors. These errors affect the system in such a way that the complex plane representations of the states are deformed and drift toward the origin of the complex plane over time.
[0056] Stabilization—also coined as confinement—protects the system against these errors by efficiently fixing the coherent states |+α> and |-α>—and their superpositions—in place in the complex plane. In particular, R. Lescanne et al. (2020) demonstrated that cat qubits can be stabilized by a nonlinear transformation between two photons of the first mode a—memory mode or cat qubit mode—and one photon of the second mode b—called buffer mode.
[0057] Stabilized cat qubits are not error-free despite their stabilization. Bit flip errors are unwanted transitions between two qubit states. Therefore, when a bound coherent state is converted to another bound coherent state: a |+α> coherent state is converted to a |-α> coherent state, or vice versa, a bit flip error occurs.
[0058] [Fig. 2] At the hardware level, an indeterminate model of the circuit consists of a linear resonator referred to as the qubit memory—where quantum information is stored as |+α>, |-α>, and superpositions thereof—and an additional resonator referred to as the qubit buffer. The buffer contains a circuit element called an asymmetrically threaded SQUID (ATS), which enables the stabilization of memory states. The ATS is a SQUID (Superconducting Quantum Interference Device) loop that is shunted by a linear inductor to create two loops in which magnetic flux can be threaded to enable different operations. A simplified circuit is shown in Fig. 2, illustrating a lumped constant model of the circuit containing the elements that enable stabilization.
[0059] Stabilization or confinement is activated by two microwave drives, one for the buffer resonator and the other for the ATS, such that the sum of these two frequencies is equal to twice the frequency of memory mode. These drives are introduced through external drives.
[0060] Cat qubits may more generally be stabilized or restrained by one or more of the following exemplary methods:
[0061] Stabilization methods
[0062] a) Jump operator As a parametric dissipation stabilization having, where is the 2-photon extinction rate, and is the photon annihilation operator, and is a complex number defining the cat qubit. This jump operator has a extinction rate A lossy buffer mode b having, and a four-wave mixed device—typically a Josephson junction, an ATS, or a DC bias coupled to one or more Josephson junctions (the applicant filed European patent application EP 23306839.4 regarding this)—by coupling to cat qubit mode a and to the frequency of the four-wave mixed device Pump and frequency at Hamiltonian by enabling buffer mode drive in (Here is the photon extinction operator of buffer mode b, and This can be realized by engineering (where is the two-photon coupling rate), and, however, am.
[0063] b) Jump operator As a dissipative squeezing stabilization having, where is the squeezed two-photon extinction rate, and is the photon annihilation operator, and is a complex number defining a cat qubit, and and is the complex squeezing parameter It is the absolute value (modulus) and argument of. This jump operator is the extinction rate By coupling a lossy buffer mode b having, and a 4-wave mixing device—typically a Josephson junction or ATS—to cat qubit mode a, and frequencies , and Various pumps and frequencies at Hamiltonian as a drive in buffer mode b (Here is the photon annihilation operator of mode b, and This can be realized by engineering (which is the squeezed two-photon coupling rate), and, however, am.
[0064] c) As a variation of the above stabilization method b), the applicant filed European patent application EP 23175147.0 regarding this.
[0065] d) Kerr Hamiltonian As, here is the amplitude of the Kerr Hamiltonian, and is the photon annihilation operator, and is the average number of photons.
[0066] e) Detuned Cur Hamiltonian As, here is the amplitude of the Kerr Hamiltonian, and is the photon annihilation operator, and is a complex number defining a cat qubit, and is the detuning coefficient.
[0067] f) Two-photon exchange (TPE) Hamiltonian As, here is the complex two-photon coupling rate, and is the photon annihilation operator, and is a complex number defining a cat qubit, and are the descent and rise operators of a two-level system. This Hamiltonian can be engineered in the same way as the parametric dissipation stabilization a).
[0068] g) Jump operator As a resonance dissipation stabilization having, where is the 2-photon extinction rate, and is the photon annihilation operator, and is a complex number defining the cat qubit. This jump operator has a extinction rate This can be realized by coupling a lossy buffer mode b and a three-wave mixed device having to a cat qubit mode a, which is Hamiltonian (Here is the photon extinction operator of buffer mode b) is engineered, provided that the mode frequencies are substantially and Satisfies, and accordingly, the frequency A drive in buffer mode is added.
[0069] In particular, dissipation stabilization schemes including resonant dissipation stabilization scheme g) (specifically schemes a) to c)) have been found to be particularly effective in realizing cat qubits with very low bit flip errors. In addition, combinations of various dissipation stabilization schemes and other stabilization schemes are also particularly effective so that some components of dissipation stabilization still exist.
[0070] Bit flip errors
[0071] Bit flip errors occur when one bound state transforms into another bound state. For example, a |+α> state transforms into a |-α> state or vice versa. Bit flip errors are quantified by measuring the occupancy rates of the stabilized coherent states |+α> and |-α>.
[0072] For example, bit flip errors induced by an operation can be quantified in the following way: a) prepare and stabilize state |+α>, b) perform an operation—an interaction sequence leading to a quantum gate or any operation, which may simply wait for a duration of time while stabilization is enabled (on), for example, c) measure the occupancy of states |+α> and |-α>.
[0073] Assuming the desired result of the operation is state |+α>, in the case of few bit flip errors, a high occupancy rate in |+α> and a low occupancy rate in |-α> are measured; whereas when many bit flips occur during the operation, the occupancy rates of |+α> and |-α> tend to become equal, so they are measured as being the same or close to the same.
[0074] The occupancy rates of states |+α> and |-α> can be measured, for example, by measuring the parity of corresponding +α and -α points in the complex plane, where even parity corresponds to high occupancy and zero parity corresponds to no occupancy.
[0075] Measuring the occupancy of states can be done by performing Wigner tomography and comparing the values at the locations of two "blobs". Then, the occupancy P(±α) of states |+α> and |-α> is 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] Since the Wigner function is proportional to parity, measuring the Wigner function is therefore equivalent to measuring parity.
[0078] Parity can be measured, for example, in the following ways: - Encode parity into the state of the auxiliary transmon and measure the state of the transmon.
[0079] Parity can also be measured by transmon-free readings. Parity measurements and photon count measurements are known. For example, see Reglade et al. in arXiv:2307.06617 2023, which describes a measurement process using transmon-free readings, and Lescanne et al. in Nature Physics volume 16, pages 509-513 (2020), which describes a measurement process using transmons.
[0080] Since two specific locations ±α are known a priori, a full Wigner tomography is not necessary: only Wigner tomography that provides a Wigner function of these locations is useful for measuring occupancy rates.
[0081] Using occupancy measures, the bit flip error rate Γ can be given / extracted by the following, and
[0082] Γ = [ P(+α) - P(-α) ] / Δt
[0083] Here, Δt is the duration of the operation.
[0084] The above calculation can be performed starting from the |-α> state.
[0085] An example of measuring bit flip times according to the above method is presented in Lescanne et al. in Nature Physics volume 16, pages 509-513 (2020).
[0086] Dissipating circuits
[0087] In dissipation stabilization schemes, the cat qubit is Jump operator represented by - or dissipator - It is stabilized through engineering. Parametric dissipation stabilization is terms and It depends on generating. term corresponds to the nonlinear transformation between two photons of the first mode a - memory - and one photon of the second mode b - buffer -. This term is frequency It requires a pump at, and here is the resonance frequency of the first mode a, and is the resonance frequency of the second mode b. Therefore, such a pump is a two-photon pump. term The frequency It requires a drive of the second mode b in. Therefore, such a drive is a two-photon drive. Resonant dissipation stabilization is It responds to specific cases.
[0088] The quantum system includes a nonlinear superconducting quantum circuit and a command circuit. The nonlinear superconducting quantum circuit is arranged to enable three-wave mixing or four-wave mixing between a first mode a and a second mode b. Hereinafter, the first mode a is used as a memory hosting a cat qubit, while the second mode b is used as a buffer between the cat qubit and the external environment.
[0089] The first mode a and the second mode b correspond to the intrinsic resonant frequencies of the nonlinear superconducting quantum circuit, respectively. Therefore, the first mode a and the second mode b each have a respective resonant frequency. The first mode a is the resonant frequency With , the second mode b is the resonant frequency has, here and are the respective angular frequencies of the first mode a and the second mode b. Thus, a nonlinear superconducting quantum circuit comprises components operating in a superconducting regime that host the modes independently or simultaneously. The first mode a and the second mode b may be hosted on components of different subsets of the superconducting circuit or on components of the same subset.
[0090] A nonlinear superconducting quantum circuit receives microwave radiation delivered by a command circuit that causes various nonlinear interactions between the first mode a and the second mode b. The frequency of each microwave radiation is tuned to select specific terms within the rotating wave approximation.
[0091] A nonlinear superconducting quantum circuit includes a nonlinear element and at least one resonant part.
[0092] Non-linear elements are jump operators for stabilizing cat qubits It generates. Alternatively, the nonlinear element and at least one resonant part are arranged to otherwise constrain the cat qubit through methods d), e), or f).
[0093] When parametric dissipation stabilization is implemented, the nonlinear element may be a 4-wave mixed nonlinear element.
[0094] A 4-wave mixed nonlinear element is, for example, an ATS. As successfully demonstrated by R. Lescanne et al. (2020), to perform parametric dissipation stabilization, the ATS is a 2-to-1 photon conversion, i.e., a jump operator. Article 1 of It is known to those skilled in the art that it can be used to engineer. More specifically, such a 2-to-1 photon conversion between the first mode a and the second mode b is a frequency It is obtained by parametrically pumping the ATS. Advantageously, the pump frequency is such that this parametric pumping operates as well as possible satisfies.
[0095] Unlike previous implementations of the stabilization method in which the superconducting circuit element used as a 4-wave mixer was a transmon with a single Josephson junction, the solution developed by R. Lescanne et al. (2020) utilizes an ATS design with cross-Kerr terms much lower than the transmon, thus allowing for exponential suppression of bit flips.
[0096] Alternatively, the nonlinear element performing the four-wave mixing may be something other than an ATS. For example, the nonlinear element may instead contain only one or more Josephson junctions.
[0097] When resonance dissipation stabilization is implemented, the nonlinear element is a three-wave mixed nonlinear element. As mentioned above, in this case of resonance dissipation ( It may be particularly advantageous to respond to specific situations. The three-wave mixed nonlinear element is used with a DC bias that causes the system to become 2-to-1 photon resonant, as described in EP4207005A1.
[0098] The resonant part of the circuit is each resonant frequency and It is arranged to be connected to a nonlinear element of the circuit to provide a first mode a and a second mode b having to the nonlinear superconducting quantum circuit. 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.
[0099] Such a nonlinear superconducting quantum system includes at least one resonant part, typically two or more resonant parts to form two electromagnetic modes including a cat qubit mode and a buffer mode.
[0100] The command circuit is arranged to deliver microwave radiation and optionally a DC bias. The command circuit is at least at a second resonant frequency It is arranged to drive the second mode b by delivering radiation of the same frequency to the resonant part, because both parametric dissipation stabilization and resonant dissipation stabilization require such a drive of the second mode b.
[0101] Bias-preserving gates
[0102] X Gate
[0103] [Fig. 3] The X(π) gate, referred to as the X gate below, transforms |+α>=|0> to |-α>=|1> and vice versa, which is equivalent to performing a rotation of π around the X-axis. This can be realized in software, or by implementing a Hamiltonian of the following form.
[0104]
[0105] Δ is the coefficient and a is the ladder operator for memory mode.
[0106] This gate can preserve noise bias where bit flips are suppressed.
[0107] The operation of the Hamiltonian of Equation 1 can be viewed as a rotation of coherent state Gaussians around the center of the complex plane with a rate Δ. Therefore, the X gate can be realized by implementing the Hamiltonian of Equation 1 for a duration such that the rotation angle is π.
[0108] Since stabilization prevents such processes from occurring, it needs to be deactivated (off) and reactivated at the precise time. Bit flip errors occur when the rotation angle differs from π.
[0109] Therefore, the time T during which stabilization remains deactivated is a key parameter of the X gate.
[0110] Other implementations of the X gate may exist, but are not described in this specification.
[0111] Z Gate
[0112] [Fig. 4] The Z(θ) gate, referred to as the Z gate below, is a single qubit gate. It performs a rotation of an angle θ around the Z-axis. This can be realized by a displacement Hamiltonian of the following form, and
[0113]
[0114] Here, ε is the displacement amplitude, φ is the displacement phase, and a is the ladder operator for the memory mode of the qubit. Such a Hamiltonian derives rotations of the qubit Bloch sphere around the Z-axis.
[0115] This gate can preserve noise bias where bit flips are suppressed.
[0116] [Fig. 5] The Hamiltonian of Equation 2 can be obtained from a microwave drive at memory frequency.
[0117] When applied to a stabilized cat qubit, this gate changes state |C by an angle θ. + State from >=|+> |C - It generates transformations from >=|-> and the reverse transformation. Stabilization prevents the coherent states from being displaced in the complex plane, and the net effect of the Hamiltonian in Equation 2 is a change in the interference pattern of the cat states. The pulse amplitude ε and duration T are selected to obtain a rotation of the interference fringes by a desired angle θ.
[0118] Better gate performance is achieved for the optimal phase φ modulo π. Otherwise, bit flip errors occur during gate, and the duration or amplitude must be increased to reach the angle θ.
[0119] Therefore, ε, T, and φ are the parameters of the Z gate.
[0120] CNOT gate (also named CX gate, meaning control X gate)
[0121] 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 keeps both bits unchanged. If the first bit is 1, the gate keeps the first bit unchanged and performs an X gate on the second bit, that is, changes the second bit to its opposite.
[0122]
[0123] This gate can preserve noise bias where bit flips are suppressed.
[0124] The natural representation of CNOT is that the control qubit is in a coherent state |+α> CWhen in, the target remains in an idle state, and the control qubit is in a coherent state |-α> C It is to state that the target rotates by π (in the complex plane) when it is in.
[0125] [Fig. 6] Otherwise, a symmetrical perspective is adopted by performing a frame rotation of the target qubit so that the target qubit is always rotated, but rotates 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:
[0126]
[0127] The physical system in which the CNOT is implemented consists of two coupled circuits. One circuit is the control qubit, and the other circuit is the target qubit.
[0128] [Fig. 7] In the form of longitudinal coupling between a control qubit and a target qubit, the Hamiltonian enabling the gate can be described as follows:
[0129]
[0130] Here, g CNOT is the CNOT interaction strength, and a and b are ladder operators for the control qubit and target qubit, respectively.
[0131] Figure 7 illustrates the simplified effect of a CNOT gate. The CAT state in control, in addition to what is illustrated, will cause the target to be split into two blobs.
[0132] CNOT pulse parameters
[0133] [Fig. 8] As can be seen from the Hamiltonian of Equation 3, the rotation of the target qubit in the complex plane is g CNOT It occurs at a rate proportional to.
[0134] Therefore, the CNOT gate time until the end point when the target qubit has rotated as desired is T CNOTIf referred to as such, this parameter of the gate is of the following form, where k is a real number.
[0135]
[0136] The Hamiltonian of Equation 3 is activated by a microwave drive that is transmitted to the ATS of the target qubit at the frequency of the control memory.
[0137] Starting with the stabilized control and target qubits, the stabilization of the target qubit is disabled to allow conditional rotation of the target qubit. In contrast, the stabilization of the control qubit remains enabled.
[0138] Then, the conditional rotation of the target qubit is activated by a microwave drive transmitted to the target's ATS at the frequency of the control's memory. This drive has a duration T CNOT It remains in an active state for a while.
[0139] When the desired conditional rotation is completed (i.e., T CNOT After the duration), stabilization on the target is reactivated at the same time as the CNOT pulse, that is, the drive on the ATS of the target qubit is deactivated.
[0140] The parameters of the CNOT pulse itself—transmitted to the ATS of 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, while the other is b) duration T CNOT It is the amplitude integrated over time.
[0141] a) CNOT pulse phase angle
[0142] In general, there are free phase degrees of freedom between the stabilized control qubit cat state and the longitudinal Hamiltonian of Equation 3. Without loss of generality, this relative phase can be described as being carried by the longitudinal Hamiltonian of Equation 3 as follows.
[0143]
[0144] Physically, the phase difference φ CNOT It arises from different optical paths taken by the stabilization drives and the CNOT pulse.
[0145] Given CNOT pulse amplitude g CNOT and duration T CNOT Regarding, φ CNOT Each value corresponds to a rotation of the target qubit cat state in the complex plane.
[0146] φ CNOT It affects the target qubit and the control qubit very differently.
[0147] - modulo π Identical to each other, φ CNOT The optimal values of It corresponds to the largest rotation angle formed by the target qubit coherent states around the origin in the complex plane.
[0148] - And, more importantly, phase φ CNOT Non-optimal values of induce bit-flip errors in the control qubit during the operation of the gate.
[0149] b) CNOT pulse amplitude integrated with respect to time over the duration of the pulse
[0150] CNOT pulse amplitude integrated over time over the pulse duration If the value of is miscalibrated, this results in an undershoot or overshoot of the conditional rotation in the target. When the control is CAT and the target is a single blob, the target blob splits into two blobs rotating in opposite directions; if undershoot or overshoot exists due to miscalibration, the final separation between the blobs will not be at the maximum of a π angle. Therefore, bit flip errors occur when stabilization is reactivated. Thus, the miscalibrated pulse amplitude integrated over time over the duration of the pulse induces bit flip errors in the target qubit.
[0151] Parameters of the additional drive
[0152] Spurious terms arise, and experimental incompleteness needs to be corrected. One such spurious term corresponds to a direct drive on a control qubit having a Hamiltonian of the following form, and
[0153]
[0154] Here, the coefficient ξ depends on the amplitude and phase of the pulse transmitted to the ATS of the target cat qubit.
[0155] The spurious Hamiltonian of Equation 6 can be offset by an additional drive of the following form, transferred to the memory of the control qubit.
[0156]
[0157] It can be made to become the following
[0158]
[0159] In fact, this is achieved by an additional pulse referred to as the cancellation pulse, which is emitted simultaneously with the main gate pulse, that is, the pulse transmitted onto the ATS of the target cat qubit (and discussed above).
[0160] The cancellation pulse is transmitted at the frequency of the control cat qubit's memory.
[0161] Tuning the amplitude and phase of the cancellation pulse involves the parameters g within the Hamiltonian of Equation 7 so that the equation in Equation 8 is satisfied. cancel and φ cancel Effectively tunes it.
[0162] Parameters for the restart of stabilization pulses on the target cat qubit
[0163] And finally, the stabilization of the target qubit is reactivated after the completion of the gate, when the previous parameters of the gate are well calibrated and a rotation of +π / 2 or -π / 2 of the target qubit's coherent states in the complex plane is obtained.
[0164] As previously explained, this stabilization includes waves transmitted to the target qubit's ATS and buffer, respectively. The two waves must be restarted at a defined phase angle.
[0165] Due to the detuning of the target qubit's memory and the influence of other spurious terms, the total rotation of the target qubit's coherent states is also added to the conditional rotation of the target qubit provided by the gate.
[0166] Therefore, the angle θ defined as the phase angle difference between the stabilization waves transmitted to the target qubit's ATS and buffer, respectively, when reactivating stabilization. T is set.
[0167] In fact, an incorrectly calibrated target stabilization differential phase angle induces bit-flip errors in the target due to the offset between the actual positions of the stabilization and coherent states.
[0168] Summary of CNOT parameters
[0169] Therefore, φ CNOT , , φ cancel , g cancel , and θ T These are the parameters of the CNOT gate.
[0170] Incorrect calibration of specific gate parameters can result in bit-flip errors. This disclosure relates to measuring bit-flip errors in various configurations that sweep parameters of the physical implementations of gates. This enables sensitive and robust optimization of gates using bit-flip errors as a measure. This disclosure also relates to a set of calibration procedures for pulse parameters. When bias-preserving gates are used, the reduction of bit-flip errors persists during the execution of the gates. The calibration procedures utilize bit-flip errors by employing stabilization and noise bias to tune hardware parameters for the gate implementations. Bit-flip-based calibrations are robust due to stabilization and noise bias.
[0171] Detailed calibration method
[0172] [Fig. 9] A method for calibrating parameters according to the present invention generally, for at least one parameter of an interaction sequence for performing a physical quantum gate defined for a single or multiple qubit system in a superconducting circuit based on cat qubits having a reduced bit flip error rate as a noise bias,
[0173] A qubit of the system affected by variations of at least one parameter, wherein the qubit is identified as a qubit under review, and for the qubit, a list of at least one state of the qubit system,
[0174] Regarding the value of at least one parameter selected in the step (S10) of selecting a parameter value, and regarding each state of the list,
[0175] - In step S20, the step of preparing that state,
[0176] - In step S30, a step of applying an interaction sequence as a parameter value to a prepared state,
[0177] - A step of measuring the unchanged occupancy rates and bit-flipped occupancy rates of the affected (subject to review) qubits at the end of the above application,
[0178] - A step of changing the state of the list in step S50 and repeating the preparation, application, and measurement steps above,
[0179] - Then, changing the parameter value in step S60, and repeating the preparation, application, and measurement steps with the changed values of the at least one parameter while sweeping the range of candidate values of the at least one parameter,
[0180] - Then, in step S70, the optimal value of the preservation of the noise bias by the gate is identified from the occupancy rates measured during the iterative steps, and in step S80, the inferred parameter value associated with the identified optimal value is implemented as the calibrated value of the at least one parameter.
[0181] The above optimal value for the preservation of noise bias by the gate may be: (i) symmetry between the measured bit flip occupancy rates of two or more states in the list; and / or (ii) minimum value of the measured bit flip occupancy rates of one or more states.
[0182] In the following paragraphs, calibration procedures utilizing bit flip errors in relation to the gates described above are given.
[0183] X Gate
[0184] [Fig. 10] The X gate can be implemented by releasing stabilization for an appropriate duration and reactivating stabilization so that the rotation angle is π. The duration of the release of stabilization can be calibrated as follows:
[0185] Prepare state |+α>
[0186] Release stabilization for the given duration
[0187] Resuming stabilization and immediately measuring the number of bit flip errors for the expected termination state |-α>. Actually, it is possible to resume stabilization, wait between 10 ns and the natural bit flip time depending on the parameters, and then measure the occupancy rates.
[0188] After that, this is performed multiple times with varying durations of release from stabilization.
[0189] Here, a list of only one state can be used—then there is no need to iterate through other states.
[0190] This can also be accomplished by preparing a |-α> state instead of a |+α> state and measuring errors with respect to the expected termination state, i.e., |+α>.
[0191] The optimal duration is the duration in which bit flip errors are minimized.
[0192] The figure illustrates a qualitative representation of the calibration of the X gate, assuming |+α> as the initial state. The release time is swept along the x-axis, and the occupancy rates of |+α> and |-α> are measured at each time step. The optimal release time maximizes the occupancy ratio by minimizing bit flips for the target state |-α>, indicated by the black vertical line.
[0193] Z(θ) gate
[0194] The Z(θ) gate can be implemented as a pulse of 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:
[0195] Prepare state |+α>
[0196] Activate stabilization and simultaneously apply a Z-gate pulse to the qubit with a given phase angle value and for a duration equal to an integer multiple of a fixed and predetermined duration (e.g., 1, 2, ... N).
[0197] At the end of the Z gate pulse, the occupancy rates of |+α> and |-α> are measured, and thus a position is obtained to infer the number of bit flip errors when necessary.
[0198] Repeat with various values of the given phase angle
[0199] Repeat optionally but advantageously in different integer multiples of the duration
[0200] Then, repeat the process of preparing state |-α> (in this embodiment, a list of two coherent states with a phase difference of π is used - however, an alternative is given below).
[0201] The optimal phase angle of the pulse is the phase angle such that the number of bit flip errors is equal for two state preparations |+α> and |-α>. The optimal phase is found from clear and sharp features within the data, reducing experimental uncertainty compared to conventional methods.
[0202] The data analysis for preparing each state is as follows:
[0203] For each value of φ, the measured share of control p(N) vs N is fitted to exponential decay of the following form, and
[0204]
[0205] Here, A is the prefactor, B is the offset, and Γ is the decay rate of the curve associated with the bit flip rate during the gate. The value of Γ for each φ is obtained.
[0206] Then, Γ vs φ is fitted to a linear function of the following form, and
[0207]
[0208] Here, m and q are fitting parameters.
[0209] The intersection of the linear fittings for the two preparations corresponds to the optimal value φ.
[0210] As described above, this scenario begins by preparing state |+α>| and repeats by preparing state |-α>|. However, the method can also be prepared for a cat state, namely state |+> or |->. In that case, the procedure would be, for example, as follows: state |+> , Prepare, execute the CNOT pulse, and measure the bit flips. The optimal phase is the phase that minimizes the bit flips. Using this alternative, the list can be a list of only one state, but that state is the cat state.
[0211] CNOT Gate
[0212] Bit flip errors can be used to calibrate different parameters of the CNOT gate.
[0213] 3.1) Additional Drive HCANCEL
[0214] The amplitude and phase of the cancellation pulse providing HCANCEL can be tuned using bit flip errors in the following way:
[0215] State |+α> C |0> T Preparing
[0216] Applying the CNOT pulse and the cancellation pulse to the qubits with the given amplitude and phase of the cancellation pulse.
[0217] Measures the number of bit flip errors in the control qubit
[0218] Repeat with various values of amplitude and phase of the cancellation pulse
[0219] States |-α> C |0>T , |+iα> C |0> T , |-iα> C |0> T Repeatedly preparing, where i is the imaginary unit (in this embodiment, the list contains 2 pairs of coherent states, and the phase difference between the 2 states of each pair is π). This is for one qubit. The other qubit is kept unchanged in the list.
[0220] The optimal amplitude and phase of the cancellation pulse are such that the number of bit flip errors is as equal as possible for all four state preparations.
[0221] States |-α> C |0> T , |+iα> C |0> T , |-iα> C |0> T Note that instead of using , this can be done with any two pairs of coherent states where the phase difference between the states within the pair is π.
[0222] 3.2) φ CNOT
[0223] [Fig. 11] The phase of the CNOT pulse can be tuned using bit flip errors in the following way:
[0224] State |+α> C |+α> T Preparing
[0225] Enables stabilization on control qubits and applies CNOT pulses to qubits with a given phase value and a given duration.
[0226] |+α> C Measuring the market shares of
[0227] Similar to the case of the Z gate above, it is possible to actually resume stabilization of the control, wait between 10 ns and the natural bit flip time of the control depending on the parameters, and then measure the occupancy rates of the control.
[0228] In FIG. 11, parts a) and c) are a) |+α> C |±α> T State when preparing |+α> C and c) |-α> C |±α> T State when preparing |-α> C The measured control shares are plotted. The x-axis plots the increased time length of the applied pulse (represented by integers from 1 to 446 of the basic predefined pulse length, where each N consists of a certain stabilization time following the CNOT pulse), and the y-axis plots the phase angle value used. In the experiment of Fig. 11, the delay (i.e., stabilization time) between the end of the CNOT pulse and the measurement of the control was approximately 2 ms.
[0229] Optionally, but advantageously, repeat with different CNOT pulse durations (which may be integer N times the predetermined duration of the pulse).
[0230] Repeat with different values of CNOT pulse phase angle
[0231] For example, the CNOT pulse phase is swept in the range of -0.3 rad. to 0.1 rad, and the number of CNOT pulses N is in the range of 1 to 446.
[0232] State |-α> C |+α> T Repeated by preparing, that is, for every preparation, the state of the target qubit does not change, but the state of the control qubit changes in the opposite way - in that case |-α> C Measures the market shares of.
[0233] The optimal phase of the CNOT pulse is the phase that makes the number of control bit flip errors the same for two state preparations.
[0234] This also means states |+α> C |-α> T Prepare and then |-α> C |-α> T It can be performed by preparing.
[0235] Calibration can be performed by running CNOT pulses multiple times (varying N), which implements optional steps that repeat with different pulse durations. By increasing N, it may be possible to "magnify" the bit flips, which increases the resolution of the bit flip measurements.
[0236] The data analysis for each state preparation is as follows:
[0237] φ CNOT For each value of , the measured share of control p(N) vs N is fitted to exponential decay of the following form and
[0238]
[0239] Here, A is the prefactor, B is the offset, and Γ is the decay rate of the curve associated with the bit flip rate during the gate. For each φCNOT value, the value of Γ is obtained. As is understood, obtaining the value of Γ for each φCNOT value is valid within experimental regions where the control stabilizes for a short time period after the end of the CNOT pulse and before measurements. This short time period is between 2 ns and 5 times the natural bit flip time of the control, and most preferably between 10 ns and the natural bit flip time of the control. In other experimental regions, for example, when the measurement of the control is performed almost immediately after the end of the CNOT pulse, the prefactor A or offset B may instead contain the necessary information, and thus the values of A and / or B are obtained for each φCNOT value.
[0240] In FIG. 11, parts b) and d) illustrate the fittings of the data of a) and c) using Equation 11, respectively.
[0241] The axes are labeled in the same way in all parts a) through d) of FIG. 11.
[0242] Γ vs φCNOT fits a linear function of the following form, and
[0243]
[0244] Here, m and q are fitting parameters.
[0245] In Fig. 11, part e) is |+α> C |±α> T Extracted Γ vs φ for the preparation (falling line) and for the |-α>C|±α>T preparation (rising line) CNOT Plot this along with the fittings for Equation 12. The optimal phase is found at the intersection of the fittings.
[0246] The intersection of linear fittings for two state preparations corresponds to the optimal φCNOT.
[0247] The optimal CNOT phase is φCNOT It is found to be -0.118 rad. The associated exponential decay rate is approximately 0.008 ns -1 am.
[0248] As mentioned above, this scenario is state |+α> C |+α> T Start by preparing and state |-α> C |+α> T The list was prepared and repeated, and repeated again. In this embodiment, the list contains two pairs of coherent states, and the phase difference between the two states of each pair is π. This is for one qubit. The other qubit remains unchanged in the list.
[0249] However, the method can also be prepared as a cat state within control, namely state |+> or |->. Using this alternative, the list may be a list of only one state, but that state is the cat state within control. The other qubits in the list remain unchanged. In that case, the procedure would be, for example, as follows: state |+> C |+α> T, Prepare, execute the CNOT pulse, and measure the bit flips within the control. The optimal phase is the phase that minimizes the bit flips.
[0250] Generally, the state within the target is not important for the calibration of the phase of the CNOT pulse.
[0251] 3.3) CNOT Pulse Amplitude and Stabilization Angle
[0252] [Fig. 12] CNOT pulse amplitude g CNOT and target qubit differential angle θ for resuming post-pulse stabilization T (Hereafter abbreviated as stabilization angle or also 'buffer phase') can be tuned together using bit flip errors within a common calibration procedure as follows.
[0253] State |+α>C |+α> T Preparing
[0254] Activate stabilization on the control qubit and apply a CNOT pulse while deactivating stabilization on the target qubit with a given CNOT pulse amplitude value.
[0255] Apply this CNOT pulse to the target while maintaining control stabilization.
[0256] Stabilization angle value θ T Activates stabilization on the given target.
[0257] Sequences in which stabilization on the target qubit follows the control qubit stabilization and the target qubit CNOT pulse can be repeated more than once—N times, i.e., an integer number—before measurement. This increases the resolution.
[0258] |+α> T and |-α> T It is possible to measure the occupancy rates and then infer the number of bit flip errors within the target qubit.
[0259] Figure 12 shows part a): control is |+α> C The target qubit p+ prepared is plotted. The x-axis plots the CNOT pulse amplitude having a relative unit of essentially 0.9 to 1.1. The y-axis plots the target qubit buffer phase (stabilization angle) of approximately -4 rad to 4 rad.
[0260] The axes are labeled in the same way in all parts a) through h) of FIG. 12.
[0261] In Fig. 12, in part b): control is |+α> C For the target p- prepared as, in part c): control is |-α> C The target p+ prepared as, and in part d): control is |-α> C The target p- prepared is plotted.
[0262] Repeat with various CNOT pulse amplitude and stabilization angle values - the CNOT pulse amplitude is swept in a range of, for example, 0.9 to 1.1 arbitrary units (au), and the target stabilization angle θ T It is taken in the full range from 0 rad to 2π rad.
[0263] State |-α> C |+α> T Repeating the preparation of
[0264] In this embodiment, a list of two coherent states with a phase difference of π is used for one qubit. The other qubit is maintained in an unchanged state in the list.
[0265] The optimal CNOT pulse amplitude and stabilization angle are the amplitude and angle such that the number of bit flips is the same for both preparations and is minimized.
[0266] This also includes states |+α> C |-α> T Prepare and then |-α> C |-α> T It can be performed by preparing.
[0267] The data analysis for each state preparation is as follows:
[0268] For each amplitude of the CNOT pulse, respectively |+α> T and |-α> T The target shares p+ and p- at are fitted to the following function, and
[0269]
[0270] Here, A is the total factor, B is the occupancy offset, θ0 is the offset of the stabilization angle, and the optimal stabilization angle θ for each amplitude of the CNOT pulse. T is the angle that maximizes the difference between p+ and p-.
[0271] In Fig. 12, parts e), f), g), and h) are the fittings for the data in a), b), c), and d), respectively. The dashed lines represent the optimal θ found by fitting for each CNOT pulse amplitude. T It represents.
[0272] Optimal stabilization angle θ T vs CNOT pulse amplitude is fitted to a linear function.
[0273] In Fig. 12, in part i), the optimal θ T vs CNOT pulse amplitude is fitted to a linear function.
[0274] Optimal θ T and CNOT pulse amplitude are 2-state preparations |+α> C |+α> T and |-α> C |+α> T It is found at the intersection of linear fittings for.
[0275] The intersection of the fittings is the optimal CNOT pulse amplitude and target stabilization angle θ T Corresponds to.
Claims
Claim 1 A method for calibrating at least one parameter of an interaction sequence for performing a defined quantum operation for a single or multiple qubit system in a superconducting circuit based on cat qubits having a reduced bit flip error rate as a noise bias, wherein the method comprises a qubit of the system affected by variations of the at least one parameter and a list of at least one states of the qubit system, wherein for each state of the list: a value of the at least one parameter and for each state of the list: - Step of preparing the above state (S20), - Step (S30) of applying the interaction sequence with the parameter values to the prepared state, A method of calibration comprising: a step (S40) of measuring bit-flipped or bit-unflipped occupancy rates of the affected qubit at the end of the above-mentioned application step; a step (S50, S60, S10, S20) of repeating the preparation, application, and measurement steps with the varied values of the at least one parameter to sweep the range of candidate values of the at least one parameter; a step (S70) of identifying an optimal value for preserving the noise bias by the operation from the occupancy rates measured during the step of repeating the above steps; and a step (S80) of implementing an inferred parameter value associated with the identified optimal value as a calibrated value of the at least one parameter. Claim 2 A method of calibration according to claim 1, wherein at least one parameter comprises the phase angle of a microwave pulse transmitted to a non-linear coupling element that couples the memory of a qubit of the system to the buffer of the qubit. Claim 3 A method of calibration according to claim 1, wherein at least one parameter comprises the phase angle of a microwave pulse transmitted at the frequency of the memory of a qubit of the system, which is a multi-qubit system. Claim 4 A method of calibration according to claim 1, wherein at least one parameter includes a duration of release of driving stabilization of the qubit of the system. Claim 5 A method of calibration according to claim 1, wherein the at least one parameter comprises the integral over time of the amplitude of a microwave pulse transmitted to a non-linear coupling element that couples the memory of a qubit of the system to the buffer of the qubit during the implementation of the operation. Claim 6 A method of calibration according to claim 1, wherein the at least one parameter comprises the integral over time of the amplitude of a microwave pulse transmitted at the frequency of the memory of the qubit of the system during the implementation of the operation. Claim 7 A method of calibration according to claim 1, wherein the at least one parameter comprises the difference in phase angles of two microwaves each transmitted to a buffer associated with a qubit of the system and a non-linear coupling element coupling the associated buffer and the memory of the qubit, as a driving stabilizer after the execution of the operation. Claim 8 A method of calibration according to any one of claims 1 to 7, wherein the step of applying the interaction sequence is performed multiple times prior to the measuring step, the number of times N in which the interaction sequence is applied also varies from Nmin to Nmax, and the measuring step is performed for each value of N used. Claim 9 A method of calibration according to any one of claims 1 to 8, wherein at least one parameter comprises two numerical real numbers, and the step of applying the interaction sequence is performed multiple times while varying the two numerical real numbers independently. Claim 10 A method of calibration according to any one of claims 1 to 9, wherein the list of states comprises one or more pairs of coherent states or cat states having a phase difference of π between two states of a given pair, or is a list of only one coherent state. Claim 11 A method of calibration according to any one of claims 1 to 10, wherein identifying the optimal value comprises identifying, from the occupancy rates measured during the step of repeating the steps, a minimum value of the bit flip rate or the number of bit flips or a minimum value of the distance between the bit flip rates or the number of bit flips associated with distinct states of the list, or finding both the minimum value of the bit flip rate or the number of bit flips and the minimum value of the distance between the bit flip rates or the number of bit flips associated with distinct states of the list. Claim 12 A method of calibration according to any one of claims 1 to 11, wherein the quantum operation is one or a combination of an X gate, a CNOT gate, a tofoli gate, a state preparation, and a parity measurement. Claim 13 A method of calibration according to any one of claims 1 to 12, wherein the cat qubits are stabilized by driven two-photon dissipation. Claim 14 A method of calibration according to any one of claims 1 to 13, wherein the measuring step is performed by Wigner tomography or any parity measurement process, in particular by direct or indirect parity measurements, and in particular by mapping different parities to different occupancy rates in the memory of the affected qubit, or the measuring step is performed by determining the number of photons in the memory of the affected qubit. Claim 15 A computer program comprising instructions, wherein the instructions cause the computer to perform the method of any one of claims 1 to 14 when the program is executed by the computer.