Methods and systems for benchmarking quantum gate groups

Abstract

This disclosure provides a system and method for benchmarking a quantum gate set. The quantum gate set may have an input domain and a fidelity function defined on that input domain. Benchmarking the quantum gate set may include determining an approximation of the fidelity function on the input domain. Such benchmarking may include determining multiple fidelity measures. Each fidelity measure may be associated with a basis function from one of a set of basis functions. The basis functions can be used to generate a probability distribution. The probability distribution can be used to determine the fidelity measure. An approximate fidelity function can be generated using the fidelity measure and the corresponding basis function.

Description

Technical Field

[0001] This disclosure relates to quantum computing, and more specifically, to a benchmarking protocol for generating approximate fidelity functions for a set of gates. Background Technology

[0002] Quantum computing can solve classically intractable computational problems. However, existing quantum computing devices are limited by various sources of error and imprecision. Benchmarking can be used to determine the fidelity of a set of gates implemented on a quantum computing device. However, traditional benchmarking techniques may be impractical when a set of gates contains a large number of gates. Furthermore, benchmarking a specific selection of gates may not be feasible. A finite set of gates may be too large to be characterized by benchmarking individual gates. A continuous set of gates can be characterized by benchmarking the gates sampled from the continuous set, but this becomes infeasible when the continuous set of gates is located in high dimensions. Improved benchmarking techniques can identify gate sets or quantum computing devices with excellent fidelity, thereby supporting the development of quantum computing. Summary of the Invention

[0003] This disclosure provides a method and system for generating an approximate fidelity function for a set of quantum gates by sampling from a set of quantum gates based on a distribution during benchmarking. The distribution can be generated using one of a set of basis functions.

[0004] Embodiments of this disclosure provide a method for benchmarking a quantum gate set. The method may include selecting a quantum gate set, wherein the quantum gates in the quantum gate set are defined over an input domain. The method may include determining an approximate fidelity function of the quantum gate set. Determining the approximate fidelity function of the quantum gate set includes selecting a set of basis functions defined over the input domain. The method may include generating a first probability distribution defined over the input domain using basis functions from one of the basis function sets. The method may include obtaining a fidelity metric of the quantum gate set under the first probability distribution by performing a randomized benchmark on the quantum components. The approximate fidelity function is a function between the fidelity metric and the basis functions of one of the basis function sets. The method may include providing the approximate fidelity function.

[0005] In some embodiments, obtaining a fidelity metric may include scaling a first fidelity value of an interleaved sequence of quantum gates with a second fidelity value of a non-interleaved sequence of quantum gates. In some embodiments, performing a randomized benchmark test on a quantum component may include determining a first fidelity value of a first sequence of quantum gates. Each first sequence may be interleaved, consisting of a sequence selected from a set of quantum gates according to at least one first probability distribution and a sequence selected from a group of quantum gates according to a second probability distribution. In some embodiments, the second probability distribution may be a uniform probability distribution over the input domain. In some embodiments, the set of quantum gates may be a subset of a group of quantum gates. In some embodiments, the set of basis functions may include a set of trigonometric basis functions; a set of polynomial basis functions; or a set of wavelet basis functions. In some embodiments, the approximate fidelity function includes two or more terms of a Fourier, Taylor, or wavelet expansion of the fidelity function of the set of quantum gates over the quantum component. In some embodiments, the input domain includes two or more variables. In some embodiments, the quantum component may include a transmon or fluxonium qubit.

[0006] Embodiments of this disclosure provide a system for benchmarking a group of quantum gates. The system may include at least one processor and at least one non-transitory computer-readable medium containing instructions. When executed by the at least one processor, the instructions cause the system to perform operations. These operations may include selecting a group of quantum gates in which quantum gates are defined over an input domain. These operations may include determining an approximate fidelity function for the quantum gate group. Determining the approximate fidelity function for the quantum gate group may include selecting a set of basis functions defined over the input domain. The determination may include generating a first probability distribution defined over the input domain using basis functions from one of the basis function sets. The determination may include obtaining a fidelity metric for the quantum gate group under the first probability distribution by performing a randomized benchmark on the quantum components. The approximate fidelity function may be a function between the fidelity metric and the basis functions from one of the basis function sets. These operations may also include providing the approximate fidelity function.

[0007] In some embodiments, obtaining a fidelity metric may include scaling a first fidelity value of an interleaved sequence of quantum gates with a second fidelity value of a non-interleaved sequence of quantum gates. In some embodiments, performing a randomized benchmark test on a quantum component may include determining a first fidelity value of a first sequence of quantum gates. Each first sequence may interleave a sequence selected from quantum gates according to at least one first probability distribution and a sequence selected from a group of quantum gates according to a second probability distribution. In some embodiments, the group of quantum gates may be a subset of a group of quantum gates. In some embodiments, the group of basis functions may include a trigonometric basis function set; a polynomial basis function set; or a wavelet basis function set. In some embodiments, the approximate fidelity function may include two or more terms of a Fourier, Taylor, or wavelet expansion of the fidelity function of the group of quantum gates on the quantum component. In some embodiments, the quantum component may include a transmon or fluxonium qubit.

[0008] The disclosed embodiments include a non-transitory computer-readable medium containing instructions. When executed by at least one processor of a system, the instructions can cause the system to perform operations. These operations can include selecting a set of quantum gates, wherein quantum gates in the set are defined over an input domain. These operations can include determining an approximate fidelity function for the set of quantum gates. The determination can include selecting a set of basis functions defined over the input domain. The determination can include generating a first probability distribution defined over the input domain using basis functions from one of the basis function sets. The determination can include obtaining a fidelity metric for the set of quantum gates under the first probability distribution by performing a randomized benchmark test on the quantum components. The approximate fidelity function can be a function between the fidelity metric and a function of one of the basis function sets. These operations can include providing an approximate fidelity function.

[0009] In some embodiments, obtaining a fidelity metric may include scaling a first fidelity value of an interleaved sequence of quantum gates with a second fidelity value of a non-interleaved sequence of quantum gates. In some embodiments, performing a randomized benchmark test on a quantum component may include determining a first fidelity value of a first sequence of quantum gates. Each first sequence may interleave a sequence selected from quantum gates according to a first probability distribution and a sequence selected from a group of quantum gates according to a second probability distribution. In some embodiments, the group of quantum gates is a subset of a group of quantum gates. In some embodiments, the group of basis functions includes: a trigonometric basis function group; a polynomial basis function group; or a wavelet basis function group. In some embodiments, the approximate fidelity function includes two or more terms of a Fourier, Taylor, or wavelet expansion of the fidelity function of the group of quantum gates on the quantum component. In some embodiments, the quantum component may include a transmon or fluxonium qubit.

[0010] It should be understood that the foregoing general description and the following detailed description are merely exemplary and illustrative, and not intended to limit the disclosed embodiments. Attached Figure Description

[0011] The accompanying drawings, which form part of this specification, illustrate several embodiments and, together with the specification, serve to explain the principles and features of the disclosed embodiments. In the drawings:

[0012] Figure 1A A fully randomized benchmark (FRB) is described according to the disclosed embodiments.

[0013] Figure 1B An interleaved fully randomized benchmark (iFRB) is described according to the disclosed embodiments.

[0014] Figure 1C An exemplary distributed staggered randomization benchmark (iRBD) according to the disclosed embodiments is described.

[0015] Figure 2 A hypothetical fidelity function is described for a set of gates with two input parameters.

[0016] Figure 3 An exemplary system for decomposing and applying quantum gate sequences to realize quantum computing, according to the disclosed embodiments, is described.

[0017] Figure 4 An exemplary method for performing iRBD according to the disclosed embodiments is described.

[0018] Figure 5 Examples are depicted that converge to a sequence of approximate fidelity functions that have a known fidelity function with one input parameter.

[0019] Figures 6A to 6D An example is given that a sequence of approximate fidelity functions converges to a known fidelity function with two input parameters. Detailed Implementation

[0020] Exemplary embodiments will now be discussed in detail with reference to the accompanying drawings. In some cases, the same reference numerals will be used throughout all the drawings and in the following description to refer to the same or similar parts. Unless otherwise defined, technical or scientific terms have the meanings commonly understood by one of ordinary skill in the art. The disclosed embodiments have been described in sufficient detail to enable those skilled in the art to practice them. It should be understood that other embodiments may be utilized and changes may be made without departing from the scope of the disclosed embodiments. Therefore, the materials, methods, and examples are illustrative only and are not intended to be limiting.

[0021] Performance characterization is a crucial component of the development and verification of quantum computing devices. It can be achieved by benchmarking a quantum computing device, a set of gates on a quantum computing device, or a specific implementation of a set of gates on a quantum computing device. Efficient and reliable benchmarking schemes enable comparisons between different quantum computing devices (e.g., those manufactured by different companies) and provide useful feedback for device calibration and error diagnosis. Therefore, such benchmarking can support future hardware design and the development of fault-tolerant quantum computing.

[0022] Benchmarking protocols include randomized benchmarking protocols, which attempt to extract fidelity information about quantum gate sets while isolating the effects of state preparation and measurement (SPAM) errors. For example... Figure 1A As shown, an FRB can be performed using multiple random gate sequences that are independently and identically distributed across a set of gates used for benchmarking. For each of the multiple sequences, a recovery gate, which is the inverse of the specific random gate sequence, can be computed. The quantum computing device can be initialized to a specific state (e.g., state |0>), the specific sequences of random gates and recovery gates can be applied, and the state of the quantum computing device can be measured.

[0023] For a gate sequence of length m, the probability p of measuring the initial state is... m The fidelity metric μ of the gate assembly, the state preparation error A, and the measurement error B can be related as follows:

[0024]

[0025] FRB can include performing multiple trials to estimate different values ​​of m. The fidelity metric μ can then be measured by... The fidelity metric μ is determined by a linear fit to the logarithm of the sequence length m. The fidelity metric μ can be normalized to the range [0, 1] to generate the gate fidelity r = 1 - (1 - u)(d - 1) / d, where d is the dimension of the quantum system.

[0026] It can be understood that the μ value obtained by FRB corresponds to the entire group of gates used for benchmarking. Conversely, iFRB can be used to determine the fidelity metric for a specific gate T within this group of gates. For example... Figure 1B As shown, a sequence of m random gates, independently and identically distributed across this group of gates, can be interleaved with m instances of gate T. A recovery gate can be computed, which is the inverse of a specific interleaved sequence of random gates and instances of gate T. A quantum computing device can be initialized to a specific state (e.g., state |0>), a specific sequence of random gates and recovery gates can be applied, and the state of the quantum computing device can be measured.

[0027] Similar to the case of FRB, multiple trials can be conducted to estimate the expected probability of measuring the initial gate with different values ​​of m. Then, the value of the fidelity metric v can be determined by linearly fitting the logarithm of the expected probability to the dependence of m. The fidelity metric of T can then be calculated as the ratio v / u.

[0028] According to the disclosed embodiments, the fidelity value μ can be calculated separately from the fidelity value v. For example, FRB can be used to estimate μ, and then iFRB can be used for each gate T in a set of i gates. i Estimate v i Each of the i doors in the group can be T. i Calculate the fidelity metric v i / u. Fidelity metrics can be standardized to the door fidelity of the target door, as... Where d is the dimension of the quantum system.

[0029] Figure 2 The fidelity function of a hypothetical gate with two input parameters is described. The fidelity function depends on the values ​​of these two input parameters. The relationship between gate fidelity and input parameter values ​​can be studied by determining the gate fidelity at sampling locations in the input domain (e.g., using iFRB). Figure 2 The sampling locations within the grid pattern are depicted, but the disclosed embodiments are not limited thereto. Other deterministic or stochastic sampling schemes may be used. It is understood that obtaining an accurate estimate of the relationship between input parameter values ​​and gate fidelity may require a very large number of trials.

[0030] Figure 1C The Interleaved Randomization (iRBD) benchmark test is shown, an improved version of iFRB that can determine the approximate fidelity function using a feasible number of trials. The first sequence and the second random gate sequence T1 to T2 are shown. m Interleaving, instead of the first random gate sequence U′1 to U′ as in traditional iFRB. inInterleaved with a single gate T. While the random gates in the first sequence are drawn independently and identically from a group of gates according to a uniform distribution, the random gates in the second sequence are drawn independently and identically from a group of gates according to a potentially non-uniform distribution. In some cases, this group of gates may be a subset of the group of gates. In some embodiments, this group of gates and the group of gates may be selected to ensure the existence of a suitable recovery gate. In some embodiments, the potential non-uniform distribution may be generated using a set of basis functions. Generating the distribution may include scaling one (or a combination of) the set of basis functions to the range [0, 1]. In some embodiments, a suitable set of basis functions limited to the range [0, 1] may be selected, and further scaling may not be necessary. Similar to iFRB, multiple trials can be performed for different sequence lengths. The fidelity value v can be determined using the results of multiple trials.

[0031] According to the disclosed embodiments, the fidelity value *v* can be the coefficients of the basis functions in the fidelity function expansion. For example, when the basis function set is a sine (or complex exponential) Fourier expansion, the fidelity value of the basis functions can be the coefficients of those basis functions in the Fourier expansion of the fidelity function. It is understood that the basis function set can be any suitable set of basis functions and is not limited to trigonometric functions. In some cases, polynomial basis functions can also be used to generate probability distributions. In this case, the fidelity value *v* can be the coefficients in the Taylor or Laurent series approximation of the fidelity function. Wavelets can be used to generate probability distributions in various cases. In particular, when the shape or characteristics of the fidelity function are generally known or a priori suspected, wavelet transform can achieve a more accurate approximation of the fidelity function using fewer wavelet expansion terms. Furthermore, wavelets can support approximations of fidelity functions at different scales and locations, providing a more precise representation where such accuracy is required.

[0032] Figure 3 A system 300 for performing iRBD according to the disclosed embodiments is depicted. System 300 may include classical components 310 (e.g., classical computing devices or collections of classical computing devices) and quantum components 320.

[0033] Quantum component 320 can be configured to process information using quantum phenomena (e.g., superposition or entanglement). Quantum component 320 can operate on units of information called "qubits." A qubit is the smallest unit of information in a quantum computer and can have any linear combination of two values, typically represented as |0> and |1>. The value of a qubit can be represented as |ψ>. Unlike digital bits, which can have values ​​"0" or "1", |ψ> can have the value α|0>+β|1>, where α and β are complex numbers (called "amplitudes"), unconstrained except for |α| 2 +|β| 2=1. A qubit can be constructed in various forms and can be represented as a quantum state of a component of quantum component 320. For example, a quantum state can be realized by using the polarization of a photon as a quantum state (e.g., in a laser); using the spin of an electron or ion as a quantum state (e.g., trapped in an electromagnetic field); using the charge, current flux, or phase of a Josephson junction as a quantum state (e.g., in a superconducting quantum system); using the point spin of a quantum dot as a quantum state (e.g., in a semiconductor structure), a topological quantum system, or any other system that can provide two or more quantum states to physically realize a qubit. Quantum component 320 can be used with quantum logic gates (or simply "quantum gates") to create, remove, or modify qubits.

[0034] Conversely, classical component 310 can be a computing system that cannot perform quantum computing, such as an electronic computer (e.g., a laptop, desktop computer, cluster, cloud computing platform, etc.). Classical component 310 can operate on binary value bits in digital logic. Classical component 310 may include one or more processors (e.g., CPU, GPU, etc.), application-specific integrated circuits, hardware accelerators, or other components for processing digital logic. Classical component 310 may include one or more memories, buffers, caches, or other components for storing binary values. Classical component 310 may include one or more I / O devices that communicate with other systems, devices (e.g., quantum component 320), users, etc.

[0035] Classical component 310 can be configured to control quantum component 320. The classical component may include compilation module 311. Compilation module 311 can be configured to obtain a description of a benchmarking task. The description of the benchmarking task may include a description of a group and a set of gates used for benchmarking. In some cases, this set of gates may be a subset of the group of gates. The description of the benchmarking task may include a description of a set of basis functions and / or obtain one or more probability distributions for benchmarking.

[0036] Based on the description of the benchmark task, the compilation module 311 can determine the gate sequence for the iRBD benchmark. In some embodiments, the description of the benchmark task may include a fidelity metric μ for a group of gates (e.g., determined as a result of previous benchmark experiments). When the description of the benchmark task does not include the fidelity metric μ, the compilation module 311 can determine the gate sequence for the FRB benchmark to determine the fidelity metric μ.

[0037] Compiler module 311 can determine several sets of gate sequences of different sequence lengths m. As described herein, the gate sequences of sequence length m can include those with the second random gate sequences T1 to T2. m Interleaved first random gate sequences U′1 to U′ mThe random gates in the first sequence are drawn independently and identically from this group of gates according to a uniform distribution. The random gates in the second sequence are drawn independently and identically from this group of gates according to a probability distribution. In some embodiments, the description of the benchmark task may indicate the probability distribution (or the basis functions used to generate the probability distribution). In some embodiments, the compilation module 311 may be pre-configured with the probability distribution (or the basis functions used to generate the probability distribution). The compilation module 311 may also determine the recovery gate based on the interleaved first and second random gate sequences.

[0038] It is understood that the quantum component 320 can be designed to implement arbitrary quantum gates using a set of local gates. The gate decomposition module 313 (which can be implemented as a submodule of the compilation module 311) can be configured to decompose the gate sequence determined by the compilation module 311 into a sequence of local gates that can be physically implemented on the quantum component 320. The sequence of local gates can then be provided to the quantum controller 315.

[0039] Quantum controller 315 can be configured to directly control quantum component 320. Quantum controller 315 can be a digital computing device (e.g., a computing device including a CPU, graphics processing unit, application-specific integrated circuit, field-programmable gate array, or other suitable processor). Quantum controller 315 can configure quantum component 320 for computation, provide quantum gates to quantum component 320, and read state information from quantum component 320.

[0040] Quantum controller 315 may include instruction generation module 316. The capabilities of instruction generation module 316 may depend on the specific implementation of quantum component 320. In some embodiments, instruction generation module 316 may be configured to directly or indirectly provide bias drive to quantum component 320 to enable or disable interaction between qubits. Instruction generation module 316 may indirectly provide bias drive by providing instructions to a bias drive source (e.g., a waveform generator, etc.) such that the bias drive source provides bias drive to quantum component 320. Instruction generation module 316 may apply local quantum gates by providing one or more microwave pulses (or other gate drives) to qubits in quantum component 320. In various embodiments, instruction generation module 316 may implement such gates by providing instructions to a computation drive source (e.g., a waveform generator, etc.) such that the computation drive source provides such microwave pulses (or other gate drives) to qubits in quantum component 320. As described herein, microwave pulses may be selected or configured to implement one or more local quantum gates. Microwave pulses may be provided to qubits using one or more coils coupled to the respective qubits. The coil can be external to the quantum component 320 or on the chip that implements the quantum component 320.

[0041] Quantum controller 315 can be configured to determine state information of quantum component 320. In some embodiments, quantum controller 315 can measure the state of one or more qubits of quantum component 320. The state can be measured when one or more quantum operation sequences are completed. In some embodiments, instruction generation module 316 can provide a probe signal (e.g., a microwave probe tone) to the coupled resonator of quantum component 320, or provide instructions to a readout device (e.g., an arbitrary waveform generator) that provides the probe signal.

[0042] In various embodiments, the quantum controller 315 may include a data processing module 317. The capabilities of the data processing module 317 may depend on the specific implementation of the quantum component 320. In some embodiments, the data processing module 317 may take an output signal (e.g., electrons / photons), convert it into a discrete signal, and perform data processing (e.g., averaging, post-processing) on ​​it to obtain a computational result. In some embodiments, the data processing module 317 may include or be configured to receive information from a detector configured to determine the amplitude and phase of an output signal received from a coupled resonator in response to the provision of a microwave probe tone. The amplitude and phase of the output signal can be used to determine the state of the probed qubit. The disclosed embodiments are not limited to any particular method for measuring the state of a qubit.

[0043] According to the disclosed embodiments, the quantum controller 315 can be configured to provide output to the compilation module 311 (or another suitable module of the classical component 310). The compilation module 311 (or other suitable module) can use this output to determine a fidelity metric for this set of gates under a probability distribution (e.g., by accumulating measurements). Using empirically estimated values ​​over sequence length m The function determines v, determines the fidelity metric v / u, or determines ).

[0044] Quantum component 320 can be configured to receive commands (e.g., bias drive, quantum gate, probe signal, etc.) from classical component 310. In some embodiments, quantum component 320 can be implemented using a superconducting quantum circuit coupled to quantum controller 315 using at least one microwave drive line. According to the disclosed embodiments, the superconducting quantum circuit can implement multiple qubits (e.g., transmon qubits, fluxonium qubits, or any other suitable type of qubit). In some embodiments, the superconducting quantum circuit can be implemented using one or more chips containing qubits, each chip including at least a portion of a microwave drive line coupling the qubits to quantum controller 315.

[0045] Figure 4An exemplary method 400 for performing iRBD according to the disclosed embodiments is depicted. In some embodiments, method 400 may be performed using system 300. Method 400 may include operations performed on a classical computing device such as classical component 310 (e.g., mobile device, laptop, desktop, workstation, computing cluster, cloud computing platform, etc.). Method 400 may include operations performed on a quantum computing device such as quantum component 320 (e.g., quantum controller managing superconducting circuits, trapped ion quantum system, topological quantum computing system, photonic quantum computing system, etc.). The iRBD gate sequence may be generated by the classical computing device. The classical computing device may provide instructions to configure the quantum computing device to apply the gate sequence to the appropriate arrangement of qubits. The quantum computing device may perform a benchmark test by applying the gate sequence. The classical computing device may then provide instructions to the quantum computing device to read out the results of the benchmark test.

[0046] Before executing method 400, a basis function set can be selected. In some embodiments, a conventional computing device can be configured to select this basis function set. In some embodiments, a classical computing device can be configured with a predetermined basis function set. In various embodiments, the classical computing device can receive or retrieve the basis function set (e.g., from another system or through interaction with a user).

[0047] According to the disclosed embodiments, a classical computing device can select a suitable set of basis functions based on the following information: the number of input variables to the gate, the domain of the input variables to the gate (e.g., 0 to 2π, -1 to 1, etc.), or the characteristics of the fidelity function known a priori (e.g., whether the fidelity function exhibits some symmetry, whether the fidelity function is spherical, whether there are discontinuities or regions of interest in the fidelity function at a specific input value or within a specific range of input values, etc.).

[0048] According to the disclosed embodiments, in step 410, a basis function can be selected from the basis function set. In some embodiments, a classical computing device can select a basis function from the basis function set. In various embodiments, the classical computing device can receive an instruction to select one of the basis functions from the basis function set. In some embodiments, the basis function set can be selected according to a sequence (e.g., the basis function corresponding to the zeroth term of the series expansion can be selected first, then the basis function corresponding to the first term of the series expansion can be selected, and so on).

[0049] In step 420, the classical computing device can generate a probability distribution based on the selected basis functions. In some embodiments, generating the probability transformation may include scaling the basis functions to the range [0, 1]. In some embodiments, generating the probability distribution may include transforming the domain of the basis functions. For example, the domain of the basis functions (e.g., the domain from 0 to 2π, etc.) may be mapped to the domain of the set of gates used for benchmarking (e.g., the domain from -1 to 1, or some other domain). In some embodiments, the basis functions may be complex-valued. In such embodiments, generating the probability distribution may include converting the complex-valued basis functions into a real-valued probability distribution (e.g., by truncating the complex part of the complex-valued function, using the amplitude or norm of the basis functions, or another suitable method).

[0050] In step 430, a fidelity metric for a set of gates under the generated probability distribution can be obtained. (See reference...) Figure 1C The fidelity metric can be obtained. A classical computing device can be configured to generate multiple sets of trials. Each set of trials can be for a specific sequence length m. Each trial may include initializing a quantum component to a specific state, applying a gate sequence to the quantum component, applying a recovery gate, and measuring the resulting state of the quantum component. The gate sequence may include m gates drawn independently and identically from a group of gates according to a first distribution (e.g., a uniform distribution), which are interleaved with m gates drawn independently and identically from a group of gates according to a generated distribution (e.g., a subset of this group of gates, etc.). The measurement states of this set of trials can be used (e.g., by the classical computing device) to estimate the probability of measuring the initial state. For multiple values ​​of m, the estimated probability of measuring the initial state can be used to determine a fidelity value v, which can be scaled by the fidelity value μ of this group of gates (e.g., to obtain...). Classical computing devices can be configured to generate fidelity values ​​μ using FRBs, or to obtain fidelity values ​​μ from a user, another system, or an accessible storage location. In some embodiments, classical computing devices can be configured to transform scaled fidelity values ​​to the range [0, 1] based on the dimension of the quantum components, as described herein.

[0051] In step 440, the classical computing device can determine whether a stopping condition has been met. The stopping condition can depend on time, the number of generated fidelity metrics, a convergence criterion, or any combination thereof. For example, the classical computing device can determine that the stopping condition is met when the elapsed benchmark time exceeds a predetermined time threshold. As another example, the classical computing device can determine that the stopping condition is met when ten fidelity metrics have been determined (e.g., corresponding to the first ten basis functions in a selected basis function set). As yet another example, the classical computing device can determine that the stopping condition is met when a metric (e.g., norm, measure, or other function) is less than a threshold. This metric can depend on a term in the series expansion corresponding to the fidelity metric determined in step 430. For example, when the basis function set is a Fourier series and the selected basis functions are the fourth basis functions in the Fourier series, the classical computing device can determine that the coefficients of the fourth basis functions (e.g., the fidelity metric determined using the fourth basis function in step 440) are less than a certain value. For example, the value could be 0.05, indicating that the fourth term in the extension will change the approximate fidelity function by less than 0.05 (e.g., when the magnitude of the fourth basis function is less than 1).

[0052] According to the disclosed embodiments, when the condition is not met, method 400 may return to step 410 and select another basis function (e.g., the basis function of the next item in the expansion). When the condition is met, method 400 may proceed to step 450.

[0053] According to the disclosed embodiments, in step 450, the classical computing device can provide an approximate fidelity function. Providing the approximate fidelity function may include displaying (e.g., on a graphical user interface associated with the classical computing device), transferring (e.g., to another system), or storing (e.g., in a storage location accessible to the classical computing device) a fidelity metric determined for each selected basis function. Such fidelity metrics may be provided together with indications of their corresponding selected basis functions. Alternatively, such fidelity metrics may be provided separately from any indications of their corresponding selected basis functions.

[0054] While the above description includes the steps of selecting basis functions and generating probability distributions based on the selected basis functions, the disclosed embodiments are not limited thereto. In some embodiments, a classical computing system may be configured with a predetermined set of probability distributions (e.g., probability distributions corresponding to the first twenty terms of a Fourier series or Taylor series). In such embodiments, the classical computing system may be configured to select a set of probability distributions rather than a set of basis functions. This set of probability distributions may be selected according to the same criteria described above for selecting basis functions. For example, a classical computing device may select a suitable set of probability distributions based on the number of input variables of a gate, the domain of the input variables of the gate (e.g., 0 to 2π, -1 to 1, etc.), or the properties of a priori known fidelity functions (e.g., whether the fidelity function exhibits some symmetry, whether the fidelity function is spherical, etc.).

[0055] As an example, method 400 can be used to benchmark a set of X rotations. In this example, A set of single-qubit gates describes rotations about the x-axis of a Bloch sphere. The "ground-real" fidelity function for simulating a quantum system is given below:

[0056]

[0057] Method 400 can be executed using this set of base functions:

[0058] (even: even numbers, odd numbers)

[0059] A set of probability distributions can be generated from these basis functions, as shown below:

[0060]

[0061] As mentioned above Figure 4 As mentioned above, iRBD can be used to generate fidelity metrics. Where k ≥ 0. Then, the approximate fidelity function can be constructed as follows:

[0062]

[0063] Figure 5The first approximate fidelity functions are plotted among some selected from the first seven basis functions. The y-axis represents the fidelity values, and the x-axis represents θ radians. Trace 510 plots the values ​​of the first approximate fidelity function (k=0), including only the first constant term of the series expansion described above. It can be understood that this first approximate fidelity function is merely the average fidelity over the input domain. Trace 520 plots the values ​​of the second approximate fidelity function (k=2), including the values ​​of the first three terms. Traces 530 and 540 plot the values ​​of the third (k=4) and fourth (k=6) fidelity functions, including the values ​​of the first five and first seven terms of the fidelity function expansion, respectively. It can be seen that the approximate fidelity functions converge quickly to the "ground truth" values ​​of the fidelity functions. In this way, a good approximation of the "ground truth" fidelity function can be obtained over the entire input domain using only iRBD experiments.

[0064] As another example, method 400 can be used to benchmark a set of spherical harmonic reflection gates. In this example, a set of two-input single-qubit gates can have the following form:

[0065]

[0066] In this hypothetical example, these doors could have a ground truth fidelity function:

[0067]

[0068] The basis functions can be selected as follows:

[0069]

[0070] Among them, Y lm It is a real spherical harmonic function, α lm The coefficient is such that:

[0071] p lm ∈[0,1]

[0072] In this example, method 400 can be executed to generate the coefficients of the series expansion.

[0073]

[0074] Figure 6A It describes the ground truth fidelity function, while Figures 6B to 6D The value of the approximate fidelity function is described, and is used for:

[0075] (use )

[0076] Figure 6B The value of the fidelity function describing l=0 (e.g., item). Figure 6C The value of the fidelity function is depicted when l = 2. Figure 6D The value of the fidelity function when l = 4 is plotted. As can be observed, this approximation becomes closer to the desired fidelity function as the number of terms in the fidelity function increases. Figure 6A The true value of the ground.

[0077] In some embodiments, a non-transitory computer-readable storage medium including instructions that can be executed by a device (e.g., the disclosed encoder and decoder) to perform the methods described above is also provided. Common forms of non-transitory media include, for example, floppy disks, hard disks, solid-state drives, magnetic tape or any other magnetic data storage media, CD-ROMs, any other optical data storage media, any physical media with a perforated pattern, RAM, PROMs and EPROMs, FLASH-EPROMs or any other flash memory, NVRAM, caches, registers, any other memory chips or cassette memories and their network versions. The device may include one or more processors (CPUs), input / output interfaces, network interfaces, and / or memory.

[0078] The foregoing description is for illustrative purposes. This description is not exhaustive and is not limited to the precise forms or embodiments disclosed. Modifications and adaptations to the embodiments will be apparent from the detailed description and practice of the disclosed embodiments. For example, the described implementations include hardware, but systems and methods conforming to this disclosure can be implemented in both hardware and software. Furthermore, while some components have been described as coupled to each other, these components may be integrated with each other or distributed in any suitable manner.

[0079] Furthermore, although illustrative embodiments have been described herein, the scope includes any and all embodiments based on this disclosure that have equivalent gates, modifications, omissions, combinations, adjustments, or alterations (e.g., aspects spanning various embodiments). Gates in the claims are to be interpreted broadly based on the language used in the claims and are not limited to the examples described in this specification or in the application process, which are to be interpreted as non-exclusive. Moreover, the steps of the disclosed method can be modified in any way, including reordering steps or inserting or deleting steps.

[0080] It should be noted that the relational terms used herein (e.g., “first” and “second”) are used only to distinguish one entity or operation from another, and do not require or imply any actual relationship or order between these entities or operations. Furthermore, the words “including,” “having,” “containing,” and “comprising,” as well as other similar forms, are intended to be semantically equivalent and open-ended, as one or more items following any of these terms do not imply an exhaustive list of those items, nor do they imply limitation to the listed items.

[0081] The features and advantages of this disclosure are readily apparent from the detailed description, and therefore the appended claims are intended to cover all systems and methods falling within the true spirit and scope of this disclosure. As used herein, the indefinite articles “a” and “an” mean “one or more”. Furthermore, since many modifications and variations will readily arise upon studying this disclosure, it is not intended to limit this disclosure to the exact constructions and operations shown and described; therefore, all suitable modifications and equivalents may be considered to fall within the scope of this disclosure.

[0082] As used herein, unless otherwise expressly stated, the term "or" includes all possible combinations unless impractical. For example, if it is specified that a database may include A or B, then unless otherwise specified or impractical, the database may include A, B, or A and B. As a second example, if it is specified that a database may include A, B, or C, then unless otherwise specified or impractical, the database may include A, or B, or C, or A and B, or A and C, or B and C, or A and B and C.

[0083] It should be understood that the above embodiments can be implemented by hardware, software (program code), or a combination of hardware and software. If implemented by software, it can be stored in the above-described computer-readable medium. When executed by a processor, the software can perform the disclosed methods. The computing units and other functional units described in this disclosure can be implemented by hardware, software, or a combination of hardware and software. Those skilled in the art will also understand that multiple of the above modules / units can be combined into one module / unit, and each of the above modules / units can be further divided into multiple sub-modules / sub-units.

[0084] In the foregoing description, numerous specific details have been described with reference to embodiments, which may vary depending on the implementation. Certain adjustments and modifications may be made to the described embodiments. Other embodiments will be apparent to those skilled in the art in light of the detailed description and practice of the invention disclosed herein. The description and embodiments are to be considered exemplary only, and the true scope and spirit of the invention are indicated by the appended claims. The sequence of steps shown in the figures is also intended for illustrative purposes only and is not intended to limit one to any particular order of steps. Therefore, those skilled in the art will understand that these steps may be performed in a different order when implementing the same method.

[0085] Exemplary embodiments have been disclosed in the accompanying drawings and description. However, many variations and modifications can be made to these embodiments. Therefore, although specific terminology has been used, it is used only in a general and descriptive sense and not to limit or constrain the scope of the embodiments as defined by the appended claims.

Claims

1. A method for benchmarking quantum gate groups, comprising: Select a quantum gate set, wherein the quantum gates in the quantum gate set are defined on the input domain; Determining the approximate fidelity function of the quantum gate group, wherein determining the approximate fidelity function of the quantum gate group includes: Select the set of basis functions defined on the input domain; A first probability distribution defined over the input domain is generated using a basis function from one of the basis function sets. A fidelity metric for the quantum gates under the first probability distribution is obtained by performing randomized benchmark tests on the quantum components; and Wherein, the approximate fidelity function is a function of the fidelity metric and one of the basis functions in the basis function set, and the approximate fidelity function includes two or more terms of the Fourier, Taylor, or wavelet expansion of the fidelity function of the quantum gate group on the quantum component; and Provide the approximate fidelity function.

2. The method according to claim 1, wherein: Obtaining the fidelity metric involves scaling the first fidelity value of the interleaved sequence of the quantum gate with the second fidelity value of the non-interleaved sequence of the quantum gate.

3. The method according to claim 1, wherein: Performing randomized benchmark tests on the quantum components includes: Determine the first fidelity value of the first sequence of quantum gates, with each first sequence interleaved: A sequence selected from the quantum gate group according to the at least one first probability distribution; and A sequence selected from a group of quantum gates based on a second probability distribution.

4. The method according to claim 3, wherein: The second probability distribution is a uniform probability distribution over the input domain.

5. The method according to claim 1, wherein, The quantum gate set is a subset of a group of quantum gates.

6. The method according to claim 1, wherein, The basis function set includes: Trigonometric basis function set; polynomial basis function set; or Wavelet basis function set.

7. The method according to claim 1, wherein, The input domain includes two or more variables.

8. The method according to any one of claims 1 to 7, wherein, The quantum components include transmon or fluxonium qubits.

9. A system for benchmarking quantum gate sets, comprising: At least one processor; as well as At least one non-transitory computer-readable medium containing instructions that, when executed by the at least one processor, cause the system to perform the method of any one of claims 1 to 8.

10. A non-transitory computer-readable medium containing instructions that, when executed by at least one processor of a system, cause the system to perform the method of any one of claims 1 to 8.

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