Estimating the Fidelity of Quantum Logic Gates and Quantum Circuits

Abstract

Methods, systems, and apparatuses for estimating the fidelity of a quantum logic gate. In one aspect, a method includes defining a plurality of sets of random quantum circuits; for each set of random quantum circuits: selecting observables for each element of the set of random quantum circuits, where each selected observable corresponds to a respective element of the set of random quantum circuits and depends on the element to which each selected observable corresponds; estimating a value of a polarization parameter for the set of random quantum circuits, including performing least mean square minimization based on a plurality of expected values, where each expected value includes an expected value of a respective selected observable with respect to an output of an experimental implementation of the random quantum circuit corresponding to the respective selected observable; and processing the estimated polarization parameter value to obtain an estimate of the fidelity of an n-qubit quantum logic gate.

Description

BACKGROUND OF THE INVENTION

[0001] This specification relates to quantum computing.

[0002] Quantum computing uses quantum mechanical phenomena, such as superposition and entanglement, to perform computations. A quantum circuit is an example model for quantum computing, in which a computation is a sequence of quantum logic gates, which is a reversible transformation on a quantum mechanical simulation of an n-qubit register. SUMMARY OF THE INVENTION

[0003] This specification describes techniques for estimating the fidelity of quantum logic gates and quantum circuits.

[0004] In general, one innovative aspect of the subject matter described in this specification can be implemented as a method for estimating the fidelity of an n-qubit quantum logic gate, the method comprising: defining a plurality of sets of random quantum circuits, wherein the plurality of sets of random quantum circuits correspond to different respective depths and each set of the plurality of sets of random quantum circuits includes random quantum circuits having the same circuit depth d, wherein defining the plurality of sets of random quantum circuits includes, for each set of random quantum circuits: defining one or more elements of the set of random quantum circuits, including: for each element, randomly sampling d*n single-qubit gates from a predefined set of single-qubit gates, wherein each single-qubit gate operates on a corresponding qubit for a corresponding period (cycle); and defining the elements of the set of random quantum circuits to be equal to n randomly sampled single-qubit gates for d periods followed by an n-qubit quantum logic gate; for each set of random quantum circuits: choosing an observable for each element in the set of random quantum circuits, wherein each chosen observable corresponds to a respective element in the set of random quantum circuits and depends on the element to which it corresponds; and estimating a value of a polarization parameter for the set of random quantum circuits, including performing least mean squares minimization based on a plurality of expected values, wherein each expected value includes an expected value of a respective chosen observable with respect to an output of an experimental implementation of the random quantum circuit corresponding to the respective chosen observable; and processing the estimated polarization parameter value to obtain an estimate of the fidelity of the n-qubit quantum logic gate.

[0005] Other implementations of this aspect include corresponding classical and quantum computer systems, devices, and computer programs recorded on one or more computer storage devices, each of which is configured to perform the actions of the method. A system of one or more computers can be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system, which, in operation, causes the system to perform the actions. One or more computer programs can be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform the actions.

[0006] Each of the foregoing and other implementations can optionally include, individually or in combination, one or more of the following features. In some implementations, estimating the value of the polarization parameter of a set of random quantum circuits further includes determining a plurality of expected values, including: for each expected value, defining the expected value of the corresponding selected observable relative to the output of the experimental implementation of the random quantum circuit corresponding to the corresponding selected observable as i) the trace of the corresponding selected observable divided by the dimension of the Hilbert space, plus ii) the polarization of the random quantum circuit multiplied by the difference between the expected value of the corresponding selected observable relative to the ideal output state of the random quantum circuit and the trace of the corresponding selected observable divided by the dimension of the Hilbert space; numerically estimating (i) the value of the expected value of the corresponding selected observable relative to the ideal output state of the random quantum circuit, and ii) the value of the trace of the corresponding selected observable divided by the dimension of the Hilbert space; and experimentally estimating the value of the expected value of the corresponding selected observable relative to the output of the experimental implementation of the random quantum circuit.

[0007] In some implementations, experimentally estimating the value of the expected value of the corresponding selected observable relative to the output of the experimental implementation of the random quantum circuit includes: repeatedly: preparing a quantum system in an initial state; applying the random quantum circuit to the quantum system prepared in the initial state to generate an evolved state of the quantum system; and measuring the corresponding selected observable relative to the evolved state of the quantum system; and averaging the measured selected observables to estimate the value of the expected value of the corresponding selected observable relative to the output of the experimental implementation of the random quantum circuit.

[0008] In some implementations, performing least mean square minimization includes: minimizing the sum of squared residuals, where each squared residual corresponds to a respective element in a set of random quantum circuits, and each squared residual includes the square of: i) the trace of the respective selected observable divided by the dimension of the Hilbert space, plus ii) the value of the polarization parameter of the set of random quantum circuits multiplied by the difference between the expected value of the respective selected observable with respect to the ideal output state of the random quantum circuit and the trace of the respective selected observable divided by the dimension of the Hilbert space, minus iii) the expected value of the respective selected observable with respect to the output of the experimental implementation of the random quantum circuit corresponding to the respective selected observable.

[0009] In some implementations, processing the estimated polarization parameter values to obtain an estimate of the fidelity of an n-qubit quantum logic gate includes: fitting the estimated polarization parameter values corresponding to each circuit depth d to an exponential decay of d; determining an estimated per-cycle polarization pn of the n-qubit quantum logic gate based on the exponential decay of d and an estimate of the polarization of single-qubit gates in the quantum circuit operating on n qubits; and using F = pn+(1 - pn) / D to determine an estimate of the fidelity of the n-qubit quantum logic gate, where D = 2 n represents the Hilbert space dimension.

[0010] In some implementations, the n-qubit quantum logic gate operates on at most 5 qubits.

[0011] In some implementations, the selected observables are diagonal in the computational basis.

[0012] In some implementations, the selected observables include cross-entropy benchmark observables, linear cross-entropy observables, or heavy output fraction observables.

[0013] In some implementations, the method further includes using the estimate of the fidelity of the n-qubit quantum logic gate to determine one or more properties of the quantum hardware implementing the n-qubit quantum logic gate.

[0014] In some implementations, the method further includes determining one or more adjustments to quantum hardware control parameters based on the determined estimate of the fidelity; and using the quantum computing hardware to implement the determined one or more adjustments to perform quantum computing.

[0015] In some implementations, the method further includes calculating the distribution of a random variable associated with the selected observables; and using the calculated distribution of the random variable to perform one or more statistical tests to obtain additional information about the n-qubit quantum logic gate.

[0016] In some implementations, performing one or more statistical tests using the distribution of the computed random variables to obtain additional information about an n-qubit quantum logic gate includes: performing a Kolmogorov-Smirnov test to validate the effectiveness of the estimated fidelity of the n-qubit quantum logic gate.

[0017] Generally, another innovative aspect of the subject matter described in this specification can be implemented as a method for estimating the fidelity of a quantum circuit, the method including: defining one or more random quantum circuits, where each of the one or more random quantum circuits has the same circuit depth d and operates on the same number of qubits n, where defining the random quantum circuits includes: randomly sampling d*n single-qubit gates from a predefined set of single-qubit gates, where each single-qubit gate operates on a corresponding qubit; and defining the random quantum circuit to be equal to d periods of n randomly sampled single-qubit gates and a plurality of multi-qubit quantum logic gates that then operate on different qubits; selecting an observable of the random quantum circuit for each defined random quantum circuit, where i) the selected observable depends on the random quantum circuit, and ii) the selected observable obeys a concentration of measure related to the expected value of the selected observable with respect to the expected value of the density matrix representing the error effects in the experimental implementation of the random quantum circuit; and determining an estimate of the fidelity of the random quantum circuit for each defined random quantum circuit, including: estimating the expected value of the corresponding selected observable with respect to the output of the experimental implementation of the random quantum circuit for each random quantum circuit.

[0018] Other implementations of this aspect include corresponding classical and quantum computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each of which is configured to perform the actions of these methods. A system of one or more computers can be configured to perform specific operations or actions by installing software, firmware, hardware, or a combination thereof on the system, which in operation causes the system to perform the actions. One or more computer programs can be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform the actions.

[0019] Each of the foregoing and other implementations may optionally include one or more of the following features, either alone or in combination. In some implementations, estimating the expected value of a selected observable with respect to the output of an experimental implementation of a random quantum circuit includes: approximating the expected value of the selected observable with respect to the output of an experimental implementation of a random quantum circuit as: the trace of the selected observable divided by the Hilbert space dimension, plus the fidelity of the quantum circuit times the sum of i) the difference between the expected value of the selected observable with respect to the ideal output state of the quantum circuit and the trace of the selected observable divided by the Hilbert space dimension; experimentally estimating the value of the expected value of the selected observable with respect to the output of an experimental implementation of a random quantum circuit; numerically or analytically estimating i) the value of the expected value of the selected observable with respect to the ideal output state of the random quantum circuit, and ii) the value of the trace of the corresponding selected observable.

[0020] In some implementations, experimentally estimating the value of the expected value of a selected observable with respect to the output of an experimental implementation of a random quantum circuit includes: repeatedly: preparing a quantum system in an initial state; applying a random quantum circuit to the quantum system prepared in the initial state to generate an evolved state of the quantum system; and measuring the corresponding selected observable with respect to the evolved state of the quantum system; and averaging the measured selected observables to estimate the value of the expected value of the selected observable with respect to the output of an experimental implementation of a random quantum circuit.

[0021] In some implementations, the method includes analytically estimating i) the value of the expected value of the selected observable with respect to the ideal output state of a random quantum circuit, and ii) the value of the trace of the corresponding selected observable, and wherein the output of the experimental implementation of the random quantum circuit is approximated by a Porter-Thomas distribution.

[0022] In some implementations, the method further includes determining an average of the determined estimates of the fidelity for each random quantum circuit to obtain an average estimate of the circuit fidelity for circuits of depth d and number of qubits n.

[0023] In some implementations, the quantum circuit operates on 10 or more qubits.

[0024] In some implementations, the selected observable is diagonal in the computational basis.

[0025] In some implementations, the selected observable includes a cross-entropy benchmark observable, a linear cross-entropy observable, or a re-output generation fraction observable.

[0026] In some implementations, the method further includes using the determined estimate of the fidelity or the average estimate of the circuit fidelity to determine one or more properties of the quantum hardware implementing the quantum circuit.

[0027] In some implementations, the method further includes determining one or more adjustments to quantum hardware control parameters based on the determined estimate of fidelity; and using quantum computing hardware to implement the determined one or more adjustments to perform quantum computing.

[0028] In some implementations, the method further includes computing the distribution of a random variable associated with a selected observable; and using the computed distribution of the random variable to perform one or more statistical tests to obtain additional information about the quantum circuit.

[0029] In some implementations, using the computed distribution of the random variable to perform one or more statistical tests to obtain additional information about the quantum circuit includes: performing a Kolmogorov-Smirnov test to confirm the validity of the estimated fidelity of the quantum circuit.

[0030] The subject matter described in this specification can be implemented in a particular manner so as to achieve one or more of the following advantages.

[0031] The currently described techniques for estimating the fidelity of a quantum circuit can be applied to a wider range of quantum circuits and numbers of qubits compared to known techniques. For example, the currently described techniques do not have to assume that the quantum circuit approximates a Porter-Thomas distribution. Additionally, the currently described techniques are not limited to a particular observable (e.g., the cross-entropy observable), but can be applied in combination with different observables that can provide a more accurate fidelity estimate for a particular quantum circuit. Additionally, the currently described techniques are applicable to any quantum logic gate and are not limited to Clifford gates. Additionally, the currently described techniques provide increased scalability, e.g., up to 40 qubits or more.

[0032] The currently described techniques can be applied to improve quantum computing hardware. For example, circuit fidelity can be used to calibrate quantum computing hardware or confirm the validity of quantum computing hardware, or to determine adjustments that can improve the accuracy or efficiency of existing quantum computing hardware. Since the circuit fidelity estimated using the techniques described in this specification can be more accurate and more tailored for a particular quantum circuit and / or quantum computing hardware, the adjustments determined using the estimated circuit fidelity can be more effective. Additionally, high-fidelity gates are crucial for quantum computers. High-fidelity gates require high-precision control, and the currently described techniques can be used to improve the precision of control. Additionally, high-fidelity gates are crucial for quantum computers. High-fidelity gates require high-precision control, and the currently described techniques can be used to improve the precision of control.

[0033] Details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the specification, the drawings, and the claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 An example system for benchmarking quantum computing hardware is depicted.

[0035] Figure 2 Is a flowchart of a first example process for estimating the fidelity of a quantum logic gate.

[0036] Figure 3 Is a flowchart of a second example process for estimating the fidelity of a quantum circuit.

[0037] Figure 4 Is a flowchart of an example process for determining properties of a quantum circuit. DETAILED DESCRIPTION

[0038] OVERVIEW

[0039] A quantum circuit is a model for quantum computing, in which quantum logic gates are applied to a register of qubits in a specific sequence to encode quantum information. In theory, any quantum algorithm can be implemented with high precision by applying a correctly chosen sequence of quantum logic gates. However, in practice, quantum logic gates are prone to errors - instead of implementing the unitary quantum operation representing the ideal quantum logic gate, a corresponding noisy quantum operation is implemented.

[0040] Quantum logic gate fidelity is a measure of how closely the noisy quantum operation ε approximates the ideal unitary quantum operation For a given quantum state ρ, the quantum logic gate fidelity between ε and Can be given by the following formula:

[0041]

[0042] Estimating quantum logic gate fidelity is an important process for tuning or correcting the quantum hardware that physically implements the quantum logic gate, and thus is an important process for performing successful quantum computing. This specification describes general techniques for obtaining a statistical fidelity estimator that does not assume a Porter-Thomas distribution and is not limited to one type of observable.

[0043] EXAMPLE HARDWARE

[0044] Figure 1Illustrates an example system 100 for benchmarking quantum computing hardware. The example system 100 is an example of a system implemented as classical and quantum computer programs on one or more classical and quantum computers at one or more locations, where the systems, components, and techniques described below may be implemented.

[0045] System 100 includes a classical processor 102 that communicates data with quantum computing hardware 104. For convenience, classical processor 102 and quantum computing hardware 104 are shown as separate entities. However, in some implementations, classical processor 102 may be included within quantum computing hardware 104. For example, quantum computing hardware 104 may include one or more components for performing classical computing operations.

[0046] Quantum computing hardware 104 includes components for performing quantum computing using quantum circuits. For example, quantum computing hardware 104 includes a quantum system 120 and a control device 122. Quantum system 120 includes one or more multi-level quantum subsystems for performing algorithmic operations or quantum computing, such as qubits. The specific implementation of the multi-level quantum subsystems included in quantum computing hardware 104 and how the multi-level quantum subsystems interact with each other depend on a number of factors, including the type of quantum computing being performed by the quantum computing hardware. For example, the multi-level quantum subsystems may include qubits implemented via atomic, molecular, or solid-state quantum systems. In other examples, qubits may include, but are not limited to, superconducting qubits or semiconductor qubits.

[0047] The multi-level quantum subsystem can be frequency-tunable. For example, each qubit can have an associated operation frequency, which can be adjusted, for example, by applying voltage pulses via one or more drivelines coupled to the qubit, using one or more control devices 122. Example operation frequencies include the qubit idling frequency, the qubit interaction frequency, and the qubit readout frequency. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operation frequency to the corresponding idling frequency can place the qubit in a state where the qubit does not strongly interact with other qubits and where the qubit can be used to perform single-qubit gates. As another example, in the case where qubits interact with a fixed coupling via a coupler, the qubits can be configured to interact with each other by setting their respective operation frequencies to some gate-dependent frequencies that are detuned from their common interaction frequency. In other cases, such as when qubits interact via a tunable coupler, the qubits can be configured to interact with each other by setting the parameters of their respective couplers to enable the interaction between the energy qubits, and then by setting the respective operation frequencies of the qubits to some gate-dependent frequencies that are detuned from their common interaction frequency. Such interactions can be performed in order to perform two-qubit or many-qubit gates.

[0048] The control device 122 can also include a measurement device, such as a readout resonator. Measurement results obtained via the measurement device can be provided to a classical processor included in the quantum computing hardware 104 or to the classical processor 102 for processing and analysis.

[0049] The classical processor 102 receives input data 106 representing a quantum logic gate or a quantum circuit to be benchmarked. For example, the input data 106 can include data representing an n-qubit quantum logic gate that the quantum computing hardware 104 is configured to implement. The input data 106 can also specify the type of observable to be used when benchmarking the quantum logic gate or the quantum circuit. Example observables are described in detail below with reference to Figure 2 and Figure 3 Example observables are described in detail below with reference to

[0050] The classical processor 102 processes the received input data 106 to generate output data 108 representing the benchmarking results (e.g., properties of the implementation of the quantum logic gate or the quantum circuit). For example, the output data 108 can include data representing the estimated fidelity of the implementation of the n-qubit quantum logic gate.

[0051] The classical processor 102 includes multiple components for processing received input data. For example, the classical processor 102 may include a random quantum circuit generator 110, a fidelity estimation module 112, and a post-processing module 114.

[0052] The random quantum circuit generator 110 may be configured to define multiple random quantum circuits. The multiple random quantum circuits may include sets of random quantum circuits corresponding to different respective circuit depths, and the sets of random quantum circuits include random quantum circuits having the same circuit depth d.

[0053] To define a random quantum circuit of depth d, the random quantum circuit generator 110 is configured to randomly sample single-qubit gates from a predefined set of single-qubit gates (e.g., the set of single-qubit gates that can be implemented by the quantum hardware 120). For example, the predefined set of single-qubit gates may include quantum gates representing π / 2 rotations about axes in the (x,y) plane (whose orientation is uniformly randomly sampled) and d-phase quantum gates representing non-Clifford diagonal matrices {0,e if}, where f is randomly sampled from the interval (0,2π). Other sets of single-qubit gates may also be used as long as they are sufficiently random such that the depolarizing channel model can be used to model the effect of noise in the defined random quantum circuits.

[0054] Then, the random quantum circuit generator 110 defines the random quantum circuit to be equal to d cycles of randomly sampled single-qubit gates followed by the quantum logic gates or circuits to be benchmarked. For example, each cycle includes a respective randomly sampled single-qubit gate followed by the quantum logic gates to be benchmarked. The classical processor 102 may also define observables corresponding to each defined random quantum circuit, e.g., based on the type of observable specified by the input data 106.

[0055] The classical processor 102 may be configured to send data 116 representing the defined random quantum circuits and observables to the quantum computing hardware 104. As described above, the quantum computing hardware 104 is configured to use the control device 122 to implement the defined random quantum circuits on the quantum system 120 and provide output data representing the results of the circuit implementation, e.g., data 124 representing the measured values of the observables.

[0056] The fidelity estimation module 112 may be configured to estimate the value of a fidelity parameter of the random quantum circuits generated by the random quantum circuit generator 110 and implemented by the quantum computing hardware 104. For example, the fidelity estimation module 112 may be configured to perform numerical simulations, least mean square minimization routines, and the following see Figures 2 - 4Other classical computations described.

[0057] The post - processing module 114 can be configured to process the estimated fidelity parameter values generated by the fidelity estimation module 112 based on the random quantum circuit generated by the random quantum circuit generator 110 to obtain an estimate of the fidelity of a particular component of the random quantum circuit (e.g., a particular quantum logic gate specified by the input data 106). The post - processing module 114 can generate output data representing the fidelity estimate of the component of the random quantum circuit, e.g., output data 108.

[0058] In addition, in some implementations, the post - processing module 114 can be configured to process or analyze the estimated fidelity parameter values to determine properties of the quantum computing hardware 104 (e.g., its performance), or to calibrate, validate, or benchmark the quantum computing hardware 104. The post - processing module 114 can also be configured to perform additional statistical tests to obtain additional information about the quantum computing hardware, e.g., the variance of the quantum circuit fidelity. Refer to the following Figure 4 for an example process for performing additional statistical tests to obtain such information.

[0059] In some implementations, the post - processing module 114 can also generate output data representing one or more adjustments 126 that can be used to tune and improve the quantum computing hardware 104. For example, the post - processing module 114 can use the estimated fidelity parameter values to determine how to control the adjustments to the quantum computing hardware when implementing a particular quantum circuit or type of quantum circuit, e.g., determining corrections to the programming of the control device 122 to implement higher - fidelity quantum gates. A parameterized control model can be used to determine the corrections, where the parameterized control model relates the parameters of the quantum gate (e.g., phase, qubit rotation angle, etc.) to the physical parameters of the system / multiple systems used to implement / control the quantum gate (e.g., voltage, pulse shape, frequency, etc.). Then an outer loop can be executed to find the optimal experimental control to improve the performance of the quantum computing hardware 104.

[0060] Programming the hardware

[0061] Figure 2 is a flowchart of a first example process 200 for estimating the fidelity of an n - qubit quantum logic gate G n The example process 200 can be applied to determine the fidelity of a quantum logic gate G operating on any number (e.g., for any n≥1) of qubits. However, the example process 200 is particularly suitable for estimating the fidelity of a quantum logic gate G operating on a small number of qubits (e.g., n < 5). n n ​The fidelity. For convenience, process 200 will be described as being performed by a system of one or more classical and quantum computing devices located at one or more locations. For example, system 100 Figure 1 appropriately programmed according to this specification can perform process 200.

[0062] Operating a single multi-qubit gate in isolation is different from operating a single multi-qubit gate in the context of a complex algorithm on a large device because non-idealities (such as crosstalk and unwanted interactions) affect how the multi-qubit gate is implemented. Thus, to estimate the fidelity of quantum logic gate G n , the gate is embedded in a quantum circuit that includes multiple quantum logic gates. Then, the fidelity of each of the quantum circuits is estimated, and the fidelity of the single gate G n can be extracted from the estimated circuit fidelity.

[0063] The quantum circuit in which gate G n is embedded takes the same form: each quantum circuit includes one or more single-qubit gates selected from a random set and quantum logic gate G n . For example, each quantum circuit can include one or more cycles of n randomly sampled single-qubit gates (e.g., one per qubit) followed by quantum logic gate G n . The depth of the quantum circuit is equal to the number of cycles. For example, a quantum circuit with depth d = 2 can include two cycles, where each cycle includes n randomly sampled single-qubit gates, and each randomly sampled single-qubit gate operates on the corresponding qubit and the subsequent quantum logic gate G n . Due to the random sampling of the single-qubit gates, quantum circuits of this form are herein referred to as random quantum circuits.

[0064] The system defines multiple sets of random quantum circuits (step 202). The multiple sets of random quantum circuits correspond to different respective depths, and each set of the multiple sets of random quantum circuits includes random quantum circuits having the same circuit depth d. To define a set of random quantum circuits having circuit depth d, the system defines the elements of the set of random quantum circuits, i.e., the random quantum circuits included in the set. Defining the elements of the set of random quantum circuits having circuit depth d includes randomly sampling d*n single-qubit gates from a predefined set of single-qubit gates, where each single-qubit gate operates on the corresponding qubit of n qubits, and defining the elements of the set of random quantum circuits to be equal to n randomly sampled single-qubit gates for d cycles followed by an n-qubit quantum logic gate.

[0065] The effect of noise in a random quantum circuit with depth d = 1 can be represented by the depolarizing channel given in the following equation (1).

[0066]

[0067] In equation (1), ρ represents the state of n qubits, represents the identity matrix, and p represents the random circuit polarization and includes the fidelity from the quantum logic gate G n and the fidelity of the single-qubit gates included in the random circuit. The system uses data from multiple ensembles of the random circuit to estimate the parameter p. Each ensemble includes random circuits with the same circuit depth d, and the multiple ensembles of random circuits correspond to different respective depths, e.g., approximately 100 different depths between 10 and 700.

[0068] The effect of noise in a random circuit with depth d > 1 and an instance of the parameter p (corresponding to the random circuit with depth d = 1) can be represented by the depolarizing channel given in the following equation (2).

[0069]

[0070] In equation (2), ρ represents the state of n qubits and p d represents the random circuit. The system estimates the parameter p of the set of experimental implementations of the random circuit with depth d d , as described in more detail below with reference to step 206.

[0071] The system selects an observable O (step 204). The selected observable is an observable that depends on a particular instance of the random quantum circuit U (i.e., O = O U ). In some implementations, the selected observable O U can be chosen to be diagonal in the computational basis, e.g., to simplify obtaining an experimental estimate of the expected value of the output of the observable O U with respect to the experimental noisy implementation of the random circuit.

[0072] For example, in some implementations, the system can select the linear cross-entropy observable. The linear cross-entropy observable is given by the following equation (3).

[0073]

[0074] In equation (3), N = 2 n , p U (z) represents the probability that the circuit U outputs the specific bitstring z, and |z> represents the output state corresponding to the specific bitstring z.

[0075] As another example, in some implementations, the system may choose to re - output the generated fractional observables. The re - output generated fractional observables are given by the following equation (4).

[0076]

[0077] In equation (4), if Np U (z)≥log2, then equals 1, otherwise equals 0, and |z> represents the output state corresponding to the specific bit string z.

[0078] Since the selected observables depend on a particular instance of the random quantum circuit (e.g., the corresponding random sampling of single - qubit gates and the circuit depth), the system defines corresponding observables for each element in each set of random quantum circuits.

[0079] The observable O U with respect to the output ρ U of the noisy implementation of the experiment on the random quantum circuit U can be given by the following equation (5).

[0080] Trρ U O U =R U =p d <ψ U |O U |ψ U >+(1 - p d )TrO U / 2 n (5)

[0081] =p d (V U -N U )+N U

[0082] In equation (5), |ψ U > represents the state of the qubit after the random quantum circuit U is applied to the initial state of the qubit. For example, |ψ U >=U|ψ0>, V U =<ψ U |O U |ψ U >, R U =Trρ U O U and N U =TrO U / 2 n .

[0083] The system estimates the polarization parameter p for each ensemble of random quantum circuits (e.g., each ensemble of depth d). d The system estimates the value of the polarization parameter p for the corresponding ensemble of random quantum circuits by performing least mean square minimization based on Equation (5). d First, the system determines the values of V, N, and R for the corresponding random quantum circuit U in the ensemble of random quantum circuits in Equation (5). U N U and R U The system can numerically determine the values of V and N, for example, via V = ∑ p(z)<z|O|z> and N = ∑ <z|O|z> / N, and experimentally determine the value of R. U V z p U (z)<z|O U |z> and N U N z <z|O U |z> / N to numerically determine V U and N U and experimentally determine the value of R U .

[0084] The system can experimentally estimate R in Equation (5) by performing multiple measurements on the selected observable O U R = α(V U - N U ) + N U to determine the estimate U This can include repeatedly preparing the quantum system (qubit register) in the initial state, applying the random circuit U to the quantum system prepared in the initial state to generate the evolved state of the quantum system, and measuring the observable with respect to the evolved state of the quantum system to obtain multiple measurement results that can be used to determine the estimate For example, by averaging the multiple measurement results to determine the estimate

[0085]

[0085] For example, the action of the selected observable O U = ∑ z O U (z)|z><z| maps each bitstring or measurement result z to the real value O U (z). Each value O U (z) depends on the random circuit U, and the calculation of the experimental values {O U (z j )} requires classical simulation of U. For example, for the linear cross-entropy observable Np U (z) (where p U (z) = |<z|U|0>| 2 ), if the initial state is |0>, the calculation of the specific bitstring z measured in the experimentj The value p U (z j ) = |<z j |U|0>| 2 A simulation of U is required. For a set {z j} of M measurement results, the estimate of R U can be given by the following equation (6), which in turn gives the following equation (7).

[0086]

[0087] To obtain an estimate of the polarization parameter p for the corresponding set of random quantum circuits

[0088]

[0089] the system performs least mean square minimization by minimizing the sum of the squared residuals (over the random circuits), where each residual is given by setting equation (5) equal to zero, i.e., each residual is given by d which is (in other words) the difference between i) the polarization parameter p of the corresponding set of random quantum circuits multiplied by V d (the expected value of the corresponding selected observable with respect to the ideal output state of the random quantum circuit) and N U (the trace of the corresponding selected observable divided by the dimension of the Hilbert space), and ii) U (the expected value of the estimate of the observable O with respect to the output ρ U of the experimental noisy realization of the random quantum circuit) and N U (the trace of the corresponding selected observable divided by the dimension of the Hilbert space). That is, the system determines the following minimum: U where ΔV

[0090]

[0091] where ΔV U = V U - N U and Then, the minimum of LMS(p d ) is obtained using the following equation

[0092]

[0093] That is, the system obtains the polarization p d to be

[0094]

[0095] where ΔV U = V U - N U and

[0096] The system processes the estimated polarization parameter value p d to obtain an estimate of the value of the polarization parameter p for equation (1) (step 208). For example, the system may fit the estimated p d at different depths d to an exponential decay of d and extrapolate to obtain an estimate of p for d = 1. Fitting these estimates in this way distinguishes the polarization parameter p for a single application of the quantum circuit from the state preparation and measurement (SPAM) errors. More specifically, the SPAM error can be modeled as a constant depolarizing fidelity S independent of d, and fitting the exponential decay of Sp d as a function of d enables the system to fit p independently of the SPAM error.

[0097] The system can use the estimate of p (polarization per cycle) for d = 1 to determine the fidelity of a single gate G n For example, the system can obtain an estimate of the polarization p 1,n of a single-qubit gate in a circuit with n qubits from a previously randomized benchmarking or cross-entropy benchmarking experiment. Then, the system can estimate the polarization p n per cycle of the gate G n as It can use F = p n + (1 - p n ) / D to convert this polarization p n per cycle to a measure of the fidelity of the gate G n where D = 2 n represents the Hilbert space dimension.

[0098] In some implementations, the system can also perform statistical tests to gather more information about the random circuit or estimates of the fidelity parameters, as described in more detail below with reference to Figure 4

[0099] In some implementations, the system can also use the estimate of the fidelity parameter p and / or information about using the following reference Figure 4 ​Additional information of the quantum circuit obtained from the described example process 400 is used to determine one or more attributes of the random circuit. For example, an estimated value of the polarization parameter p or the corresponding fidelity can be used to determine the performance of the random circuit. As another example, the estimated fidelity can be used for (i) calibration, (ii) validation, or (iii) benchmarking of the quantum computing hardware implementing the quantum circuit.

[0100] In some implementations, the system can use the estimated fidelity and / or additional information about the quantum circuit to determine one or more adjustments to the quantum computing hardware. For example, the system can determine an adjustment to the control parameters of the control model used by the quantum computing hardware to implement a quantum operation to improve the fidelity of the quantum operation. The system can use the adjusted control model to implement the quantum operation with increased fidelity in future quantum computations.

[0101] Figure 3 is a flowchart of a second example process 300 for estimating the fidelity of an n-qubit quantum circuit. Example process 300 can be applied to estimate the fidelity of a quantum circuit operating on any number (e.g., for any n ≥ 1) of qubits. However, example process 300 is particularly suitable for estimating the fidelity of a quantum circuit operating on a large number (e.g., n ≥ 10) of qubits. For convenience, process 300 will be described as being performed by a system of one or more classical and / or quantum computing devices located at one or more locations. For example, a system 100 Figure 1 appropriately programmed according to this specification can perform process 300.

[0102] The system defines a plurality of quantum circuits (step 302). The defined quantum circuits have the same circuit depth d and operate on the same number of qubits n. Each defined quantum circuit includes one or more single-qubit gates selected from a sufficiently random set and a plurality of multi-qubit quantum gates, such as multiple instances of two-qubit gates. For example, each of the quantum circuits can include d cycles of n randomly sampled single-qubit gates (e.g., one single-qubit gate per qubit) followed by a plurality of multi-qubit quantum logic gates operating on different qubits. For example, a quantum circuit with depth d = 2 can include two cycles of n randomly sampled single-qubit gates (each operating on the corresponding qubit) followed by multi-qubit quantum gates. Due to the random sampling of the single-qubit gates, quantum circuits of this form are referred to herein as random circuits.

[0103] The output of the experimental (noisy) implementation of the random quantum circuit U defined in step 302 can be given by the following equation (9).

[0104] ρU = α|ψ U ><ψ U |+(1 - α)χ U (9)

[0105] In equation (9), |ψ U > represents the state of the quantum system (qubit register) after the application of a random quantum circuit, α = <ψ U |ρ U |ψ U > represents the random quantum circuit fidelity, and χ U represents the density matrix that represents the error effects in the experimental implementation of the random quantum circuit. In some implementations, the output of the implementation of the random quantum circuit U can approximate a Haar random state or a Porter - Thomas distribution. For completeness, it should be noted that there is a small difference of order 2 -n between the fidelity α and the parameter p of equation (1) above, which can be neglected when n >> 1.

[0106] The system selects an observable O (step 304). The selected observable is an observable that depends on a particular instance of the random quantum circuit U (i.e., O = O U ). In some implementations, the selected observable O U can be chosen to be diagonal in the computational basis, for example, to simplify obtaining an experimental estimate of the expected value of the observable O U with respect to the output of the experimental noisy implementation of the random circuit.

[0107] For example, in some implementations, the system can select a linear cross - entropy observable for the simulation of the quantum circuit. The linear cross - entropy observable is given by:

[0108]

[0109] where N = 2 n , p U (z) represents the probability that the circuit U outputs the specific bitstring z, and |z> represents the output state corresponding to the specific bitstring z.

[0110] As another example, in some implementations, the system can select a re - output generation fraction observable for the simulation of the quantum circuit. The re - output generation fraction observable is given by:

[0111]

[0112] where, if Np U (z) ≥ log2, then Equal to 1, otherwise equal to 0, and |z> represents the output state corresponding to a specific bit string z.

[0113] Since the selected observable depends on a specific instance of the random quantum circuit (e.g., the corresponding random sampling of single-qubit gates and the circuit depth), the system defines a corresponding observable for each defined random quantum circuit.

[0114] Observable O U The expectation value of the output ρ with respect to the noisy experimental implementation of the random quantum circuit U U Can be given by the following equation (12).

[0115] Trρ U O U = α <ψ U |O U |ψ U > + (1 - α) Trx U O U (12)

[0116] In equation (12), x U Represents the density matrix that represents the error effects in the experimental implementation of the random quantum circuit.

[0117] The selected observable O U Has the following property: The expectation value of the selected observable with respect to the density matrix representing the error effects in the experimental implementation of the quantum circuit (i.e., Trχ in the second term on the right-hand side of equation (12)) U O U ) obeys concentration of measure related to the expectation value of the selected observable. That is, the expectation value Trχ U O U Gives

[0118]

[0119] Where Represents a typical value independent of χ U N = 2 n Represents the dimension of the corresponding Hilbert space, and ∈ represents correction. The verification of the above property can be numerically performed by introducing digital errors and calculating the corresponding observables, or by analyzing the corresponding Pearson correlation coefficient between the ideal output distribution and the distribution with one digital error.

[0120] The system determines an estimate of the fidelity of each defined random quantum circuit (Step 306). The system estimates the fidelity of the corresponding random quantum circuit by solving equation (12) of the random quantum circuit That is, the system estimates the selected observable O U with respect to the output ρ U of an experimental implementation of a random quantum circuit U O U (left - hand side of Equation 12).

[0121] To estimate the expected value Trρ U O U of the corresponding circuit U U , the system approximates the expected value Trρ U Ox as N U = TrO U / N (the trace of the selected observable <O U > = TrO U divided by N) plus the fidelity α of the quantum circuit times the sum of the differences between i) V U (the expected value of the selected observable with respect to the ideal output state of the quantum circuit <ψ U |O U |ψ U >)) and ii) N U = TrO U / N (the trace TrO U O U is approximated by assuming that the correction ∈ tends to zero to determine an estimate of the fidelity of the quantum circuit

[0122]

[0123] The assumption that the correction value ∈ tends to zero can be numerically verified. For example, in a model that accounts for bit - flip (Pauli matrix σ ) or phase - flip (Pauli matrix σ x ) errors after each gate z ), it can be numerically verified

[0124]

[0125] where denotes the circuit obtained when σ j is added after the gate g. This assumption works well when the output |ψ U > of the random circuit U is approximated by a Haar - random state or a Porter - Thomas distribution, and typical errors can be expected to have statistical fluctuations

[0126] The system can experimentally estimate R in Equation (14) by performing multiple measurements on the selected observable O U ​U to determine an estimate This can include repeatedly preparing a quantum system (qubit register) in an initial state, applying a quantum circuit U to the quantum system prepared in the initial state to generate an evolved state of the quantum system, and measuring an observable with respect to the evolved state of the quantum system to obtain multiple measurement results that can be used to determine an estimate For example, by averaging the multiple measurement results to determine the estimate

[0127] For example, a selected observable O U = ∑ z U U (z)|z><z| maps each bitstring or measurement result z to a real value O U (z). Each value O U (z) depends on the random circuit U, and the calculation of the experimental values {O U (z j )} requires a classical simulation of U. For a set of M measurement results {z j}, the estimate of R U can be given by which in turn gives as described above with reference to equations (6) and (7).

[0128] The system can determine estimates of V U and N U in equation (14) numerically or analytically and For example, in some implementations, the output state ρ U given by equation (9) above can approximate the Porter-Thomas distribution. In these implementations, the values of V U and N U in equation (14) can be determined analytically. The values of V U and N U in equation (14) can be determined numerically by using classical simulation to calculate V U = <ψ U |O U |ψ U > and N U = TrO U / N

[0129] The system uses the determined estimates and to determine an estimate of the fidelity of the random quantum circuit U That is, the system uses and to solve for in Equation (14) Then, the system can determine the average of the determined estimates for each random quantum circuit to obtain an estimate of the circuit fidelity for a circuit with depth d and number of qubits n.

[0130] In some implementations, the system can also perform statistical tests to gather more information about the quantum circuit or the estimate of the circuit fidelity, as described in more detail below with reference to Figure 4 as described in more detail.

[0131] In some implementations, the system can also use the individual estimates of the fidelity of the random quantum circuits the average estimate of the fidelity and / or additional information about the random quantum circuits obtained using the example process 400 described below with reference to Figure 4 to determine one or more properties of the random quantum circuits. For example, the estimated fidelity can be used to determine the performance of the random quantum circuits. As another example, the estimated fidelity can be used to (i) calibrate, (ii) validate the effectiveness of, or (iii) benchmark the quantum computing hardware implementing the quantum circuits.

[0132] In some implementations, the system can use the determined fidelity estimates and / or additional information about the random quantum circuits to determine one or more adjustments to the quantum computing hardware. For example, the system can determine adjustments to the control parameters of the control model used by the quantum computing hardware to implement quantum operations to improve the fidelity of the quantum operations. The system can use the adjusted control model to implement quantum operations with increased fidelity in future quantum computations.

[0133] Figure 4 is a flowchart of an example process 400 for obtaining additional information about a quantum circuit. For example, the example process 400 can be applied in combination with the above-described processes 200 and 300. For convenience, process 400 will be described as being performed by a system of one or more classical computing devices located at one or more locations. For example, a system 100 appropriately programmed in accordance with this specification Figure 1 can perform process 400.

[0134] The system computes the distribution of the random variable associated with the observable selected in step 204 of example process 200 or step 304 of example process 300 (step 402).

[0135] The system performs a statistical test using the calculated distribution of the random variable associated with the selected observable to obtain additional statistical information about the quantum circuit (step 404). For example, the system can perform a Kolmogorov-Smirnov test using experimental data (e.g., data obtained based on Equation (6)) to confirm the validity of the fidelity estimated using Process 200 or Process 300, to reject the null hypothesis that the fidelity estimate is 0. As another example, given an estimate of the fidelity α, the system can determine the Kolmogorov-Smirnov p-value of the cumulative distribution function of the experimental data. A large Kolmogorov-Smirnov p-value indicates that the assumptions about the model and the estimate of α are correct.

[0136] For example, one example observable selected at step 204 of Example Process 200 or step 304 of Example Process 300 is the cross-entropy benchmark observable given by the following Equation (15).

[0137]

[0138] In Equation (15), N = 2 n (where n represents the number of qubits on which the quantum circuit operates), p U (z) represents the probability that the circuit U outputs a particular bitstring z, and |z> represents the output state corresponding to the particular bitstring z.

[0139] For a noisy implementation of a given quantum circuit U, where when sampling the random circuit, the distribution of the random variable Y(z) = log N p U (z) is Thus, the random variable Y on the noisy random circuit has a probability density The expected value of this random variable is α - γ (where γ represents Euler's constant), which gives an estimator of the fidelity α where, represents the expected value of the observable in Equation (15). The estimator has a variance of π 2 / 6 - α 2 , and the estimator with M measurements has a standard deviation of

[0140] As another example, one example observable that the system may select at step 204 of example process 200 or step 304 of example process 300 is the linear cross-entropy observable given by the following equation (16).

[0141]

[0142] The random variable X(z) = Np U (z) has a distribution and when sampling on a noisy random circuit, the distribution of X(z) is Therefore, the random variable X on the noisy random circuit has a probability density (αx + 1 - α)e -x . The expected value of this random variable is 1 + α, which gives an estimator of α where denotes the expected value of the observable in equation (16). The variance of the random variable is 1 + 2α + α 2 , which gives the standard deviation of the estimator with M measurements That is, an improvement in the standard deviation obtained using the cross-entropy benchmark observable.

[0143] As another example, one example observable that the system may select at step 204 of example process 200 or step 304 of example process 300 is the re-output generation fraction observable given by the following equation (17).

[0144]

[0145] The expected value of this observable for the fidelity α output is The corresponding estimator is where denotes the expected value of the observable in equation (17), and the variance is (log2) -2 -α 2 .

[0146] As described in step 208 of process 200 and step 308 of process 300, additional statistical information about the quantum circuit obtained via statistical tests can also be used to determine the properties of the quantum circuit.

[0147] The digital and / or quantum topics described in this specification, as well as the implementation of digital functional operations and quantum operations, may be implemented in digital electronic circuits, suitable quantum circuits, or more generally in a quantum computing system, in tangibly implemented digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware (including the structures disclosed in this specification and their structural equivalents), or in a combination of one or more of these. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0148] The implementation of the digital and / or quantum topics described in this specification may be implemented as one or more digital and / or quantum computer programs (i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium) for execution by, or to control the operation of, a data processing apparatus. The digital and / or quantum computer storage medium may be a computer-readable storage device, a computer-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of these. Alternatively or additionally, the program instructions may be encoded on an artificially generated propagated signal (such as a machine-generated electrical, optical, or electromagnetic signal) that is capable of encoding digital and / or quantum information, generating the propagated signal to encode digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing apparatus.

[0149] The terms quantum information and quantum data refer to information or data carried, held, or stored in a quantum system, where the smallest non-trivial system is a qubit, i.e., the system that defines the unit of quantum information. It should be understood that the term "qubit" encompasses all quantum systems that can be suitably approximated as two-level systems in the corresponding context. Such quantum systems may include, for example, multi-level systems having two or more levels. For example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In various implementations, the computational basis states are identified with the ground state and the first excited state, but it should be understood that other settings where the computational states are identified with higher excited states are possible.

[0150] The term "data processing device" refers to digital and / or quantum data processing hardware and encompasses all types of devices, equipment, and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The device may also be or further include dedicated logic circuitry, such as FPGAs (field-programmable gate arrays), ASICs (application-specific integrated circuits), or quantum simulators, i.e., quantum data processing devices designed to simulate or generate information about a particular quantum system. Specifically, a quantum simulator is a dedicated quantum computer that does not have the ability to perform universal quantum computing. In addition to the hardware, the device may optionally include code that creates an execution environment for digital and / or quantum computer programs, such as code that constitutes processor firmware, protocol stacks, database management systems, operating systems, or a combination of one or more thereof.

[0151] Digital computer programs (which may also be referred to as or described as programs, software, software applications, modules, software modules, scripts, or code) can be written in any form of programming language (including compiled or interpreted languages or declarative or procedural languages) and can be deployed in any form (including as a stand-alone program or as a module, component, subroutine, or other unit suitable for a digital computing environment). Quantum computer programs (which may also be referred to as or described as programs, software, software applications, modules, software modules, scripts, or code) can be written in any form of programming language (including compiled or interpreted languages or declarative or procedural languages) and be translated into a suitable quantum programming language or can be written in a quantum programming language (e.g., QCL or Quipper).

[0152] Digital and / or quantum computer programs may or may not correspond to files in a file system. The program can be stored in a portion of a file that holds other programs or data, such as in a markup language document, in a single file dedicated to the program in question, or in one or more scripts in multiple coordinated files (e.g., files that store one or more modules, subroutines, or portions of code). Digital and / or quantum computer programs can be deployed to execute on one digital or one quantum computer or on multiple digital and / or quantum computers, which are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can use quantum systems (e.g., qubits) to transmit quantum data. Generally, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0153] The processes and logical flows described in this specification can be performed by one or more programmable digital and / or quantum computers that operate in conjunction with one or more digital and / or quantum processors as needed to execute one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logical flows can also be performed by, or by a combination of, special purpose logic circuitry, such as an FPGA or ASIC, or a quantum simulator, and the apparatus can also be implemented as, or as a combination of, special purpose logic circuitry, such as an FPGA or ASIC, or a quantum simulator and one or more programmed digital and / or quantum computers.

[0154] For a system of one or more digital and / or quantum computers to be “configured to” perform a particular operation or action means that the system has installed thereon software, firmware, hardware, or a combination thereof that, when operated, causes the system to perform the operation or action. For one or more digital and / or quantum computer programs to be configured to perform a particular operation or action means that the one or more programs include instructions that, when executed by a digital and / or quantum data processing apparatus, cause the apparatus to perform the operation or action. A quantum computer can receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operation or action.

[0155] Digital and / or quantum computers suitable for executing digital and / or quantum computer programs can be based on general purpose or special purpose digital and / or quantum processors, or both, or on any other kind of central digital and / or quantum processing unit. In general, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from read only memory, random access memory, or a quantum system (such as a photon) suitable for transmitting quantum data, or a combination thereof.

[0156] The basic elements of a digital and / or quantum computer are a central processing unit for executing or running instructions and one or more memory devices for storing the instructions and digital and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or a quantum simulator. In general, a digital and / or quantum computer will also include one or more mass storage devices (such as magnetic disks, magneto - optical disks, optical disks, or a quantum system suitable for storing quantum information) for storing digital and / or quantum data, or operatively coupled to one or more mass storage devices to receive digital and / or quantum data therefrom, or to send digital and / or quantum data thereto. However, a digital and / or quantum computer need not have such devices.

[0157] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memories, media, and memory devices, including, for example, semiconductor memory devices such as EPROMs, EEPROMs, and flash memory devices; magnetic disks such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems such as trapped atoms or electrons. It will be understood that a quantum memory is a device capable of storing quantum data for long periods of time with high fidelity and high efficiency, such as an optical-matter interface in which light is used for transmission and matter is used to store and preserve quantum features of the quantum data, such as superposition or quantum coherence.

[0158] Control of various systems or portions thereof described in this specification can be implemented in a digital and / or quantum computer program product including instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. Each of the systems or portions thereof described in this specification can be implemented as an apparatus, method, or system that can include one or more digital and / or quantum processing devices and memory for storing executable instructions for performing the operations described in this specification.

[0159] Although this specification includes many specific implementation details, these details should not be construed as limitations on the scope that can be claimed, but rather as descriptions of features specific to particular implementations. Certain features described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation can also be implemented separately in multiple implementations or in any suitable sub-combination. Additionally, although the features are described above as acting in certain combinations and even initially claimed as such, one or more features from the claimed combination can in some cases be deleted from the combination, and the claimed combination can be directed to a sub-combination or variation of the sub-combination.

[0160] Similarly, although operations are depicted in the figures in a particular order, this should not be understood as requiring that the operations be performed in the particular order shown or sequentially, nor that all of the shown operations be performed to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. Additionally, the separation of various system modules and components in the above implementations should not be understood as required in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0161] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the acts recited in the claims can be performed in a different order and still achieve the desired result. As one example, the processes depicted in the figures do not necessarily require the particular order or sequential order shown to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for estimating the fidelity of an n - qubit quantum logic gate, the method comprising: defining a plurality of sets of random quantum circuits, wherein the plurality of sets of random quantum circuits correspond to different respective depths and each set of the plurality of sets of random quantum circuits includes random quantum circuits having the same circuit depth d, and wherein defining the plurality of sets of random quantum circuits includes, for each set of random quantum circuits: defining one or more elements of the set of random quantum circuits, which includes: for each element, randomly sampling d*n single - qubit gates from a predefined set of single - qubit gates, wherein each single - qubit gate operates on a corresponding qubit for a corresponding period; and defining the element of the set of random quantum circuits to be equal to n randomly sampled single - qubit gates for d periods followed by an n - qubit quantum logic gate; for each set of random quantum circuits: selecting observables for each element of the set of random quantum circuits, wherein each selected observable corresponds to a corresponding element of the set of random quantum circuits and depends on the element to which it corresponds; and estimating the value of the polarization parameter of the set of random quantum circuits, which includes performing least - mean - square minimization based on a plurality of expected values, wherein each expected value includes the expected value of a corresponding selected observable with respect to the output of an experimental realization of the random quantum circuit corresponding to the corresponding selected observable; and processing the estimated polarization parameter value to obtain an estimate of the fidelity of the n - qubit quantum logic gate, wherein performing the least - mean - square minimization includes: minimizing the sum of squared residuals, each squared residual corresponding to a corresponding element of the set of random quantum circuits, and each squared residual including the square of the following: i) the trace of the corresponding selected observable divided by the dimension of the Hilbert space, plus ii) the value of the polarization parameter of the set of random quantum circuits multiplied by the difference between the expected value of the corresponding selected observable with respect to the ideal output state of the random quantum circuit and the trace of the corresponding selected observable divided by the dimension of the Hilbert space, minus iii) the expected value of the corresponding selected observable with respect to the output of the experimental realization of the random quantum circuit corresponding to the corresponding selected observable, and wherein processing the estimated polarization parameter value to obtain an estimate of the fidelity of the n - qubit quantum logic gate includes: fitting the estimated polarization parameter values corresponding to each circuit depth d to an exponential decay of d; and Determining an estimated per-cycle polarization p of an n-qubit quantum logic gate based on an exponential decay of d and an estimated polarization of single-qubit gates in a quantum circuit that operates on n qubits n ; and Use F = p n +(1 - p n ) / D to determine an estimate of the fidelity of an n-qubit quantum logic gate, where D = 2 n represents the Hilbert space dimension.

2. The method according to claim 1, wherein estimating the value of the polarization parameter of the set of random quantum circuits further includes determining a plurality of expected values, which includes: for each expected value, defining the expected value of the corresponding selected observable with respect to the output of the experimental realization of the random quantum circuit corresponding to the corresponding selected observable as i) the trace of the corresponding selected observable divided by the dimension of the Hilbert space, plus ii) the polarization of the random quantum circuit multiplied by the difference between the expected value of the corresponding selected observable with respect to the ideal output state of the random quantum circuit and the trace of the corresponding selected observable divided by the dimension of the Hilbert space; Numerically estimate (i) the value of the expected value of the corresponding selected observable relative to the ideal output state of the random quantum circuit, and (ii) the value of the trace of the corresponding selected observable divided by the dimension of the Hilbert space; and Experimentally estimate the value of the expected value of the corresponding selected observable relative to the output of the experimental implementation of the random quantum circuit.

3. The method according to claim 2, wherein, Experimentally estimating the value of the expected value of the corresponding selected observable relative to the output of the experimental implementation of the random quantum circuit includes: Repeatedly: Prepare the quantum system in an initial state; Apply the random quantum circuit to the quantum system prepared in the initial state to generate an evolved state of the quantum system; and Measure the corresponding selected observable relative to the evolved state of the quantum system; and Average the measured selected observables to estimate the value of the expected value of the corresponding selected observable relative to the output of the experimental implementation of the random quantum circuit.

4. The method according to claim 1, wherein, The n-qubit quantum logic gate operates on at most 5 qubits.

5. The method according to claim 1, wherein The selected observable is diagonal in the computational basis.

6. The method according to claim 1, wherein The selected observables include cross-entropy benchmark observables, linear cross-entropy observables, or re-output generation score observables.

7. The method according to claim 1, further comprising using an estimate of the fidelity of the n-qubit quantum logic gate to determine one or more properties of the quantum hardware implementing the n-qubit quantum logic gate.

8. The method according to claim 1, further comprising: Determine one or more adjustments to the quantum hardware control parameters based on the determined estimate of the fidelity; And Use the quantum computing hardware to implement the determined one or more adjustments to perform quantum computing.

9. The method according to claim 1, further comprising: Calculate the distribution of the random variable associated with the selected observable; And Use the calculated distribution of the random variable to perform one or more statistical tests to obtain additional information about the n-qubit quantum logic gate.

10. The method according to claim 9, wherein, Using the calculated distribution of the random variable to perform one or more statistical tests to obtain additional information about the n-qubit quantum logic gate includes: performing a Kolmogorov-Smirnov test to confirm the validity of the estimated fidelity of the n-qubit quantum logic gate.

11. An apparatus for estimating the fidelity of an n-qubit quantum logic gate, comprising: One or more classical processors; And Quantum computing hardware in data communication with the one or more classical processors; Wherein the apparatus is configured to perform the method according to any one of claims 1 to 10.

12. A method for estimating the fidelity of a quantum circuit, the method comprising: Define one or more random quantum circuits, wherein each of the one or more random quantum circuits has the same circuit depth d and operates on the same number of qubits n, wherein defining a random quantum circuit includes: Randomly sample d*n single-qubit gates from a predefined set of single-qubit gates, wherein each single-qubit gate operates on a corresponding qubit; and Define a random quantum circuit as equal to n randomly sampled single-qubit gates for d cycles followed by multiple multi-qubit quantum logic gates operating on different qubits; For each defined random quantum circuit, select an observable of the random quantum circuit, where i) the selected observable depends on the random quantum circuit, and ii) the selected observable is subject to a measure concentration related to the expected value of the selected observable with respect to the expected value of the density matrix representing the error effects in the experimental implementation of the random quantum circuit; Determine an estimate of the fidelity of the random quantum circuit for each defined random quantum circuit, including: estimating the expected value of the corresponding selected observable with respect to the output of the experimental implementation of the random quantum circuit for each random quantum circuit, including: Express the expected value of the selected observable with respect to the output of the experimental implementation of the random quantum circuit as: the trace of the selected observable divided by the Hilbert space dimension, plus the fidelity of the quantum circuit multiplied by the sum of i) the difference between the expected value of the selected observable with respect to the ideal output state of the quantum circuit and the trace of the selected observable divided by the Hilbert space dimension; Experimentally estimate the value of the expected value of the selected observable with respect to the output of the experimental implementation of the random quantum circuit; and Numerically or analytically estimate i) the value of the expected value of the selected observable with respect to the ideal output state of the random quantum circuit, and ii) the value of the trace of the corresponding selected observable; and Determine the fidelity of the quantum circuit by averaging the determined estimates of the fidelity for each random quantum circuit to obtain an estimate of the circuit fidelity for circuits with circuit depth d and number of qubits n.

13. The method according to claim 12, wherein, Experimentally estimating the value of the expected value of the selected observable with respect to the output of the experimental implementation of the random quantum circuit includes: Repeatedly: Prepare the quantum system in an initial state; Apply the random quantum circuit to the quantum system prepared in the initial state to generate an evolved state of the quantum system; and Measure the corresponding selected observable with respect to the evolved state of the quantum system; and Average the measured selected observables to estimate the value of the expected value of the selected observable with respect to the output of the experimental implementation of the random quantum circuit.

14. The method according to claim 13, wherein, The method includes analytically estimating i) the value of the expected value of the selected observable with respect to the ideal output state of the random quantum circuit, and ii) the value of the trace of the corresponding selected observable, and wherein the output of the experimental implementation of the random quantum circuit is approximated by a Porter-Thomas distribution.

15. The method according to claim 12, wherein, The quantum circuit operates on 10 or more qubits.

16. The method according to claim 12, wherein, The selected observable is diagonal in the computational basis.

17. The method according to claim 12, wherein, The selected observables include cross-entropy benchmark observables, linear cross-entropy observables, or re-output generation fraction observables.

18. The method according to claim 12, further comprising using the determined estimate of the fidelity or the average estimate of the circuit fidelity to determine one or more properties of the quantum hardware implementing the quantum circuit.

19. The method according to claim 12, further comprising: Determine one or more adjustments to the quantum hardware control parameters based on the determined estimate of the fidelity; And Use quantum computing hardware to implement the determined one or more adjustments to perform quantum computing.

20. The method according to claim 12, further comprising: Calculating the distribution of a random variable associated with a selected observable; And Using the calculated distribution of the random variable to perform one or more statistical tests to obtain additional information about the quantum circuit.

21. The method according to claim 20, wherein, Using the calculated distribution of the random variable to perform one or more statistical tests to obtain additional information about the quantum circuit includes: performing a Kolmogorov-Smirnov test to confirm the validity of the estimated fidelity of the quantum circuit.

22. The method according to any one of claims 12-21, wherein determining an estimate of the fidelity of the random quantum circuit for each defined random quantum circuit further comprises: Estimating the fidelity of the random quantum circuit by solving for the representation of the expected value of the output of the selected observable with respect to the experimental implementation of the random quantum circuit using the experimentally estimated value of the expected value of the output of the selected observable with respect to the experimental implementation of the random quantum circuit and the estimated values of i) the expected value of the selected observable with respect to the ideal output state of the random quantum circuit, and ii) the trace value of the corresponding selected observable.

23. The method according to any one of claims 12-21, further comprising verifying that the error effects in the experimental implementation of the random quantum circuit obey the metric concentration.

24. An apparatus for estimating the fidelity of a quantum circuit, comprising: One or more classical processors; And Quantum computing hardware in data communication with the one or more classical processors; Wherein the apparatus is configured to perform the method according to any one of claims 12 to 23.

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