Mitigation of qubit errors via subtracting correlated signals

Abstract

A method for operating a quantum computing system is disclosed. The method includes executing a set of quantum circuits to generate a set of noisy signals. A set of extrapolation datasets is generated. Each extrapolation dataset includes a set of additional errors in a corresponding quantum circuit. A value for each expansion coefficient of a set of expansion coefficients is determined. The set of expansion coefficients parameterizes a linear combination of the set of extrapolation datasets. An approximation of a noiseless measurement of the observable is determined based on the set of noisy signals, the set of extrapolation datasets, and the set of determined values.

Description

[0001] MITIGATION OF QUBIT ERRORS VIA SUBTRACTING CORRELATED SIGNALS

[0002] PRIORITY

[0003] [1] This application claims priority to the U. S. Provisional Application No 63 / 714,674 entitled MITIGATION OF QUBIT ERRORS VIA SUBTRACTING CORRELATED SIGNALS, filed on October 31, 2024, the contents of which are herein incorporated in their entirety.

[0004] FIELD

[0005] [2] The present disclosure relates generally to quantum computing and information processing systems, and more particularly to mitigation of qubit errors via subtracting correlated signals.

[0006] BACKGROUND

[0007] [3] Quantum computing is a computing method that takes advantage of quantum effects, such as superposition of basis states and entanglement to perform certain computations more efficiently than a classical digital computer. In contrast to a digital computer, which stores and manipulates information in the form of bits, e.g., a “1” or “0,” quantum computing systems can manipulate information using quantum bits (“qubits”). A qubit can refer to a quantum device that enables the superposition of multiple states, e.g., data in both the “0” and “1” state, and / or to the superposition of data, itself, in the multiple states. In accordance with conventional terminology, the superposition of a “0” and “1” state in a quantum system may be represented, e.g., as a |0) + b 11). The “0” and “1” states of a digital computer are analogous to the |0) and 11) basis states, respectively of a qubit.

[0008] SUMMARY

[0009] [4] Aspects and advantages of embodiments of the present disclosure will be set forth in part in the following description, or can be learned from the description, or can be learned through practice of the embodiments.

[0010] [5] One non-limiting embodiment includes a method for operating a quantum computing system (QCS). The method includes executing each quantum circuit of a set of quantum circuits to generate a set of noisy signals. Executing each quantum circuit of the set of quantum circuits generates a separate and corresponding noisy signal of the set of noisy signals. Each noisy signal of the set of noisy signals corresponds to a separate noisy measurement of an observable associated with the set of quantum circuits. A set of extrapolation datasets is generated. Each extrapolation dataset of the set of extrapolation datasets corresponds to a separate quantum circuit of the set of quantum circuits and includes a set of additional errors in a corresponding quantum circuit of the set of quantum circuits. A value for each expansion coefficient of a set of expansion coefficients is determined. A set of determined values for the set of expansion coefficients is generated by determining the value for each expansion coefficient of the set of expansion coefficients. The set of expansion coefficients parameterizes a linear combination of each extrapolation dataset of the set of extrapolation datasets. An approximation of a noiseless measurement of the observable is determined based on the set of noisy signals, the set of extrapolation datasets, and the set of determined values.

[0011] [6] These and other features, aspects, and advantages of various embodiments of the present disclosure will become better understood with reference to the following description and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate example embodiments of the present disclosure and, together with the description, explain the related principles.

[0012] BRIEF DESCRIPTION OF THE DRAWINGS

[0013] [7] Detailed discussion of embodiments directed to one of ordinary skill in the art is set forth in the specification, which refers to the appended figures, in which:

[0014] [8] FIG. 1 depicts an example quantum computing system, according to various embodiments.

[0015] [9] FIG. 2 shows a flowchart for a method for operating a quantum computing system, according to various embodiments.

[0016]

[0010] FIG. 3 shows a flowchart for another method for operating a quantum computing system, according to various embodiments.

[0017]

[0011] FIG. 4 shows a flowchart for still another method for operating a quantum computing system, according to various embodiments. DETAILED DESCRIPTION

[0018]

[0012] The embodiments disclosed herein provide methods and systems for mitigating quantum errors in the estimation of observable expectation values from a set of quantum circuits executed on a noisy quantum computing system (QCS). The quantum circuits in the set (or ensemble) of quantum circuits may be directed towards a quantum algorithm or quantum computation. The quantum algorithms may be directed towards measuring and / or determining the observable expectation values.

[0019]

[0013] Quantum computing holds the promise of solving certain computational problems that are intractable with classical computers. This potential is realized through the execution of quantum algorithms, which are implemented as sequences of quantum gate operations, or quantum circuits, on a set of quantum bits (qubits). A central task in many quantum algorithms is the estimation of an expectation value of a physical observable. For an ensemble of quantum circuits, denoted by

[0020]

[0021] where 0 is a parameter indexing the circuits, the goal is often to estimate the ideal (or noiseless) expectation value

[0022]

[0023] for agivenobservable O.

[0024]

[0014] A challenge in the practical realization of quantum computation is the inherent susceptibility of quantum processors to noise. Physical qubits are delicate systems that interact with their environment, leading to phenomena such as decoherence and relaxation Furthermore, the control hardware used to implement quantum gates is imperfect, resulting in gate operations that deviate from their ideal (or noiseless) unitary descriptions Measurement apparatuses are also prone to errors, further corrupting the final output. Consequently, the signal measured from a physical quantum computer may not be the ideal (or noiseless) value

[0025]

[0026] but rather a noisy, corrupted signal At This discrepancy between the ideal (or noiseless) and measured values represents a significant obstacle to achieving high-fidelity quantum computation and realizing a quantum advantage. The accumulation of these errors, especially in deep and complex circuits, can render the output of the computation essentially random and useless.

[0027]

[0015] Quantum computing holds the promise of solving certain computational problems that are intractable with classical computers. This potential is realized through the execution of quantum algorithms, which are implemented as sequences of quantum gate operations, or quantum circuits, on a set of quantum bits (qubits), A central task in many quantum algorithms is the estimation of an expectation value of a physical observable. For an ensemble of quantum circuits, denoted by where (9 is a parameter indexing the circuits, the goal is often to estimate the ideal, noiseless expectation values

[0028]

[0029] for a given observable O.

[0030]

[0016] The embodiments include methods for error mitigation. In contrast to quantum error correction (QEC) codes, which aim to perfectly fix errors as they happen, error mitigation uses the noisy results from the QCS and processes them to produce a more accurate estimate of the ideal, noiseless value. The error mitigating embodiments work for a set (or ensemble) of quantum circuits. As used herein, a set (or ensemble) of quantum circuits may include a collection of many different quantum circuits. In some embodiments, a set (or ensemble) of quantum circuits may include a plurality of quantum circuits.

[0031]

[0017] The embodiments are discussed using a framework referred to as the k-error resolution formalism. This formalism provides techniques for mathematically breaking down the noisy output of a quantum computer.

[0032]

[0018] One concept that may be used in this formalism is that the final noisy signal may be modeled as a sum of different possible outcomes, each corresponding to a specific number of errors having occurred during the computation. The formalism expresses the noisy signal as a weighted average:

[0033]

[0034] where k represents the number of discrete error events that occurred, k = 0 means no errors occurred, k = 1 means one error occurred, and so on.

[0035]

[0036] is the "k-error component," representing the average result of the computation given that exactly k errors happened. The ideal (or noiseless) signal that the embodiments determine is therefore f̃₀,₀,

[0037]

[0038] the probability that a computation will finish with exactly k errors. This formalism provides a set of powerful tools to analyze the noisy output and forms the basis for the error mitigation embodiments described below.

[0039]

[0019] The embodiments include at least three different families of embodiments. One family of embodiments is directed towards error mitigation with extrapolation datasets. A second family of embodiments is directed towards error mitigation by classical signal subtraction. A third family of embodiments is directed towards hybrid quantum-classical error mitigation.

[0040]

[0020] The error mitigation with extrapolation dataset embodiments may be use extrapolation, wherein measurements obtained at multiple, known noise levels may be used to determine an estimated result at zero noise. In these embodiments, additional noise is intentionally and controllably introduced to the quantum circuits to generate additional data points, which serve as the basis for the extrapolation. These extrapolation dataset embodiments may include at least three operations: (1) generating an extrapolation dataset, (2) forming a linear combination of the noisy data and the extrapolation dataset, and (3) optimizing the linear combination coefficients.

[0041]

[0021] The generating the extrapolation dataset step includes operating the quantum circuits and collecting the noisy output and several additional datasets, called extrapolation datasets. These may be generated by running the same circuits but with more errors deliberately introduced.

[0042]

[0022] The embodiments include at least four families of techniques for generating the extrapolation datasets: (a) increasing the error rate, (b) inserting hard errors, (c) using locationspecific error rates, and (d) using related circuits. Increasing the error rate techniques include amplifying the inherent noise of the quantum device Inserting hard errors techniques include randomly injecting a fixed number of additional errors into the circuit operations. Using location-specific error rates techniques includes applying different amounts of artificial noise to different parts of the circuit Using related circuits techniques include running circuits that are structurally similar but not identical to the original ones. An extrapolation dataset may be referred t

[0043]

[0023] The forming the linear combination step includes combining the original noisy data and the extrapolation datasets to create a better estimate of the noiseless signal. This may be done by taking a weighted sum (a linear combination) of the collected data:

[0044]

[0045]

[0024] The next step includes determining an optimized set of weights (or coefficients) U such that this approximation is close to the true value.

[0046]

[0025] The optimizing the coefficients step includes determining an optimal (or at least somewhat optimal) set of expansion coefficients. Optimizing the set of expansion coefficients may include minimizing (or at least decreasing) the Mean Square Error (MSE) across the ensemble (or set) of circuits. The MSE measures the average difference between the estimated value and the true, noiseless value. By using the k-error resolution formalism, the paper provides a direct mathematical solution for the optimal coefficients that minimize this error: c — v'4This equation allows for the calculation of the at least somewhat optimized weights to use in the linear combination, providing an optimized, error-mitigated result.

[0047]

[0026] The family of extrapolation datasets includes a non-limiting method for operating a quantum computing system (QCS). The method includes executing each quantum circuit of a set of quantum circuits to generate a set of noisy signals. Executing each quantum circuit of the set of quantum circuits generates a separate and corresponding noisy signal of the set of noisy signals. Each noisy signal of the set of noisy signals corresponds to a separate noisy measurement of an observable associated with the set of quantum circuits. A set of extrapolation datasets is generated. Each extrapolation dataset of the set of extrapolation datasets corresponds to a separate quantum circuit of the set of quantum circuits and includes a set of additional errors in a corresponding quantum circuit of the set of quantum circuits. A value for each expansion coefficient of a set of expansion coefficients is determined. A set of determined values for the set of expansion coefficients is generated by determining the value for each expansion coefficient of the set of expansion coefficients. The set of expansion coefficients parameterizes a linear combination of each extrapolation dataset of the set of extrapolation datasets. An approximation of a noiseless measurement of the observable is determined based on the set of noisy signals, the set of extrapolation datasets, and the set of determined values.

[0048]

[0027] Another method of the extrapolation datasets family of embodiments includes executing, on a QCS, a first set of quantum circuits corresponding to an ensemble (or set) of circuits to obtain a first set of measurement data. The method may further involve executing one or more additional sets (or ensembles) of quantum circuits on the QCS to obtain one or more additional sets of measurement data, where each additional set of quantum circuits is a modified version of the first set. The modifications may involve, for example, increasing a native error rate of the circuits, inserting a predetermined number of additional quantum gates at random locations, or executing related but non-equivalent circuits. These additional sets of data are referred to as extrapolation datasets.

[0049]

[0028] The method may then proceed with post-processing. In some embodiments, the post processing may include classical post-processing. A set of expansion coefficients may be determined for forming a linear combination of the first set of measurement data and the one or more additional sets of measurement data. This determination can be based on minimizing an error metric, such as the mean square error (MSE), evaluated over the ensemble of circuits. An improved estimate of the ideal, noiseless observable value is then calculated by forming said linear combination using the determined coefficients

[0050]

[0029] The determination of the coefficients may be informed by a novel formalism for modeling noise, referred to as the k-error resolution formalism. This formalism expands the noisy quantum evolution into a sum of components, each corresponding to a specific number, k. of discrete error events occurring during the circuit execution. The optimization of the coefficients can then be performed using autocorrelation matrices of these k-error components, which capture the statistical properties of the errors across the circuit ensemble.

[0051]

[0030] The error mitigation by classical signal subtraction embodiments decomposes the total signal into a “quantum” part (e.g., a first component) and a “classical” part (e.g., a second component). These embodiments propose that the total signal can be split into two parts: a "quantum" part that is difficult to simulate on a classical computer and a "classical" part that may be calculated or measured.

[0052]

[0053] where Q& is the "quantum" signal, which captures the complex quantum correlations and C is the "classical" signal.

[0054]

[0031] The goal i s then redefined: instead of trying to measure the full

[0055]

[0056] the experiment focuses on accurately determining the quantum signal Qe. The error mitigation by classical signal subtraction embodiments may include operations: (1) obtaining the classical signal, (2) decomposing the noisy signal, and (3) subtracting and rescaling.

[0057]

[0032] In obtaining the classical signal step, the classical part, Cf, can be estimated using a classical computer or by running a specially designed experiment on a quantum computer that measures this "classical" behavior.

[0058]

[0033] These classical signal subtraction embodiments assume that the total noisy signal measured from the quantum computer can also be decomposed. In the decomposing the noisy signal step, the total noisy signal may be modeled as a decayed version of the quantum signal plus a noisy version of the classical signal:

[0059] St) • XQo C,<

[0060] where A is a decay factor that represents how much the quantum signal is weakened by noise.

[0061]

[0034] By running experiments to measure both 9and Ctf, these embodiments can subtract the noisy classical signal from the total noisy signal. This isolates the decayed quantum part, ^-Qo. This is performed in the subtracting and rescaling step of the classical signal subtraction embodiments. This result is then rescaled by dividing by the decay factor to get an estimate of the true quantum signal, Qo. The final, full result is reconstructed by adding back the known classical part C&.

[0062]

[0035] Methods of classical signal subtraction involve redefining the target observable. The ideal (or true) observable, », may be decomposed into a "quantum" signal, Q& and a "classical" signal, ■'&. The classical signal may be chosen such that it can be efficiently estimated, either through classical simulation or by executing a specific type of noisy quantum circuit on the QCS The experimental goal is redefined to be the estimation of the quantum signal Qu. A noisy measurement of the total signal, Sf), is performed, and a noisy estimate of the classical signal, C, is obtained. The difference between these two is calculated, which isolates a decayed version of the quantum signal. This isolated signal can then be rescaled to provide an improved estimate of Qo.

[0063]

[0036] The family of classical signal subtraction embodiments includes a non-limiting method for operating a quantum computing system (QCS). The method includes accessing a classical signal. In this method, a noiseless signal of a quantum circuit is decomposable into a first component and a second component. The first component corresponds to a noiseless quantum signal. The second component corresponds to the classical signal. The noiseless signal of the quantum circuit corresponds to an observable of the quantum circuit. A first version of the quantum circuit is executed to generate a first noisy measurement of the observable. An approximation of the noiseless quantum signal is determined based on the first noisy measurement of the observable. An approximation of the observable is determined based on the approximation of the noiseless quantum signal and the classical signal.

[0064]

[0037] The error mitigation by hybrid quantum -class! cal embodiments combine quantum and classical computation. These embodiments may be useful when a fast but approximate ("heuristic") classical method exists for estimating the desired observable. The hybrid embodiments may include operations: (1) performing a heuristic classical simulation, (2) running quantum experiments (2), and (3) constructing a hybrid estimator,

[0065]

[0038] In the performing a heuristic classical simulation step, a classical algorithm is used to produce a rough estimate of the answer, denoted This estimate is computationally cheap but has an "uncontrolled" error, meaning its accuracy may not be systematically improved.

[0066]

[0039] In the running quantum experiments step, an actual quantum circuit is run on the noisy QCS to get the experimental result So. In addition, another quantum experiment is performed that is designed to mimic the classical heuristic simulation, yielding a noisy version of that classical result

[0067]

[0040] In the constructing a hybrid estimator step, three pieces of data — the classical heuristic the noisy quantum result ^0, and the noisy quantum-mimic of the classical result '-’0 — are combined to produce a "hybrid" estimate. One possible way to combine them is:

[0068]

[0069]

[0041] The QCS may be used to calculate a difference between the true noisy signal and the noisy version of the classical simulation. This difference may represent the "quantum" correction that the heuristic classical method missed. This correction may then be properly scaled and added to the original classical estimate to produce a final result that is more accurate than either the classical simulation or the raw quantum experiment alone.

[0070]

[0042] In these hybrid embodiments, a heuristic classical algorithm, such as a Monte Carlo or mean-field simulation, is used to produce a classical estimate,

[0071]

[0072] , ot the ideal observable. A noisy quantum computation may be performed to obtain a noisy estimate, *-•'?. These two pieces of

[0073] of information may then be combined to form a hybrid estimator, ‘"o, that has a lower mean squared error (MSE) than either the classical estimate or the noisy quantum estimate alone. This combination can be achieved, for example, by rescaling the classical estimate based on the ratio of the noisy quantum value to a noisy quantum simulation of the classical algorithm, or by treating the classical estimate as a baseline and adding a corrected quantum-derived difference term.

[0074]

[0043] The family of hybrid embodiments includes a non-limiting method for operating a quantum computing system (QCS). The method includes accessing a heuristic classical estimation of an observable of a quantum circuit. A first version of the quantum circuit is executed to generate a first noisy signal that corresponds to a first noisy measurement of the observable. A hybrid estimate of the observable is determined based on a combination of the heuristic classical estimation and the first noisy signal.

[0075]

[0044] The error mitigation schemes described herein are designed for an entire ensemble of circuits, rather than for a single, isolated circuit. Some methods generally treat each circuit instance independently. In contrast, the disclosed embodiments leverage the statistical properties of the ensemble as a whole. By defining a cost function, such as the mean square error (MSE), across the entire ensemble, the mitigation strategy can be optimized for the characteristics of the hardware noise and for the specific computational problem being solved, as represented by the circuit ensemble. This ensemble-centric approach allows for a more holistic and powerful form of error mitigation. It enables the use of correlations between different error contributions across the ensemble to find an optimal correction, a capability that may not be present in certain circuit-by-circuit correction schemes.

[0076]

[0045] The embodiments described herein provide several advantages over other approaches. By optimizing the mitigation strategy over an ensemble of circuits, the embodiments can achieve a lower overall error in the estimated observables for the given computational task. The methods can be more resource-efficient, in some cases offering more favorable scaling with circuit size and fidelity compared to techniques like probabilistic error correction (PEC).

[0077]

[0046] Furthermore, the embodiments introduce a flexible and extensible framework. This framework allows for the generation of various types of auxiliary data, referred to as extrapolation datasets, by controllably manipulating the noise in the quantum computer. An optimal linear combination of these datasets can then be formed to extrapolate to the ideal, noiseless result. This process is guided by minimizing the ensemble-wide MSE, which can be optimized using knowledge of error correlations within the ensemble. Other embodiments redefine the computational goal by subtracting a classically tractable portion of the signal, thereby simplifying the error mitigation task to correcting at least the non-trivial "quantum" part of the signal. Still other embodiments create a hybrid estimator that synergistically combines data from a noisy quantum processor and a heuristic classical simulation to produce a result more accurate than either component alone. These approaches provide a versatile toolkit for improving the performance of near-term and future quantum computers.

[0078] Quantum Computing Systems

[0079]

[0047] FIG. 1 depicts an example quantum computing system 100. The system 100 is an example of a system of one or more classical computers and / or quantum computing devices in one or more locations, in which the systems, components, and techniques described below can be implemented. Those of ordinary skill in the art, using the disclosures provided herein, will understand that other quantum computing devices or systems can be used without deviating from the scope of the present disclosure.

[0080]

[0048] The system 100 includes quantum hardware 102 in data communication with one or more classical processors 104. The classical processors 104 can be configured to execute computer-readable instructions stored in one or more memory devices to perform operations, such as any of the operations described herein. The quantum hardware 102 includes components for performing quantum computation. For example, the quantum hardware 102 includes a quantum system 110, control device(s) 112, and readout device(s) 114 (e.g., readout resonator(s)). The quantum system 110 can include one or more multi-level quantum subsystems, such as a register of qubits (e.g., qubits 120). In some implementations, the multi-level quantum subsystems can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, spin-based qubits, and the like. In some implementations, the superconducting qubits may be located in a cryostat to cool the qubits to superconducting temperatures (e.g., less than about 3 Kelvin). However, aspects of the present disclosure are not limited to superconducting qubits. In some examples, any suitable qubit structure may be used without deviating from the scope of the present disclosure, such as photonic qubits, trapped ion qubits, spin qubits, neutral atom qubits, quantum dot qubits, molecular qubits, or other qubits.

[0081]

[0049] The type of multi-level quantum subsystems that the system 100 utilizes may vary. For example, in some cases the system may include one or more readout device(s) 114 coupled (e.g., electromagnetically coupled) to one or more qubits, e.g., transmon, flux, gmon, xmon, or other qubits. In other cases, ion traps, photonic devices or superconducting cavities (e.g., with which states may be prepared without employing qubits) may be used. Further examples of realizations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dot, or phosphorus impurity qubits.

[0082]

[0050] Quantum circuits may be constructed and applied to the register of qubits included in the quantum system 110 via multiple control lines that are coupled to one or more control devices 112. Example control devices 112 that operate on the register of qubits can be used to implement quantum gates or quantum circuits having a plurality of quantum gates, e.g., Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc. The one or more control devices 112 may be configured to operate on the quantum system 110 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystems may be superconducting qubits and the control devices 112 may be configured to provide control pulses to control lines to generate magnetic fields to control the qubits. For example, in some implementations, the multi-level quantum subsystems may be neutral atom qubits and the control devices 112 may be configured to provide control pulses to control lines to generate magnetic fields to control the qubits.

[0083]

[0051] The quantum hardware 102 may further include readout devices 114 (e.g., readout resonators). Measurement results 108 obtained via readout devices 114 may be provided to the classical processors 104 for processing and analyzing. In some implementations, the quantum hardware 102 may include a quantum circuit and the control device(s) 112 and readout device(s) 114 may implement one or more quantum logic gates that operate on the quantum system 110 through physical control parameters (e g., microwave pulses) that are sent through wires included in the quantum hardware 102. The readout device(s) 114 may be configured to perform quantum measurements on the quantum system 110 and send measurement results 108 to the classical processors 104.

[0084]

[0052] In addition, the quantum hardware 102 may be configured to receive data specifying physical control qubit parameter values 106 from the classical processors 104. The quantum hardware 102 may use the received physical control qubit parameter values 106 to update the action of the control device(s) 112 and readout device(s) 114 on the quantum system 110. For example, the quantum hardware 102 may receive data specifying new values representing voltage strengths of one or more digital to analog converters (DACs) included in the control devices 112 and may update the action of the DACs on the quantum system 110 accordingly. The classical processors 104 may be configured to initialize the quantum system 110 in an initial quantum state, e.g., by sending data to the quantum hardware 102 specifying an initial set of parameters 106.

[0085]

[0053] In some implementations, the readout device(s) 114 can take advantage of a difference in the impedance for the |0) and 11) states of an element of the quantum system, such as a qubit, to measure the state of the element (e.g., the qubit). For example, the resonance frequency of a readout resonator can take on different values when a qubit is in the state |0) or the state |1), due to the nonlinearity of the qubit. Therefore, a microwave pulse reflected from the readout device 114 carries an amplitude and phase shift that depend on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device(s) 114 to impede microwave propagation at the qubit frequency.

[0086]

[0054] In some embodiments, the quantum system 110 can include a plurality of qubits 120 arranged, for instance, in a two-dimensional grid 122. For clarity, the two-dimensional grid 122 depicted in FIG. 1 includes 4x4 qubits; however, in some implementations the system 110 may include a smaller or a larger number of qubits. In some embodiments, the multiple qubits 120 can interact with each other through multiple qubit couplers, e.g., qubit coupler 124. The qubit couplers can define nearest neighbor interactions between the multiple qubits 120. In some implementations, the strengths of the multiple qubit couplers are tunable parameters. In some cases, the multiple qubit couplers included in the quantum computing system 100 may be couplers with a fixed coupling strength.

[0087]

[0055] In some implementations, the multiple qubits 120 may include data qubits, such as qubit 126 and measurement qubits, such as qubit 128. A data qubit is a qubit that participates in a computation being performed by the system 100. A measurement qubit is a qubit that may be used to determine an outcome of a computation performed by the data qubit. That is, during a computation an unknown state of the data qubit is transferred to the measurement qubit using a suitable physical operation and measured via a suitable measurement operation performed on the measurement qubit.

[0088]

[0056] In some implementations, each qubit in the multiple qubits 120 can be operated using respective operating frequencies, such as an idling frequency and / or an interaction frequency and / or readout frequency and / or reset frequency. The operating frequencies can vary from qubit to qubit. For instance, each qubit may idle at a different operating frequency. The operating frequencies for the qubits 120 can be chosen before a computation is performed. In some examples, the operating of the frequencies for the qubits 120 may be adjusted using AC Stark shift according to examples of the present disclosure before a quantum computation, quantum gate, and / or a quantum algorithm is performed.

[0089]

[0057] FIG. 1 depicts one example quantum computing system that can be used to implement the methods and operations according to example aspects of the present disclosure. Other quantum computing systems can be used without deviating from the scope of the present disclosure.

[0058] In various implementations, the example system 100 can be implemented as a client device, a server device, or both. The example system 100 can be implemented as part of a distributed computing system. The example system 100 can be implemented along with other example systems, which may be the same or different. The example system 100 can be implemented in a server farm or other facility that operates multiple computing systems to provide computational services to or on behalf of a plurality of client systems. Techniques according to example aspects of the present disclosure can be applied to the calibration and maintenance of computing facilities, for example to increase service uptime or decrease failure rates.

[0090] Introduction to the Embodiments

[0091]

[0059] As noted above, the embodiments are directed towards estimating the expectation value of an observable O on quantum states prepared by acting unitary circuits on an initial state h-The embodiments consider the case where an ensemble of such circuits U labeled by 0 are available. This defines the observable values:

[0092]

[0093]

[0060] To measure

[0094]

[0095] for a fixed 9, one typically prepares -

[0096]

[0097] on a quantum device many times, measures

[0098]

[0099] on each ‘shot’, and averages the result.

[0100]

[0061] Designing an error mitigation scheme for an ensemble of circuits, as opposed to developing a scheme for a single circuit, is an example aspect of the present disclosure. Various embodiments introduce error mitigation strategies for this problem. The embodiments include at least three different families of embodiments. One family of embodiments is directed towards error mitigation with extrapolation datasets. A second family of embodiments is directed towards error mitigation by classical signal subtraction. A third family of embodiments is directed towards hybrid quantum-classical error mitigation.

[0101]

[0062] Error mitigation with extrapolation datasets embodiments generate a set of additional extrapolation datasets by artificially adding more errors in a controlled manner. Then they find good (optimal) linear combination coefficients to extrapolate the ideal observable value (signal) from the extrapolation datasets.

[0102]

[0063] Error mitigation via classical signal subtraction embodiments redefine the experimental goal to estimate a different “quantum” signal. The quantum signal is defined as the measured signal minus a ‘‘classical” signal, which can itself be measured or estimated by other means.

[0103]

[0064] The hybrid quantum-classical error mitigation embodiments may perform a heuristic classical simulation of the circuit, which converges in polynomial time but with an uncontrollable error, and use this alongside the quantum computer to construct a “hybrid” scheme with higher accuracy than either component.

[0104] Noise Model; K-Error Resolution

[0105]

[0065] The embodiments are discussed using a formalism for noisy quantum circuits which may be referred to as the k-error resolution formalism. The output of a noisy quantum computer may not be for values θ, but rather a noisy signal S̃θ. In various embodiments, a digital error model for the device is considered; i.e. where noise comes as discrete events that occur with some probability P. A unitary U 9 may be written as a product of individual sub-unitaries

[0106]

[0107]

[0066] The label of the circuit θ may be dropped from the gates g to keep the notation simpler. The environment inserts noise with probability after gate g.

[0108]

[0109] are CPTP noise maps, (1 ~ P9) is the gate (process) fidelity and

[0110]

[0111] . Gates can also be the identity gate to account for idle qubit errors.

[0112]

[0067] This effect on U = Up can be expanded

[0113]

[0114] where in the last equation the product implies repeated composition of the channels ^9 and ^9. This shows a sum of CPTP channels £v being applied with probability

[0115]

[0116] the standard definition of circuit fidelity (being attached to the noiseless channel

[0117]

[0118] ^)

[0119]

[0068] These can be separated into different orders by the Hamming weight I h of the binary vector t’, which counts the number of ‘error events’. This may be referred to as the k-error resolution

[0120]

[0121]

[0069] Note that Σ_k F_k = 1, and w =

[0122]

[0123] is the noiseless signal. This may be formalism for noisy quantum circuits. Consider as an example a noise model consisting of single qubit depolarizing channels for each qubit and cycle with depolarizing probability P

[0124]

[0125] and Ek averages over k Pauli’s inserted randomly on the circuit. This model can approximate some experiments well enough As seen in the previous equation, the coefficientv! ^k typically f G \1

[0126] scales as V

[0127]

[0071] In some models applying the same error channel twice in the same gate gives the identity, that is ε² = I. This can be used to simplify some expressions in some methods of error mitigation. For example, this can be done in the case of single qubit depolarizing noise model, Eq. (6). This noise model applies a disjoint X, Y or Z error with equal probability. This may be equivalent to applying uncorrelated errors, that is, a quantum map with error A ■ followed by a quantum map with error Y, followed by a quantum map with error Z, The probabilities of these uncorrelated channels may be adjusted for the two models to be equivalent. These uncorrelated errors satisfy the desired property as Eg~ —

[0128]

[0129] — 1 forPauli <7. Mean Square Error (MSE)

[0130]

[0072] This section discusses a non-limiting quantification of the measurement error over the set of circuits in the ensemble. For example, the performance of the noisy approximation across the entire ensemble can be quantified by the mean square error (MSE):

[0131]

[0132] where denotes the number of circuit instances in the ensemble, when applicable.

[0133]

[0073] The mean-squared error may be written using the k-error resolution. As this is a linear map, it propagates immediately to the estimation of any expectation value

[0134]

[0135]

[0074] The fidelity vector (F) includes k components denoted as:

[0136]

[0137] , and the k-error components are denoted as

[0138]

[0139] The autocorrelation matrix may be used:

[0140]

[0141] Error Mitigation with Extrapolation Datasets Embodiments

[0142]

[0076] The family of extrapolation dataset embodiments mitigate errors in the noisy signal ‘-s by taking linear combinations of the noisy signal and related extrapolation datasets Fi (for J —!> •••> -J ). These extrapolation dataset embodiments include operations: (1) generating an extrapolation dataset, (2) forming a linear combination of the noisy data and the extrapolation dataset, and (3) optimizing the linear combination coefficients.

[0143] Generating an Extrapolation Dataset

[0077] The embodiments may generate these extrapolation datasets by artificially introducing more errors in a controlled manner The embodiments include at least four families of techniques for generating the extrapolation datasets

[0144]

[0145] ): (a) increasing the error rate, (b) inserting hard errors, (c) using location-specific error rates, and (d) using related circuits.

[0146]

[0078] In the embodiments that employ increasing the error rate:

[0147]

[0148] generated by increasing the error rate of the circuit,

[0149]

[0150] 1 + b)p > P). An approximation can be made that the k-error components A are independent of the choice of 9, and the fidelity vector F changes with 6. This is exact if the error probabilities are equal Pg ~ P- The relative correction scales with Ged where is the difference between the probabilities Pg. Under this assumption, the k-error resolution of is identical to the k-error resolution of S but with a changed fidelity vector F^’ G

[0151]

[0079] As noted above, some embodiments employ inserting(or injecting) hard errors

[0152]

[0153] ) generated by artificially inserting a fixed number of of hard errors in positions randomly chosen across locations across the circuit. This can again be treated as the same k-error resolution o but with a changed fidelity vector F under the same assumption as above. That is

[0154]

[0155]

[0080] Consider the case where the error probabilities are equal Pg ~ P and applying the same p<

[0156] error twice gives the identity. For this case, an expression for Fc is given, which is the probability to end up with exactly k errors after artificially introducing ri' additional errors. At a first step for each sample, some random I positions are chosen initially for errors. The probability that m of those errors are cancelled by additional errors is the binomial probability

[0157]

[0158]

[0081] In order to have a final k errors, k — I + m more errors on the other G •••• I sites may be employed, which has probability

[0159]

[0160] m' — kp), The final probability to have k errors may then be written as:

[0161]

[0162]

[0082] The embodiments that employ using location-specific error rates apply errors with different error rates per location to e.g isolate an effective volume that is smaller than the entire circuit, or to capture an inhomogeneous error rate Pg "S P. In this case it may make more sense to further expand the signal over individual errors. Assuming an error model that applies the same error twice gives the identity (see above), the result is

[0163]

[0164] where is the probability of the string of errors labeled by the binary vector u occurring by random chance,FvSu is the chosen probability of applying the string of errors labeled byv&u- and @ denotes binary addition. The change of the labels k — v does not change the results in this text significantly

[0165]

[0083] For the embodiments that employ using related circuits,

[0166]

[0167] may be generated by implementing a related but non-equivalent quantum circuit. In this case the k-error resolution of -A will not be equivalent to the k-error resolution of A. This can be written as

[0168]

[0169] Forming a Linear Combination of the Noisy Data and the Extrapolation Dataset

[0170]

[0084] Given some choice of theseFg<a linear combination is considered to yield a better estimate of SO than Sg That is, the error mitigation scheme entails approximating

[0171]

[0172] p(θ), g.

[0173]

[0085] For ease of notation, assume that the first datasetIis A. The MSE in this approximation over the chosen ensemble can be calculated:

[0174]

[0175]

[0086] Note that in cases (1) and (2) above where

[0176]

[0177] this equation can be simplified, and the J-J indices in the autocorrelation can be dropped. This equation is quadratic in the coefficientsci. Assuming that the autocorrelations are known, it can be minimized exactly

[0178]

[0179]

[0087] Equation (21) gives the bias in the converged estimator. However, it may not include the sampling variance inherent in quantum computing; each RF estimate may employ repeating a quantum circuit Ml times. The optimal distribution of the can be found. The result is

[0180]

[0181] is the total number of shots. The total MSE is given by the bias plus the variance. This yields:

[0182]

[0183] Optimizing the Linear Combination Coefficients

[0184]

[0088] The optimizing the linear combination coefficients step may include estimating autocorrelations. In many cases, the k-error resolved covariance matrices will not be exactly known to the user, which precludes precisely optimizing Eq. (24). In this situation, one could consider estimating covariances using similar circuit ensembles on smaller system sizes (where simulation becomes feasible).

[0185] !■■(. / )...

[0186]

[0089] Consider the case where ‘■' as in cases (1) and (2) above. Furthermore, typically (. i

[0187] the values of 'A are known with some precision. The embodiments may estimate the autocorrelation between k-error components with relatively small k < K because the probability of each error is small to start with. K extrapolation datasets M'' can be measured. This gives the autocorrelations

[0188]

[0189]

[0090] The autocorrelations

[0190]

[0191]

[0091] Note that this can also be used to obtain an estimate of the MSE obtained by the extrapolation datasets error mitigation method.

[0192] Error Mitigation by Classical Signal Subtraction Embodiments

[0193]

[0092] The error mitigation by classical signal subtraction embodiments take a different approach by redefining the problem. These embodiments propose that the total signal can be split into two parts: a "quantum" part that is difficult to simulate on a classical computer and a "classical" part that may be calculated or measured. The error mitigation by classical signal subtraction embodiments may include operations. (1) obtaining the classical signal, (2) decomposing the noisy signal, and (3) subtracting and rescaling.

[0194]

[0093] assume the case where multiple errors generate the same response as a single error, that is, errors saturate. This is the assumption

[0195]

[0196] Under this assumption, one can approximate

[0197] Ss(F FSkfi+ (1 - F)Sk;>() (27)

[0198]

[0094] Here, F may be some effective fidelity coming from an effective circuit volume instead of the total circuit fidelity. This can be estimated for instance by doing Loschmidt echo, or removing a few gates from the circuit and doing a Loschmidt echo.

[0199]

[0095] Under these assumptions, if one takes two values F and F < F, one can solve Eq (27) circuit-wise at two points to yield

[0200]

[0201]

[0096] This gives a simple approximate formula that is a first correction to the rescaling by fidelity. This result can be further standardized to deal with under- or over-corrections from the difference between Loschmidt fidelity and the true effective fidelity.

[0202] Circuit-Wise K-Error Mitigation

[0203]

[0097] In some cases, error mitigation can be performed for each circuit in the ensemble using the k-error resolution formalism. The k-error resolution is truncated for each circuit at some maximum value K. K extrapolation datasets are also taken. Considering the case where

[0204]

[0205] asjn cases)anj (2) above. Also assuming a good approximation to F: exists. For each circuit 6, the system of equations is solved K

[0206] R? - E C"st / ,

[0207] (29)

[0208]

[0098] This gives an estimate of the noiseless signal

[0209]

[0210] One example of extrapolation datasets where this technique can be applied is in inserting a fixed number of hard errors.

[0211] Error Mitigation By Classical Signal Subtraction

[0212]

[0099] The observable So may be decomposed into two signals (or two components):

[0213] S.9 — Qt) + C'. (30)

[0214] where CD is a "classical" signal that can be estimated with classical algorithms and Q& is a “quantum” signal which is harder to estimate with classical methods. In this case, the goal of the experiment can be redefined as measuring the quantum signal defined as

[0215] Qe So -■ Cf>. (3 i)

[0216]

[0100] Note that this is an important difference with the method of extrapolation datasets, in that in this case what is the noiseless signal to be measured, namely Qo, is being redefined.

[0217]

[0101] The method of classical subtraction can also be extended to the case where it is defined

[0218] (32)

[0219]

[0220] for some coefficientsni and the same assumptions for w as for C '& above. For clarity, the discussion is on this case here. Although the embodiments are not so limited, as other cases are applicable for the embodiments.

[0221]

[0102] As the experimental goal is redefined to measure Qo, the classical signal Co may be subtracted from the measured observable E-r The classical signal can be obtained numerically or experimentally.

[0222]

[0103] A possible assumption for the classical signal subtraction method is that the noisy observable can be experimentally measured

[0223]

[0224] and that the related noisy observable * can also be measured. Note that this equation assumes a simple decay for the quantum signal Qo. Under this assumption, the difference of the noisy observables and Co can be measured, which is and therefore easy to correct by rescaling. One intuition is that the experimental value S can have a more ‘quantum’ signal which decays quickly, and a more ‘classical’ signal (7 that decays slower. Furthermore, the ‘more classical’ signal can be measured independently by injecting ‘noise’ in some particular way into the experiment. The corresponding assumption in the k-Pauli expansion is that the approximation can be used

[0225]

[0226] and that the ‘more classical’ value can also be measured

[0227]

[0228]

[0104] Under these assumptions, the experiment can measure the difference of the observables

[0229]

[0230] A -52^

[0231] where k. The ideal value Qo is then obtained with a trivial re-scaling.

[0232]

[0105] Note that the method of classical subtraction can be combined with the method of dataset extraction. That is, the definition of the “quantum” signal Qo from Eq. (31) is kept, but error mitigation with extrapolation datasets is used to estimate (or even

[0233]

[0234] Hybrid Quantum-Classical Error Mitigation Embodiments

[0235]

[0106] In some quantum computing applications, it may be possible to obtain a heuristic classical estimation

[0236]

[0237] ' of the appropriate Nj values, via e.g. Monte Carlo simulation or meanfield methods. By “heuristic” here is meant an algorithm which scales polynomially in the system size, but which has a mean square error that is “uncontrolled” — it may not be reduced to an arbitrarily small number (or the cost of achieving a fixed MSE scales exponentially in

[0238]

[0239] In this case, some embodiments may construct a “hybrid” estimator that uses both the data from the heuristic classical estimator and a noisy simulation t) (and possibly extra data) to make a new estimate 'E that may reduce the MSE below either alternative. This can be achieved for instance if a quantum simulation can be performed that mimics the classical

[0240] C)

[0241] algorithm, yielding an estimate ‘E. Two possible non-limiting ways of constructing theq(h),

[0242] estimator w,> from the hybrid are direct rescaling

[0243]

[0244] or subtraction and standardization of the quantum term, and re-adding to the classical signal

[0245]

[0246]

[0107] The latter can be observed to be equivalent to the error mitigation by classical signal subtraction embodiments, where the ‘quantum signal’ has been defined Q#

[0247]

[0248] These embodiments may improve on the classical estimate

[0249]

[0250] for at least some classes of circuits (e.g. out-of-time-ordered correlators) on physical grounds.

[0251] Modification to Specific Embodiments

[0252]

[0108] The embodiments described herein can be modified and extended in various ways These modifications represent alternative embodiments that may be advantageous in different contexts.

[0253]

[0109] For instance, while the optimization of linear combination coefficients is described using the mean square error (MSE) as the cost function, other error metrics or cost functions could be employed. One could, for example, minimize the mean absolute error, or use a weighted MSE that assigns greater importance to achieving accuracy for specific circuits within the ensemble that are deemed more relevant to the overall computation.

[0254]

[0110] The mitigation schemes can also be applied in an iterative fashion. An initial mitigated signal, obtained using one of the described methods, can be treated as a new baseline noisy signal. A second round of mitigation, perhaps using a different technique or a different set of extrapolation datasets, can then be applied to correct for any residual errors, potentially leading to a more accurate final result.

[0255]

[0111] Furthermore, the different mitigation techniques can be combined in various ways. As an example, the classical signal subtraction method can be used as a first step to define and isolate a "quantum" signal Qe. Subsequently, the error mitigation method using extrapolation datasets can be applied not to the original signal

[0256]

[0257] but to the task of finding an improved estimate of Qs. This hierarchical approach may be particularly effective if the noise structure affecting the quantum signal Qt> is simpler or more amenable to extrapolation than the noise affecting the total signal.

[0258]

[0112] In some embodiments, the optimal (or somewhat optimized) set of expansion coefficientsej for the linear combination of datasets may not be constant across the entire ensemble but may depend on the circuit parameters 8. The coefficients could be calculated dynamically. For example, the ensemble could be partitioned into several sub-ensembles, and a separate set of optimal coefficients could be determined for each. In more advanced embodiments, a machine learning model, such as a neural network, could be trained to predict the optimal coefficients

[0259]

[0260] as a function of the circuit parameters (9, allowing for a highly adaptive, circuit-specific mitigation strategy.

[0261]

[0113] Additionally, while the k-error resolution formalism is presented in the context of a digital, stochastic error model, the overarching principles of the mitigation schemes can be adapted to other types of noise. For coherent errors, which correspond to systematic unitary' rotations rather than stochastic processes, one could develop an analogous expansion of the noisy evolution in a basis of error generators (e.g., Pauli operators). The mitigation techniques could then be adapted to find linear combinations of datasets that are designed to cancel the leadingorder coherent error terms, providing a pathway to extend these powerful mitigation strategies to a broader class of noise models.

[0262] Methods

[0263]

[0114] FIG. 2 shows a flowchart for a method 200 for operating a quantum computing system (QCS), according to various embodiments. FIG. 2 depicts operations performed in a particular order for purposes of illustration and discussion. Those of ordinary skill in the art, using the disclosures provided herein, will understand that the operations of any of the methods provided herein can be omitted, modified, rearranged, include operations not illustrated, and / or adapted in various ways without deviating from the scope of the present disclosure.

[0264]

[0115] The QCS of method 200 may be equivalent and / or similar to QCS 100 of FIG. 1. The QCS may include a quantum processor device (e.g., quantum system 100 of FIG. 1) that has a set of qubits (e.g., qubits 120 of FIG. 1) and classical processor device (e.g., classical processor(s) 104 of FIG. 1) with a set of classical bits. The QCS 100 of FIG. 1 may perform at least some of the operations and / or steps of method 200. Method 200 begins at block 202, where each quantum circuit of a set of quantum circuits is executed to generate a set of noisy signals.

[0265] Executing each quantum circuit of the set of quantum circuits generates a separate and corresponding noisy signal of the set of noisy signals. Each noisy signal of the set of noisy signals corresponds to a separate noisy measurement of an observable associated with the set of quantum circuits. At block 204, a set of extrapolation datasets is generated. Each extrapolation dataset of the set of extrapolation datasets corresponds to a separate quantum circuit of the set of quantum circuits and includes a set of additional errors in a corresponding quantum circuit of the set of quantum circuits. At block 206, a value for each expansion coefficient of a set of expansion coefficients is determined. A set of determined values for the set of expansion coefficients is generated. The set of expansion coefficients parameterizes a linear combination of each extrapolation dataset of the set of extrapolation datasets. At block 208, an approximation of a noiseless measurement of the observable is determined based on the set of noisy signals, the set of extrapolation datasets, and the set of determined values.

[0266]

[0116] In some embodiments, the method further includes re-executing each quantum circuit of the set of quantum circuits to generate a corresponding extrapolation dataset of the set of extrapolation datasets. Re-executing a quantum circuit of the set of quantum circuits may include amplifying noise in the quantum circuit to generate the set of additional errors.

[0267]

[0117] In other embodiments, the method further includes re-executing each quantum circuit of the set of quantum circuits to generate a corresponding extrapolation dataset of the set of extrapolation datasets. Re-executing a quantum circuit of the set of quantum circuits may include randomly injecting the set of additional errors in the quantum circuit.

[0268]

[0118] In still other embodiments, the method further includes re-executing each quantum circuit of the set of quantum circuits to generate a corresponding extrapolation dataset of the set of extrapolation datasets. Re-executing a quantum circuit of the set of quantum circuits may include applying varying amounts of noise in different locations of the quantum circuit to generate the set of additional errors in the quantum circuit.

[0269]

[0119] In additional embodiments, the method further includes executing each modified quantum circuit of a set of modified quantum circuits to generate an extrapolation dataset of the set of extrapolation datasets. Each modified quantum circuit of the set of modified quantum circuits may be a modified version of a corresponding quantum circuit of the set of quantum circuits. Executing the modified version of the corresponding circuit may generate the set of additional errors in the quantum circuit.

[0270]

[0120] In at least one embodiment, the method further includes determining the set of values for the set of expansion coefficients based on the set of noisy signals, the set of extrapolation datasets, and an error metric. The error metric may indicate a difference between a noiseless signal and an estimate of the noiseless signal and the noiseless signal corresponds to the observable associated with the set of quantum circuits.

[0271]

[0121] In various embodiments, the error metric indicates an average difference, across the set of quantum circuits and the set of noisy signals, between the noiseless signal and estimate of the noiseless signal.

[0272]

[0122] In some embodiments, the estimate of the noiseless signal may be based on the linear combination of the set of extrapolation datasets for a particular choice of values for the set of expansion coefficients.

[0273]

[0123] The error metric may be a mean squared error (MSE) metric. Determining the set of values for the set of expansion coefficients may include determining the value for each expansion coefficient of the set of expansion coefficients. The difference between the noiseless signal and the estimate of the noiseless signal may be minimized (or at least decreased).

[0274]

[0124] In at least one embodiment, determining the approximation of a noiseless measurement of the observable is based on generating a linear combination of the set of noisy signals and the set of extrapolation datasets in accordance with the set of determined values.

[0275]

[0125] FIG. 3 shows a flowchart for a method 300 for operating a quantum computing system (QCS), according to various embodiments. The QCS of method 300 may be equivalent and / or similar to QCS 100 of FIG. 1. The QCS may include a quantum processor device (e.g., quantum system 100 of FIG. 1) that has a set of qubits (e.g., qubits 120 of FIG. 1) and classical processor device (e.g., classical processor(s) 104 of FIG. 1) with a set of classical bits. The QCS 100 of FIG. 1 may perform at least some of the operations and / or steps of method 300. Method 300 begins at block 302, where a classical signal is accessed. In method 300, a noiseless signal of a quantum circuit is decomposable into a first component and a second component. The first component corresponds to a noiseless quantum signal. The second component corresponds to the classical signal. The noiseless signal of the quantum circuit corresponds to an observable of the quantum circuit. At block 304, a first version of the quantum circuit is executed to generate a first noisy measurement of the observable. At block 306, a second version of the quantum circuit is executed to generate a second noisy measurement of the observable. At block 308, an approximation of the noiseless quantum signal is determined based on a difference between the first noisy measurement and the second noisy measurement of the observable. At block 310, an approximation of the observable is determined based on the approximation of the noiseless quantum signal and the classical signal.

[0126] In at least one embodiment, the method further includes generating the classical signal via executing a classical algorithm on a classical processing device. The classical signal may be a noiseless classical signal and the noiseless signal is a linear sum of the noiseless quantum signal and the noiseless classical signal.

[0276]

[0127] The method may further include determining the approximation of the noiseless quantum signal based on a difference between the first noisy measurement and the second noisy measurement of the observable.

[0277]

[0128] In some embodiments, the method further includes determining a noisy quantum signal based on the difference between the first noisy measurement and the second noisy measurement of the observable. The approximation of the noiseless quantum signal may be determined based on the noisy quantum signal.

[0278]

[0129] The method may further include rescaling the noisy quantum signal. The approximation of the noiseless quantum signal may be determined based on rescaling the noisy quantum signal.

[0279]

[0130] In at least one embodiment, the second version of the quantum circuit is configured to suppress the noiseless quantum signal.

[0280]

[0131] FIG. 4 shows a flowchart for a method 400 for operating a quantum computing system (QCS), according to various embodiments. The QCS of method 400 may be equivalent and / or similar to QCS 100 of FIG. 1. The QCS may include a quantum processor device (e.g., quantum system 100 of FIG. 1) that has a set of qubits (e g., qubits 120 of FIG. 1) and classical processor device (e.g., classical processor(s) 104 of FIG. 1) with a set of classical bits. The QCS 100 of FIG. 1 may perform at least some of the operations and / or steps of method 400. Method 400 begins at block 402, where a heuristic classical estimation of an observable of a quantum circuit is accessed. At block 404, a first version of the quantum circuit is executed to generate a first noisy signal that corresponds to a first noisy measurement of the observable. At block 406, a second version of the quantum circuit is executed to generate a second noisy signal that corresponds to a second noisy measurement of the observable. The second version of the quantum circuit is configured to mimic the classical algorithm. At block 408, a hybrid estimate of the observable is determined, via a classical processor device, based on a combination of the heuristic classical estimation, the first noisy signal, and the second noisy signal.

[0281]

[0132] In at least one embodiment, the method further includes generating the heuristic classical estimation of the observable via executing a classical algorithm, via a classical processing device. The classical algorithm may scale polynomially with a system size.

[0282]

[0133] The method may further include determining, via the classical processor device, the hybrid estimate of the observable based on a combination of the heuristic classical estimation, the first noisy signal, and the second noisy signal.

[0283]

[0134] In at least one embodiment, the method further includes determining, via the classical processor device, the combination of the heuristic classical estimation, the first noisy measurement of the observable, and the second noisy measurement of the observable by multiplying the heuristic classical estimation by a ratio of the first noisy signal to the second noisy signal.

[0284] Additional Embodiments

[0285]

[0135] The embodiments include error mitigation schemes for quantum circuits from an ensemble of quantum circuits (e.g., as opposed to developing a scheme for a single circuit). This can be done by minimizing the MSE, Eq. (24), or related quantities across this given ensemble.

[0286]

[0136] Some embodiments use a k-error resolution formalism, to enable the error mitigation schemes.

[0287]

[0137] Some embodiments generate an (at least somewhat) optimized linear combination of extrapolation datasets to approximate the noiseless signal. The linear combination can be (at least somewhat) optimized to minimize (or at least decrease) the ensemble (or set) MSE, Eq. (24), either by exact inversion, Eq (23), or by numerical optimization.

[0288]

[0138] The extrapolation datasets can be of different forms. For instance, the extrapolation datasets may be any combination of: (a) an output of noisy quantum circuit with increased error rate, (b) an output of noisy quantum circuits with a fixed number of additional errors inserted (or injected), (c) an output of a noisy quantum circuit with different artificial error rates per location, and / or (d) an output of related but non-equivalent quantum circuits.

[0289]

[0139] Various embodiments may estimate the k-error resolved autocorrelations, for example, by extrapolation from small system sizes where these can be calculated, or by solving a system of equations using extrapolation datasets.

[0290]

[0140] Some embodiments may mitigate errors by doing an extrapolation assuming saturation of errors.

[0291]

[0141] Other embodiments perform circuit-wise k-error mitigation via the k-error resolution to solve for the noiseless signal.

[0292]

[0142] Some embodiments employ a definition of a “quantum” signal to be measured by subtracting a “classical” signal. The classical signal can be obtained numerically or experimentally. For instance, one can subtract a “classical piece” consisting of inserting a line of randomly chosen Pauli operators, and standardize the result. This can be combined with different error mitigation techniques, such as the extrapolation dataset techniques.

[0293]

[0143] Some embodiments use a classical heuristic simulation of the quantum circuit, alongside noisy quantum data, to develop a “hybrid” estimator with better performance (e.g., a lower MSE) than either component.

[0294]

[0144] In an aspect, the present disclosure provides an example implementation including:

[0295] A method for operating a quantum computing system (QCS), the method comprising: executing each quantum circuit of a set of quantum circuits to generate a set of noisy signals, wherein executing each quantum circuit of the set of quantum circuits generates a separate and corresponding noisy signal of the set of noisy signals and each noisy signal of the set of noisy signals corresponds to a separate noisy measurement of an observable associated with the set of quantum circuits; generating a set of extrapolation datasets, wherein each extrapolation dataset of the set of extrapolation datasets corresponds to a separate quantum circuit of the set of quantum circuits and includes a set of additional errors in a corresponding quantum circuit of the set of quantum circuits; determining a value for each expansion coefficient of a set of expansion coefficients such that a set of determined values for the set of expansion coefficients is generated, wherein the set of expansion coefficients parameterizes a linear combination of each extrapolation dataset of the set of extrapolation datasets; and determining an approximation of a noiseless measurement of the observable based on the set of noisy signals, the set of extrapolation datasets, and the set of determined values.

[0296]

[0145] In some implementations, the example method includes re-executing each quantum circuit of the set of quantum circuits to generate a corresponding extrapolation dataset of the set of extrapolation datasets, wherein re-executing a quantum circuit of the set of quantum circuits includes amplifying noise in the quantum circuit to generate the set of additional errors.

[0297]

[0146] In some implementations, the example method includes re-executing each quantum circuit of the set of quantum circuits to generate a corresponding extrapolation dataset of the set of extrapolation datasets, wherein re-executing a quantum circuit of the set of quantum circuits includes randomly injecting the set of additional errors in the quantum circuit.

[0298]

[0147] In some implementations, the example method includes re-executing each quantum circuit of the set of quantum circuits to generate a corresponding extrapolation dataset of the set of extrapolation datasets, wherein re-executing a quantum circuit of the set of quantum circuits includes applying varying amounts of noise in different locations of the quantum circuit to generate the set of additional errors in the quantum circuit.

[0299]

[0148] In some embodiments, the method further comprises executing each modified quantum circuit of a set of modified quantum circuits to generate an extrapolation dataset of the set of extrapolation datasets, wherein each modified quantum circuit of the set of modified quantum circuits is a modified version of a corresponding quantum circuit of the set of quantum circuits, wherein executing the modified version of the corresponding circuit generates the set of additional errors in the quantum circuit.

[0300]

[0149] In some implementations, the example method includes determining the set of values for the set of expansion coefficients based on the set of noisy signals, the set of extrapolation datasets, and an error metric, wherein the error metric indicates a difference between a noiseless signal and an estimate of the noiseless signal and the noiseless signal corresponds to the observable associated with the set of quantum circuits.

[0301]

[0150] In some implementations of the example method, the error metric indicates an average difference, across the set of quantum circuits and the set of noisy signals, between the noiseless signal and estimate of the noiseless signal.

[0302]

[0151] In some implementations of the example method, the estimate of the noiseless signal is based on the linear combination of the set of extrapolation datasets for a particular choice of values for the set of expansion coefficients.

[0303]

[0152] In some implementations of the example method, the error metric is a mean squared error (MSE) metric and determining the set of values for the set of expansion coefficients includes determining the value for each expansion coefficient of the set of expansion coefficients such that the difference between the noiseless signal and the estimate of the noiseless signal is decreased.

[0304]

[0153] In some implementations of the example method, determining the approximation of a noiseless measurement of the observable is based on generating a linear combination of the set of noisy signals and the set of extrapolation datasets in accordance with the set of determined values.

[0305]

[0154] In an aspect, the present disclosure provides an example method. In some implementations, the example method includes accessing a classical signal, wherein a noiseless signal of a quantum circuit is decomposable into a first component and a second component, the first component corresponding to a noiseless quantum signal, the second component corresponding to the classical signal, and the noiseless signal of the quantum circuit corresponds to an observable of the quantum circuit. In some implementations, the example method includes executing a first version of the quantum circuit to generate a first noisy measurement of the observable. In some implementations, the example method includes determining an approximation of the noiseless quantum signal based on the first noisy measurement of the observable. In some implementations, the example method includes determining an approximation of the observable based on the approximation of the noiseless quantum signal and the classical signal.

[0306]

[0155] In some implementations, the example method includes generating the classical signal via executing a classical algorithm on a classical processing device, wherein the classical signal is a noiseless classical signal and the noiseless signal is a linear sum of the noiseless quantum signal and the noiseless classical signal.

[0307]

[0156] In some implementations, the example method includes executing a second version of the quantum circuit to generate a second noisy measurement of the observable. In some implementations, the example method includes determining the approximation of the noiseless quantum signal based on a difference between the first noisy measurement and the second noisy measurement of the observable.

[0308]

[0157] In some implementations, the example method includes determining a noisy quantum signal based on the difference between the first noisy measurement and the second noisy measurement of the observable. In some implementations, the example method includes determining the approximation of the noiseless quantum signal based on the noisy quantum signal.

[0309]

[0158] In some implementations, the example method includes rescaling the noisy quantum signal. In some implementations, the example method includes determining the approximation of the noiseless quantum signal based on rescaling the noisy quantum signal.

[0310]

[0159] In some implementations of the example method, the second version of the quantum circuit is configured to suppress the noiseless quantum signal.

[0311]

[0160] In an aspect, the present disclosure provides an example method. In some implementations, the example method includes accessing a heuristic classical estimation of an observable of a quantum circuit. In some implementations, the example method includes executing a first version of the quantum circuit to generate a first noisy signal that corresponds to a first noisy measurement of the observable. In some implementations, the example method includes determining a hybrid estimate of the observable based on a combination of the heuristic classical estimation and the first noisy signal.

[0312]

[0161] In some implementations, the example method includes generating the heuristic classical estimation of the observable via executing a classical algorithm, via a classical processing device, wherein the classical algorithm scales polynomially with a system size.

[0313]

[0162] In some implementations, the example method includes executing a second version of the quantum circuit to generate a second noisy signal that corresponds to a second noisy measurement of the observable, wherein the second version of the quantum circuit is configured to mimic the classical algorithm. In some implementations, the example method includes determining, via the classical processor device, the hybrid estimate of the observable based on a combination of the heuristic classical estimation, the first noisy signal, and the second noisy signal.

[0314]

[0163] In some implementations, the example method includes determining, via the classical processor device, the combination of the heuristic classical estimation, the first noisy measurement of the observable, and the second noisy measurement of the observable by multiplying the heuristic classical estimation by a ratio of the first noisy signal to the second noisy signal.

[0315]

[0164] One embodiment includes a quantum computing system. The quantum computing system may include any of the following: a quantum processing device, a classical processing device, a set of qubits, and one or more memory devices The one or more memory devices store computer-readable instructions that when executed by the processing unit cause the processing unit to perform operations. The operations may include executing each quantum circuit of a set of quantum circuits to generate a set of noisy signals. Executing each quantum circuit of the set of quantum circuits generates a separate and corresponding noisy signal of the set of noisy signals. Each noisy signal of the set of noisy signals corresponds to a separate noisy measurement of an observable associated with the set of quantum circuits. A set of extrapolation datasets is generated. Each extrapolation dataset of the set of extrapolation datasets corresponds to a separate quantum circuit of the set of quantum circuits and includes a set of additional errors in a corresponding quantum circuit of the set of quantum circuits. A value for each expansion coefficient of a set of expansion coefficients is determined. A set of determined values for the set of expansion coefficients is generated by determining the value for each expansion coefficient of the set of expansion coefficients. The set of expansion coefficients parameterizes a linear combination of each extrapolation dataset of the set of extrapolation datasets. An approximation of a noiseless measurement of the observable is determined based on the set of noisy signals, the set of extrapolation datasets, and the set of determined values.

[0316]

[0165] In other embodiments, the operations of the QCS include accessing a classical signal, wherein a noiseless signal of a quantum circuit is decomposable into a first component and a second component, the first component corresponding to a noiseless quantum signal, the second component corresponding to the classical signal, and the noiseless signal of the quantum circuit corresponds to an observable of the quantum circuit. In some implementations, the example method includes executing a first version of the quantum circuit to generate a first noisy measurement of the observable. In some implementations, the example method includes determining an approximation of the noiseless quantum signal based on the first noisy measurement of the observable. In some implementations, the example method includes determining an approximation of the observable based on the approximation of the noiseless quantum signal and the classical signal.

[0317]

[0166] In still other embodiments, the operations of the QCS include accessing a heuristic classical estimation of an observable of a quantum circuit. In some implementations, the example method includes executing a first version of the quantum circuit to generate a first noisy signal that corresponds to a first noisy measurement of the observable. In some implementations, the example method includes determining a hybrid estimate of the observable based on a combination of the heuristic classical estimation and the first noisy signal.

[0318]

[0167] Implementations of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs (e.g., 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, data processing apparatus). The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits / qubit structures, or a combination of one or more of them.

[0319] Alternatively or in addition, the program instructions can be encoded on an artificially-generated propagated signal that is capable of encoding digital and / or quantum information (e.g., a machine-generated electrical, optical, or electromagnetic signal) that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0320]

[0168] The terms quantum information and quantum data refer to information or data that is carried by, held, or stored in quantum systems, where the smallest non-trivial system is a qubit (i.e., a system that defines the unit of quantum information). It is understood that the term “qubit” encompasses at least some quantum systems that may be suitably approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states; however, it is understood that other setups where the computational states are identified with higher level excited states (e.g., qubits) are possible.

[0321]

[0169] The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses at least some kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable QCS, a digital computer, a quantum computer, or multiple digital and QCSs or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), or an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0170] A digital or classical computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL, Quipper, Cirq, etc..

[0322]

[0171] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub-programs, or portions of code. A digital and / or quantum computer program can be deployed to be executed on one digital or one quantum computer or on multiple digital and / or quantum computers that 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 may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network does not transmit quantum data; however, a quantum data communication network may transmit both quantum data and digital data.

[0323]

[0172] The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers, operating with one or more digital and / or QCSs, as appropriate, executing 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 logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers.

[0173] For a system of one or more digital and / or quantum computers or processors to be “configured to” or “operable to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more digital and / or quantum computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by digital and / or quantum data processing apparatus, cause the apparatus to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions.

[0324]

[0174] Digital and / or quantum computers suitable for the execution of a digital and / or quantum computer program can be based on general or special purpose digital and / or quantum microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, or a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0325]

[0175] Some example elements of a digital and / or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing 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 quantum simulators. Generally, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to, or both, one or more mass storage devices for storing digital and / or quantum data, e.g., magnetic, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0326]

[0176] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include at least some forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magnetooptical disks; and CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

[0327]

[0177] Control of the various systems described in this specification, or portions of them, can be implemented in a digital and / or quantum computer program product that includes instructions that are stored on one or more tangible, non-transitory machine-readable storage media, and that are executable on one or more digital and / or quantum processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or electronic system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification.

[0328]

[0178] While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a subcombination.

[0329]

[0179] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood that such operations be performed in the particular order shown or in sequential order, or that at least some of the illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as such separation in at least some 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.

[0180] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily employ the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

[0330]

[0181] Aspects of the disclosure have been described in terms of illustrative implementations thereof. Numerous other implementations, modifications, or variations within the scope and spirit of the appended claims can occur to persons of ordinary skill in the art from a review of this disclosure. Any and at least some features in the following claims can be combined or rearranged in any way possible. Accordingly, the scope of the present disclosure is by way of example rather than by way of limitation, and the subject disclosure does not preclude inclusion of such modifications, variations or additions to the present subject matter as would be readily apparent to one of ordinary skill in the art. Moreover, terms are described herein using lists of example elements joined by conjunctions such as “and,” “or,” “but,” etc. It should be understood that such conjunctions are provided for explanatory purposes. Lists joined by a particular conjunction such as “or,” for example, can refer to “at least one of” or “any combination of’ example elements listed therein, with “or” being understood as “and / or” unless otherwise indicated. Also, terms such as “based on” should be understood as “based at least in part on.”

[0331]

[0182] Those of ordinary skill in the art, using the disclosures provided herein, will understand that the elements of any of the claims, operations, or processes discussed herein can be adapted, rearranged, expanded, omitted, combined, or modified in various ways without deviating from the scope of the present disclosure. Some of the claims are described with a letter reference to a claim element for exemplary illustrated purposes and is not meant to be limiting. The letter references do not imply a particular order of operations. For instance, letter identifiers such as (a), (b), (c),..., (i), (ii), (iii),..., etc. can be used to illustrate operations. Such identifiers are provided for the ease of the reader and do not denote a particular order of steps or operations. An operation illustrated by a list identifier of (a), (i), etc. can be performed before, after, or in parallel with another operation illustrated by a list identifier of (b), (ii), etc.

Claims

WHAT IS CLAIMED IS:

1. A method for operating a quantum computing system (QCS), the method comprising:executing each quantum circuit of a set of quantum circuits to generate a set of noisy signals, wherein executing each quantum circuit of the set of quantum circuits generates a separate and corresponding noisy signal of the set of noisy signals and each noisy signal of the set of noisy signals is a separate noisy measurement of an observable associated with the set of quantum circuits;generating a set of extrapolation datasets, wherein each extrapolation dataset of the set of extrapolation datasets corresponds to a separate quantum circuit of the set of quantum circuits and includes a set of additional errors in a corresponding quantum circuit of the set of quantum circuits;determining a value for each expansion coefficient of a set of expansion coefficients such that a set of determined values for the set of expansion coefficients is generated, wherein the set of expansion coefficients parameterizes a first linear combination of each extrapolation dataset of the set of extrapolation datasets; anddetermining an approximation of a noiseless measurement of the observable based on the set of noisy signals, the set of extrapolation datasets, and the set of determined values.

2. The method of claim 1, the method further comprising:re-executing each quantum circuit of the set of quantum circuits to generate a corresponding extrapolation dataset of the set of extrapolation datasets, wherein re-executing a quantum circuit of the set of quantum circuits includes amplifying noise in the quantum circuit to generate the set of additional errors.

3. The method of claims 1 or 2, the method further comprising:re-executing each quantum circuit of the set of quantum circuits to generate a corresponding extrapolation dataset of the set of extrapolation datasets, wherein re-executing a quantum circuit of the set of quantum circuits includes randomly injecting the set of additional errors in the quantum circuit.

4. The method of any of claims 1-3, the method further comprising:re-executing each quantum circuit of the set of quantum circuits to generate a corresponding extrapolation dataset of the set of extrapolation datasets, wherein re-executing a quantum circuit of the set of quantum circuits includes applying varying amounts of noise in different locations of the quantum circuit to generate the set of additional errors in the quantum circuit.

5. The method of any of claims 1-4, the method further comprising:executing each modified quantum circuit of a set of modified quantum circuits to generate an extrapolation dataset of the set of extrapolation datasets, wherein each modified quantum circuit of the set of modified quantum circuits is a modified version of a corresponding quantum circuit of the set of quantum circuits, wherein executing the modified version of the corresponding circuit generates the set of additional errors in the quantum circuit.

6. The method of any of claims 1-5, the method further comprising:determining the set of values for the set of expansion coefficients based on the set of noisy signals, the set of extrapolation datasets, and an error metric, wherein the error metric indicates a difference between a noiseless signal and an estimate of the noiseless signal and the noiseless signal corresponds to the observable associated with the set of quantum circuits.

7. The method of claim 6, wherein the error metric indicates an average difference, across the set of quantum circuits and the set of noisy signals, between the noiseless signal and the estimate of the noiseless signal.

8. The method of claims 6 or 7, wherein the estimate of the noiseless signal is based on the first linear combination of the set of extrapolation datasets for a particular choice of values for the set of expansion coefficients.

9. The method of any of claims 6-8, wherein the error metric is a mean squared error (MSE) metric and determining the set of values for the set of expansion coefficients includes determining the value for each expansion coefficient of the set of expansion coefficients suchthat the difference between the noiseless signal and the estimate of the noiseless signal is decreased.

10. The method of any of claims 1-9, wherein determining the approximation of the noiseless measurement of the observable is based on generating a second linear combination of the set of noisy signals and the set of extrapolation datasets in accordance with the set of determined values.

11. A method for operating a quantum computing system (QCS), the method comprising:accessing a classical signal, wherein a noiseless signal of a quantum circuit is decomposable into a first component and a second component, the first component corresponding to a noiseless quantum signal, the second component corresponding to the classical signal, and the noiseless signal of the quantum circuit corresponds to an observable of the quantum circuit;executing a first version of the quantum circuit to generate a first noisy measurement of the observable;determining an approximation of the noiseless quantum signal based on the first noisy measurement of the observable; anddetermining an approximation of the observable based on the approximation of the noiseless quantum signal and the classical signal.

12. The method of claim 11, the method further comprising:generating the classical signal via executing a classical algorithm on a classical processing device, wherein the classical signal is a noiseless classical signal and the noiseless signal is a linear sum of the noiseless quantum signal and the noiseless classical signal.

13. The method of claims 11 or 12, the method further comprising:executing a second version of the quantum circuit to generate a second noisy measurement of the observable, anddetermining the approximation of the noiseless quantum signal based on a difference between the first noisy measurement and the second noisy measurement of the observable.

14. The method of claim 13, the method further comprising:determining a noisy quantum signal based on the difference between the first noisy measurement and the second noisy measurement of the observable; anddetermining the approximation of the noiseless quantum signal based on the noisy quantum signal.

15. The method of claim 14, the method further comprising:rescaling the noisy quantum signal; anddetermining the approximation of the noiseless quantum signal based on rescaling the noisy quantum signal.

16. The method of any of claims 13-15, wherein the second version of the quantum circuit is configured to suppress the noiseless quantum signal.

17. A method for operating a quantum computing system (QCS), the method comprising:accessing a heuristic classical estimation of an observable of a quantum circuit; executing a first version of the quantum circuit to generate a first noisy signal that corresponds to a first noisy measurement of the observable;determining a hybrid estimate of the observable based on a first combination of the heuristic classical estimation and the first noisy signal.

18. The method of claim 17, the method further comprising:generating the heuristic classical estimation of the observable via executing a classical algorithm, via a classical processing device, wherein the classical algorithm scales polynomially with a system size.

19. The method of claim 18, the method further comprising:executing a second version of the quantum circuit to generate a second noisy signal that corresponds to a second noisy measurement of the observable, wherein the second version of the quantum circuit is configured to mimic the classical algorithm; anddetermining, via the classical processor device, the hybrid estimate of the observable based on a second combination of the heuristic classical estimation, the first noisy signal, and the second noisy signal.

20. The method of claim 19, the method further comprising:determining, via the classical processor device, the second combination of the heuristic classical estimation, the first noisy measurement of the observable, and the second noisy measurement of the observable by multiplying the heuristic classical estimation by a ratio of the first noisy signal to the second noisy signal.

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