Error mitigation for quantum computing

Abstract

One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS) that includes a set of qubits. The method includes generating a set of noisy data by repeatably measuring a noisy observable of a quantum circuit operating on the set of qubits. The set of noisy data and the noisy observable are subject to noise associated with the quantum circuit and each datapoint of the set of noisy data is associated with an eigenvalue of the noisy observable. The noisy data is modeled as a hybrid distribution, which includes a combination of a noiseless distribution and a noise-only distribution. Each datapoint of the set of noisy data is assumed to have a hidden label as coming from either the noiseless distribution or the noise-only distribution. This label is estimated for each datapoint, based on the eigenvalue associated with the datapoint, and a constructed model for any of the hybrid distribution, the noise-only distribution, and the noiseless distribution. This may require additional data to be taken from the QCS to model the noise-only distribution. A target quantity is determined based on either the model for the noiseless distribution or the datapoints combined with their attached labels.

Description

ERROR MITIGATION FOR QUANTUM COMPUTINGPRIORITY

[0002] This application claims priority to U. S. Provisional Application No 63 / 548,721, entitled ERROR MITIGATION FOR QUANTUM COMPUTING, filed on February 1, 2024, the contents of which of herein incorporated in their entirety.FIELD

[0003] The present disclosure relates generally to quantum computing and information processing systems, and more particularly to efficient quantum memories via shadow tomography.BACKGROUND

[0004] 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 |1〉 The “0” and “1” states of a digital computer are analogous to the |0〉 and |1〉 basis states, respectively of a qubit.SUMMARY

[0005] 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.

[0006] One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The method includes generating a set. of noisy data by repeatably implementing a quantum circuit on the QCS. Each data point of the set of noisy data corresponds to a sample from a noisy distribution. The noisy distribution is associated with an observable of a quantum state that corresponds to the quantum circuit. The noisy distribution includes a weighted combination of a noiseless distribution and a noise-only distribution. Eachdata point of the set of noisy data. That is, the set of noisy data s a labeled set of noisy data. A label for each data point indicates an estimate of whether the data point corresponds to a sample from the noiseless distribution or the noise-only distribution. A target quantity that is associated with the observable of the quantum state is estimated based on the labeled set of noisy data.

[0007] Other aspects of the present disclosure are directed to various systems, methods, apparatuses, non-transitory computer-readable media, computer-readable instructions, and computing devices.

[0008] 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.BRIEF DESCRIPTION OF THE DRAWINGS

[0009] 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:

[0010] FIG. 1 depicts an example quantum computing system according to example embodiments of the present disclosure.

[0011] FIG. 2A provides a schematic representation of quantum circuits being employed to prepare quantum states, according to various embodiments.

[0012] FIG. 2B shows a schematic for a noisy quantum experiment that includes a process of sampling from a noisy distribution, according to various embodiments.

[0013] FIG. 3 illustrates an example three-step protocol for estimating a target quantity corresponding to a noiseless quantum state, according to various embodiments.

[0014] FIG. 4A shows a method 400 for operating a quantum computing system, according to various embodiments.

[0015] FIG. 4B shows another method 420 for operating a quantum computing system, according to various embodiments.DETAILED DESCRIPTION

[0016] Example aspects of the present disclosure are directed to the mitigation of errors on noisy quantum devices. Currently quantum devices accumulate errors with a probability of around0.1-1% per gate in a quantum circuit. Without a fault- tolerant quantum computer, these errors must be mitigated to achieve high-accuracy quantum computing results. The embodiments include novel approaches to quantum error mitigation. Previous attempts at quantum error mitigation include symmetry verification, echo verification, probabilistic error cancellation, zero noise extrapolation, or learning from fermionic linear optics. However, all of these prior methods have either large biases or large sampling costs. These previous methods have been unable to achieve lower bounds on a possible error mitigation technique. Furthermore, previous error mitigation methods have been applicable only to some types of estimation problems, e.g., expectation value estimation.

[0017] To overcome the limitations of previous error mitigation strategies, the embodiments include methods, systems, and architectures for error mitigation based on generating a set of noisy data. Generating the set of noisy data is accomplished by repeatedly employing a quantum circuit (that is subject to noise) to prepare multiple copies (or instantiations) of a quantum state. Thus, at least some of the copies of the quantum state may be noisy copies. An observable of each prepared copy of the quantum state is measured. The measurements of the observable are accumul ated to generate the set of noisy data (which may be a classical data set). Each data point in the set of noisy data is assumed to be sampled from a noisy distribution associated with the observable and the quantum state. The noisy distribution includes a weighted linear combination of a noiseless distribution and a noise-only distribution. Thus, the noisy distribution may be a “hybrid distribution.” The weighted combination of the noisy distribution is weighted by a scalar quantity (referred to as a circuit fidelity parameter) that characterizes a level of noise that the quantum circuit is subjected to. Each data point in the set of noisy data is labeled with a label (e.g., each data point is assigned a label). The label indicates an estimate of whether the data point corresponds to a sample from the noiseless distribution or the noise-only distribution. Labeling the data points may be based on prior knowledge of the noiseless distribution, the noise-only distribution, and / or the circuit fidelity. This prior knowledge may be gained by various measurements on the quantum circuit (or related quantum circuits) that, at least partially characterize the noise that the circuit (or quantum logic gates implemented by the circuit) is subject to. Depending on the information available to the user, this labeling may only be able to be made probabilistically. These labels can be used to mitigate errors on the device. That is, a target quantity associated with the observable may be estimated. Although two specific implementations of this mitigation, one via a maximum-likelihood estimator specific to quantum phase estimation,and one via a Bayesian estimator specific to expectation value estimation are discussed herein, the embodiments are not limited to these two implementations.

[0018] Some embodiments consider a quantum algorithm. These embodiments enable preparing a quantum state i via a circuit U, measuring w in some basis, and estimating some quantity on the output. The quantum state may be a pure state or a mixed state, may represent a density matrix corresponding to (or associated with) the quantum state. In some embodiments, may represent a wavefunction corresponding to the quantum state. For reasons discussed below, may be referred to as a noiseless quantum state. In the absence of noise, a measurement of draws a measurement output m from a distribution P(m) = Tr[Emip], where Emis a corresponding element of a positive operator-valued measure (POVM) and m is an index (or label) corresponding to the possible measurement outcomes (e.g., eigenvalues). P(m) may be referred to as a noiseless distribution. To solve a computational task, the embodiments may repeatedly samples from this distribution P(m) and processes the output data to yield a target quantity φ = φ[P]. In practice, quantum computers are prone to error, so applying the noisy version of a circuit Λ ≠ U leads to a noisy quantum state ρ ≠ ψ. Throughout, U may be a noiseless unitary operator representing a noiseless quantum circuit, may be a noisy operator representing a noisy quantum circuit, and p may be a noisy quantum state. Similar to measuring the noiseless quantum state, measuring ρ draws from a distribution Q(m) = Tr[Emρ], where Q(m) is a noisy distribution. Error mitigation, as provided by some of the embodiments, includes solving a computational task given access to Q(m) ≠ P(m) (and perhaps related data), and a classical algorithm to estimate φ given access to P(m).

[0019] The embodiments represent the noisy distribution Q(m) (or the “hybrid” distribution) as a probabilistic weighted sum of the true (or noiseless) distribution P(m) and a ‘noise-only’ distribution R(m): Q(m) = F × P(m) + (1 − F) × R(m). Here, Fis a scalar quantity that represents a ‘circuit fidelity’ parameter. Drawing from the noisy distribution Q(m) (the hybrid distribution) can be modeled as a noisy quantum experiment (e.g., a quantum experiment that is subject to noise and / or errors). The quantum experiment is a two-step process; first a coin is tossed with probability F of landing heads; if the coin does land heads, it is assumed that a draw from the ‘noiseless’ distribution P(m) (in which case it can be said that the experiment ran ‘without error’) is performed, if the coin lands tails it is assumed that a draw from the ‘pure- noise’ (or noise-only) distribution Rpm) (in which it can be said that the experiment ran ‘with error’) is performed. As some quantum circuits are robust to some kinds of error (e.g. an error thatsits outside the light cone of an observable will not affect the measurement of the observable), the above may be adjusted to read Q(m) = F × P(m) + (1 − F) × R(m), where P(m) is any distribution that can be substituted for P(m') to yield an unbiased estimate of the target quantity d.

[0020] Examples of channels for which the above decompositions are physically motivated are amplitude and phase damping or Pauli noise channels. In these examples, F = Π_j (1 − p_j), where p_j is the error rate on the jth gate, where Pj is the error rate on the jth gate of the unitary circuit. Some embodiments employ methods (e.g., randomized circuit compiling methods) that can twirl a larger set of noise channels to stochastic Pauli noise (with potential overhead when using non-Clifford t.wo~qubit gates).

[0021] One example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS). The method includes generating a set of noisy data by repeatably implementing a quantum circuit on the QCS. Each data point of the set of noisy data corresponds to a sample from a noisy distribution. The noisy distribution is associated with an observable of a quantum state that corresponds to the quantum circuit. The noisy distribution includes a weighted combination of a noiseless distribution and a noise-only distribution. Each data point of the set of noisy data. That is, the set of noisy data is a labeled set of noisy data. A label for each data point indicates an estimate of whether the data point corresponds to a sample from the noiseless distribution or the noise-only distribution.target quantity that is associated with the observable of the quantum state is estimated. The target quantity is estimated based on the labeled set of noisy data.

[0022] Another example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS) that includes a set of qubits. The method includes generating a set of noisy data by repeatedly measuring a noisy observable of a quantum circuit operating on the set of qubits. The set of noisy data and the noisy observable are subject to noise associated with the quantum circuit and each datapoint of the set of noisy data is associated with an eigenvalue of the noisy observable. A hybrid distribution is generated based on the set of noisy data, wherein the hybrid distribution includes a combination of a noiseless distribution and a noise- only distribution. Each datapoint of the set of noisy data is assigned a label based on the eigenvalue associated with the datapoint.. The label for a datapoint associates the datapoint with at least one of the noiseless distribution or the noise-only distribution. An estimate is generated for each of the noiseless distribution and the noise-only distribution based on the hybrid distribution, the label assigned to each datapoint of the set of noisy data, and a circuit fidelity parameter that indicates the noise associated with the quantum circuit. A target quantity is determined based on the estimate for the noiseless distribution and the noise-only distribution.

[0023] Another example aspect of the present disclosure is directed to another method for operating a quantum computing system (QCS) that includes a set of qubits. The method includes generating a set of noisy data by repeatedly measuring a noisy observable of a quantum circuit operating on the set of qubits. The set of noisy data and the noisy observable are subject to noise associated with the quantum circuit and each datapoint of the set of noisy data is associated with an eigenvalue of the noisy observable. The noisy data is modeled as a hybrid distribution, which includes a combination of a noiseless distribution and a noise-only distribution. Each datapoint of the set of noisy data is assumed to have a hidden label as coming from either the noiseless distribution or the noise-only distribution. This label is estimated for each datapoint, based on the eigenvalue associated with the datapoint, and a constructed model for any of the hybrid distribution, the noise-only distribution, and the noiseless distribution. This may require additional data to be taken from the QCS to model the noise-only distribution. A target quantity is determined based on either the model for the noiseless distribution or the datapoints combined with their attached labels.

[0024] Aspects of the present disclosure provide a number of technical effects and benefits. For instance, the embodiments improve over the sampling costs of previous work, enable error mitigation for additional classes of quantum algorithms (those not based on estimators), and allow for various prior error mitigation techniques to be combined on the same footing.Quantum Computing Systems

[0025] 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.

[0026] 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 ofqubits (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. 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.

[0027] The type of multi-level quantum subsystems that the system 100 utilizes may vary. For example, in some cases it may be convenient to include one or more readout device(s) 114 attached to one or more superconducting 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 requiring qubits) may be used. Further examples of realizations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots or phosphorus impurity qubits.

[0028] 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 adjust the frequency of the qubits.

[0029] The quantum hardware 102 may further include readout devices 114 (e.g., readout resonators). Measurement results 108 obtained via measurement devices 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 102 through physical control parameters (e.g., microwave pulses) that are sent through wires included in the quantum hardware 102. Further examples of control devices include arbitrary waveform generators, wherein a DAC (digital to analog converter) creates the signal.

[0030] 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. 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 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.

[0031] In some implementations, the readout device(s) 114 can take advantage of a difference in the impedance for the |0〉 and |1〉 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.

[0032] 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.

[0033] 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 asuitable physical operation and measured via a suitable measurement operation performed on the measurement qubit.

[0034] 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.

[0035] 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 sy stems can be used without deviating from the scope of the present disclosure.Framework for Quantum Error Mitigation via Probabilistic Post-Selection

[0036] The embodiments are directed towards methods, systems, and architectures for providing error mitigation of the output of a quantum circuit. As used herein, a quantum circuit may include one or more single-qubit and / or one or more multi-qubit quantum logic gates. A quantum circuit receives (as input) one or more input quantum states encoded in one or more qubits. The quantum circuit operates on the one or more input quantum states (e.g., via the one or more quantum logic gates) and transforms the one or more input quantum states into one or more output quantum states. As such, a quantum circuit may be said to “prepare” an output quantum state of interest. The operations of a quantum circuit that operate to prepare a quantum state of interest may be unitary’ operations (or unitary transformations). Accordingly, a quantum circuit of the embodiments may be a unitary circuit. Note that throughout, the term “quantum state,” may refer to a “pure” state (e.g., as described by a wavefunction or a vector in a Hilbert space, e.g., a ket) or a “mixed” state (e.g., a probabilistic ensemble of multiple pure states). Both pure states and mixed states may be described via a density matrix. A density matrix of a pure state may include a single projection operator (onto the pure state), while a density matrix of a mixed state may include a weighted sum of projection operators, where the weights are based on the probabilities of the separate pure states in the ensemble and each projection operators corresponds to a separate pure state of the ensemble of pure states.

[0037] As noted above, a quantum circuit may be designed to prepare a quantum state of interest. A quantum computing system (QCS) may include a set of qubits and one or more quantum circuits. In practice, a QCS, including the qubits and the quantum circuits may be noisyGsuch that the prepared quantum state may be a “noisy” quantum state. That is, when run multiple times to prepare multiple copies of the output quantum state, some of the prepared copies may be "noise-corrupted" (or “noisy”) copies and other copies may be “noiseless” copies. Note that the multiple copies may be prepared in parallel (e.g., with multiple copies of the circuit) or serially (with the same circuit executed multiple times).

[0038] FIG. 2A provides a schematic representation of quantum circuits being employed to prepare quantum states, according to various embodiments. More particularly, FIG. 2A shows a noiseless circuit 200 (e.g., referred to as U for a unitary circuit). The noiseless circuit 200 receives an input state 202 (e.g., referred to a 0) as input. Noiseless circuit 200 performs one or more unitary (and noiseless) operations that transform the input state 202 into a “noiseless” output state 204 (e.g., referred to throughout as ) / / ). FIG. 2A also shows a “noisy” circuit 206 (e.g., referred to as / I) that also receives the input state 202. Noisy circuit 206 performs “noisy” operations that transform the input state 202 into a “noisy” output state 208 (e.g., referred to throughout as p). Note that the symbol y may refer to a density matrix characterizing either a noiseless pure or noiseless mixed quantum state (e.g., the noiseless output state 204). In some embodiments, y may refer to a wavefunction characterizing the noiseless quantum state. The symbol p may refer to a density matrix characterizing either a noisy pure or noisy mixed quantum state (e.g., the noisy output state 208). In some embodiments, p may refer to a wavefunction characterizing the noisy quantum state.

[0039] In the embodiments, a noisy quantum circuit may be implemented multiple times to generate multiple copies of the quantum state of interest. As noted above, some of the copies of the state of interest may be noiseless copies (e.g., noiseless output state 204) and some of the copies may be noisy copies (e.g., noisy output state 208). Classical data may be generated via measuring (along one or more bases) one or more observables of the multiple noiseless and noisy copies. In some embodiments, the measurements may be positive-operator valued measurements (POVMs). In other embodiments, at least some of the measurements may be projective measurements.

[0040] As described in more detail, the embodiments label classical data output (e.g., generated from the measurements of the noise-corrupted states and / or the noiseless states) as “noise-corrupted” (e.g., noisy) classical data, or “noiseless” classical data via knowledge of the noiseless and noisy distributions that underlie the one or more observables. This section discusses a mathematical framework utilized by the embodiments. The next section discussesimplementations for two non-limiting example applications. Note that these two example applications are non-limiting and the embodiments may be employed in other applications.

[0041] A mathematical framework employed by the embodiments assumes a quantum algorithm (e.g., a quantum algorithm that is implemented via a quantum circuit) that aims to prepare a quantum state of interest » (e.g., the noiseless output state 204) via a (unitary) quantum circuit U (e.g., the noiseless circuit 200). After the state is prepared via the circuit U, an observable associated with / is measured in one or more bases. One or more (target) quantities are estimated based on the output of the measurements. In the absence of noise, a measurement of / draws (or samples) a measurement output m from a "noiseless" distribution P(m):P(m) = Trace[Emψ] (1)where Emis the corresponding element of a positive operator-valued measure (POVM). Note that in an n-qubit system (e.g., a 2ndimensional quantum state), there is 2npossible measurement outcomes (and 2nelements of a POVM). Thus, in eq (1), (m = 0., 1, 2,.. 2n— 1), Thus, each possible measurement outcome may be labeled via an integer index: (m = 0, 1, 2,..., 2n— 1). Also, the random variable (e.g., m) in the noiseless distribution P(m) has 2npossible values. To solve a computational task (e.g., estimate a target quantity d corresponding to the quantum state of interest), one may repeatedly sample from this distribution P(m) and processes the output data to yield target quantity d = d[P] associated with a noiseless quantum state (e.g., ip).

[0042] As noted above, QCSs are prone to noise (e.g., resulting in errors) and thus a quantum circuit may be a noisy circuit. Throughout, a noiseless unitary circuit is referred to as U (e.g., noiseless circuit 200), while a noise-corrupted (or noisy) version of the circuit is referred to as A (e.g., noisy circuit 206), where Λ ≠ U. As shown in FIG. 2A, the noiseless version of the circuit (U) (e.g., noiseless circuit 200) prepares (or outputs) the noiseless state ψ (e.g., noiseless output state 204), while the noisy version of the circuit / I (e.g., noisy circuit 206) prepares (or outputs) the noisy (or imperfect) state (ρ) (e.g., noisy output state 208), where ρ ≠ ψ.Measurements of p draws from a “noisy” distribution Q(m)'.Q(m) = Trace[Emρ], (2)where Q(m) ≠ P(m). That is, Q( ) is a “noisy” distribution and P(m) is a “noiseless” distribution. In some embodiments, at least partial access to Q(m) and P(m) (as well as other information) is assumed and the task of error mitigation includes performing a classical algorithm to estimate φ. As discussed below, access to these distributions may include modeling the distributions based on prior knowledge (e.g., priors) or expectations of these distributions. In someembodiments, modeling these distributions may be based on experimental measurements made on the noisy quantum hardware.

[0043] The mathematical framework of the embodiments represents the noisy distribution Q(m) as a probabilistic sum of the noiseless distribution ( ) and a “noise-only” distribution R(m)Q(m) = F × P(m) + (1 − F) × R(m), (3)where 0 ≤ F ≤ 1 is a scalar quantity referred to as circuit fidelity. Note that when F = 1, the circuit is a noise-free circuit and when F = 0, the circuit generates pure noise (or noise-only) with no discernable signal. FIG. 2B shows a schematic for a noisy quantum experiment that includes a process 210 of sampling from a noisy distribution, according to various embodiments. A set of noisy data (e.g., set of noisy data 238) may be generated by repeating process 210 numerous times. As shown in FIG. 2B, process 210 for sampling (or drawing) from the noisy distribution Q(m) is a two-step process. More particularly, FIG. 2B demonstrates the preparation of the noisy output state 208 (e.g., p) as described in conjunction with FIG. 2A. That is, an input state 202 (e.g., 0) is provided to the noisy circuit 206 (e.g., Λ). The noisy circuit 206 transforms the input state 202 into the noisy output state 208.

[0044] In process 210, a sample from a noisy distribution (e.g., noisy distribution 232) is generated. The noisy distribution 232 is comprised of a combination (e.g., a weighted sum) of two sub-distributions: a noiseless distribution (e.g., noiseless distribution 234) and a noise-only distribution (e.g., noise-only distribution 236). The weighted sum of the two sub-distributions may be weighted via the circuit fidelity parameter (e.g., F).. As discussed below in the weighted sum, the noiseless distribution 234 may be weighted as F, and the noise-only distribution 236 may be weighted as (1-F). Each sample sampled from the noisy distribution 232 is added to a set of noisy data (e.g., set of noisy data 238). Thus, by repeatedly performing the noisy quantum experiment that includes process 210, a set of noisy data (e.g., set of noisy data 238) may be generated.

[0045] In the first step 220 of process 210, a stochastic determination is made as to when sampling from the noisy distribution, whether to sample from the noiseless distribution (e.g., noiseless distribution 234) of the noisy distribution 232 or the noise-only distribution (e.g., noise- only distribution 236) of the noisy distribution 232. The stochastic determination may be modeled as a flip (or toss) of a weighted “coin.” The weighted coin has a probability of F (e.g., p ~ F) to land on “heads” and a probability of p ~ (1 — F) to land on “tails.” Landing on “heads” indicates a sampling from the noiseless distribution (P) and landing on tails indicates a sampling from the noise-only distribution (R). In a non-limiting embodiment, the tossing the weighted coin may beimplemented classically via a call to a pseudo-random number generator function that returns a rational number (e.g., a floating point number) r from the uniform interval [0,1]. If 0.0 < r < F, then the coin toss is a “heads” toss and if F < r < 1.0, then the coin toss is a “tails” loss.

[0046] In the second step 220 of process 210, the noisy distribution 232 (e.g., Q(m) is schematically shown. The schematic of the noisy distribution 232 includes a weighted combination of the noiseless distribution 234 (e.g., P(m) and the noise-only distribution 236 (e.g., R(m)). As shown in Eq (3), the weighting of the combination of the noiseless distribution 234 and the noise-only distribution 236 is parameterized via the selection of the value of the circuit fidelity (e.g., F).

[0047] Thus, in the second step 230 of process 210 directed to the sampling from Q(m), if 0,0 < r < F, then a sample from the noiseless distribution 234 (e.g., P(m)) is drawn, and if F < r < 1.0, then a sample from the noise-only distribution 236 (e.g., R(m)) is drawn. The sample may be included in the set of noisy data 238. Each coin toss (and sampling from the respective distribution) may be referred to as an “experiment,” (e.g., a noisy quantum experiment). The experiment may be run multiple times. A single experiment that samples from the noiseless distribution 234 may be said to run “without error,” and a single experiment that samples from the noise-only distribution 236 may be said to a “with error.” As shown in FIG. 2B, any single experiment has a probability of 0.0 < p ~ F < 1.0 to run without error and a probability 0.0 < 1 — F < 1.0 to run with error. Thus, the set of noisy data 238 may be generated by performing the noisy quantum experiment multiple times.

[0048] In review of the noisy quantum experiment (or measurement), the first step 220 of process 210, a stochastic determination is made as to whether to sample from the noiseless distribution 234 (with probability F), or whether to sample from the noise-only distribution 236 (with probability (1-F)). The first step 220 may be hidden from the “observer” of the experiment. In the second step 230 of process 210, the corresponding distribution is sampled (e.g., the noiseless distribution 234 or the noise-only-distribution 236), and an element m = 0,...,2N−1, is returned and inserted into the set of noisy data 238. The “observer” of the experiment has access to the set of noisy data 238.

[0049] Some quantum circuits are robust to some kinds of error (e.g. an error that sits outside the light cone of an observable will not affect the measurement of the observable). As such, eq (3) may be rewritten as:Q(m) = F x P(m) + (1 - F) X ft(m) (4)where P̃(m) is any distribution that can be substituted for P(m) to yield an unbiased estimate of the target quantity φ.

[0050] Examples of error (or noise) channels for which the above decompositions are physically motivated include amplitude and / or phase damping, as well as Pauli noise channels. In these examples, F = flj (1 Pj), where p;- is the error rate on the jth gate of the unitary circuit. Some embodiments employ methods (e.g., randomized circuit compiling methods) that can twirl a larger set of noise channels to stochastic Pauli noise (with potential overhead when using non¬ Clifford two-qubit gates).

[0051] If R(m) in Eq. (3) contains no information about (i.e. the Fisher information of R(m) with respect to φ is 0), a hypothetically optimal error mitigation tool would label each sample from Q(m) as coming from either P(m) or R(m). In practice this may be difficult even if exact information about the P(m) and R(m) distributions are known.

[0052] FIG. 3 illustrates an example three-step protocol 300 for estimating a target quantity corresponding to a noiseless quantum state, according to various embodiments. Protocol 300 begins at step 302, where a set of noisy data (e.g., the set of noisy data 238 of FIG. 2B) is generated by repeatedly performing a noisy quantum experiment, where each run of the experiment samples from a noisy distribution Q(m) (e.g., the noisy distribution 232 of FIG. 2B). Various embodiments for generating the set of noisy data via repeatedly performing a noisy quantum experiment that samples from a noisy distribution are discussed at least in conjunction with FIG.2B. Each data point in the set of noisy data is a sample from the noisy distribution, which includes a weighted sum of a noiseless distribution P(m) (e.g., the noiseless distribution 234 of FIG. 2B) and a noise-only distribution R(m) (e.g., the noise-only distribution 236 of FIG. 2B). The weighted sum is weighted via the circuit fidelity parameter F. For instance, see eq (3).

[0053] In step 304, each sample of the set of noisy data is labeled (e.g., / ) as either being sampled from the noiseless distribution or the noise-only distribution. Labeling the samples from the set of noisy data may be based on prior knowledge of the noiseless distribution, the noise-only distribution, and / or additional data acquired from the noisy quantum circuit. Throughout, the noisy quantum circuit may be referred to as a noisy quantum device. For instance, the noise-only distribution may be approximated (or estimated) on the noisy quantum device, or by implementing other quantum circuits that can be used to infer a model for the noise-only distribution. In some embodiments, the labels may be either definite (e.g., either the binary' values of 1=0 or 1), or probabilistic (e.g., I E [0,1]). In some embodiments, rather than labeling the samples in the set of noisy data, parameters (e.g., 0) for the distributions may be learned (e.g., P(m 0)).

[0054] In step 306, the target quantity <p of the noiseless quantum state may be estimated (or determined) based on the labeled set of noisy data.Two Non-Limiting Examine Applications of Various Embodiments

[0055] One example application of the various embodiments includes applying one or more of the various methods, processes, steps, and / or protocols discussed throughout to a quantum phase estimation (QPE) algorithm. In one example form, a QPE algorithm performs time evolution on a system register controlled by an ancilla register (prepared in an input state). The ancilla register may then be Fourier transformed. Next, the ancilla register may be measured in the computational basis to obtain the phase 0. The resulting distribution depends on the input state, but assuming a system register contains an eigenstate, it is a peaked distribution parameterized by the phase P(m) = P(m| ). Thus, given access to the distribution P(m (j)), one option is to estimate 4* as the argmax of the log-likelihood, as shown in eq ( 5 ).<p ~ argmax^%jlogP^(mj), (5)where the sum over j is a sum over the sampled data (e.g., the set of noisy data). Following the three-step protocol 300 of FIG. 3, one method includes sampling J datapoints {mj} from Q(m) and then use the quantum device to faithfully sample K datapoints {mfc} from R(m. This can be achieved by assuming knowledge of individual noise channels, by repeating the noisy circuit with additional errors added probabilistically. A noisy circuit (A) (e.g., noisy circuit 206 of FIGS. SA¬ B) may be assumed to take on the form:where F is the circuit fidelity, pj is the probability of a single error occurring during gate j (where the corresponding process the corresponding process is denoted asis the probability of errors occurring during gate j and j′ (with corresponding process εj,j′. Some embodiments simulate:noting that the distribution drawn from following the action of Ê is precisely R(m). E can be implemented by inserting the error channels Ej, Ej′, etc. by hand with probability qj, qj′, and fine-tuning qj to achieve the above distribution. This may require the subtraction of additional distributions to compensate for some qj, being negative. This sampling may be further simplified by assuming that higher-order elements of the channel S contribute to the uniform distribution to J?, which reduces the need to simulate them. The value of φ may be selected and / or determined to maximize:which is a maximum likelihood estimator assuming no prior knowledge of R. Note that eq (8) can be generalized into a combination of multiple distributions (e.g., see eq (8a)). In eq (8a), the expansion coefficients (e.g., cj) are defined via eq (8b). In eq (8b), Λ is a noisy unitary representing a noisy circuit that can be executed (and characterized) on the noisy QCS andare “basis operations,” e.g., channels that can be run on the noisy QCS.. The values of the expansion coefficients in eq (8a) can be determined by measuring the noisy unitary and “fitting” the expansion of eq (8b) to the determination of the noisy unitary. A small constant may be added to regularize the distribution, resulting in a more robust algorithm (e.g., see eq (8c)).(8b)Λ = ΣjcjεjM - - — - - — >(8c)

[0056] A second non-limiting example of an application of the embodiments includes estimating an expectation value of some observable O from a quantum circuit operating on qubits. The definition P̃ → P may be expanded to include any error that lies outside the light cone of O. Such embodiments assume the ability to sample from the distribution R by the same construction as discussed above. Given K » N samples from R, an approximate model for R may be generated. One such model could be to take a Gaussian over the set of input parameters:where μ and Σ are the mean and covariance of the model to be fitted. J sample points (e.g., mj) may then be sampled from Q using the noisy quantum device. A similar model may be generated for Q. These models may take into account any underlying knowledge about P or R (e.g. a physical conservation law, or the energy of the output string under a target Hamiltonian). Then, Bayes’ rule (e.g., see FIG. 2B) may be followed to estimate the likelihood that each point mj is drawn from R. One such estimate of the likelihood that, each point m}is drawn from R is:Additional experimental data may be gathered, such as the confidence a QEC decoder has of correctly decoding an experiment, or the probability from a readout trace to have correctly measured a 0 or 1. The observable O may be estimated as:

[0057] Note that another possible estimator of O is:

[0058] However, in this case the numerator is the difference between two potentially large terms, which blows up the statistical noise. For a chosen estimator, eq. (11) avoids this in the situation where the overlap between R and P is small (allowing us to discriminate well between the two distributions).Methods

[0059] FIG. 4A-4B show flowcharts describing methods that may be implemented via a quantum computing system (e.g., quantum computing system 100 of FIG. 1), a classical computing system, or a combination thereof. More particularly, FIG. 4A shows a method 400 for operating aquantum computing system, according to various embodiments. The quantum computing system (QCS) may include a set of qubits (e.g., the plurality of qubits 120 of FIG. 1). Method 400 begins at block 402, where a set of noisy data (e.g., the set of noisy data 238 of FIG. 2B) is generated by repeatedly measuring a noisy observable of a quantum circuit operating (e.g., the noisy circuit 206 of FIGs. 2A-2B) on the set of qubits. The set of noisy data and the noisy observable are subject to noise associated with the quantum circuit. Each datapoint of the set of noisy data is associated with an eigenvalue (e.g., indexed via m) of the noisy observable. At block 404, a hybrid distribution (e.g., the noisy distribution 232 of FIG. 2B) is generated based on the set of noisy data. The hybrid distribution (e.g., Q(m)) includes a combination of a noiseless distribution (e.g., the noiseless distribution 234 of FIG. 2B) and a noise-only distribution (e.g., the noise-only distribution 236 of FIG. 2B). At block 406, each datapoint of the set of noisy data is assigned a label based on the eigenvalue associated with the datapoint. The label for a datapoint associates the datapoint with at least one of the noiseless distribution (e.g., P(m)) or the noise-only distribution (e.g., R(m)). At block 408, an estimate for each of the noiseless distribution and the noise-only distribution is generated based on the hybrid distribution, the label assigned to each datapoint of the set of noisy data, and a circuit fidelity parameter that indicates the noise associated with the quantum circuit. At block 410, a target quantity (e.g., d = d[P]) is determined based on the estimate for the noiseless distribution and the noise-only distribution.

[0060] FIG. 4B shows another method 420 for operating a quantum computing system, according to various embodiments. The quantum computing system (QCS) may include a set of qubits (e.g., the plurality of qubits 120 of FIG. 1). Method 420 begins at block 422, where a set of noisy data (e.g., the set of noisy data 238 of FIG. 2B) is generated by repeatably measuring a noisy observable of a quantum circuit operating (e.g., noisy circuit 206 of FIG. 2B) on the set of qubits, wherein the set of noisy data and the noisy observable are subject to noise associated with the quantum circuit and each datapoint of the set of noisy data is associated with an ei genvalue of the noisy observable. At block 424, the noisy data is modeled as coming from a hybrid distribution (e.g., the noisy distribution 232 of FIG. 2B), wherein the hybrid distribution (Q(m)) includes a combination of a noiseless distribution (e.g., the noiseless distribution 234 of FIG. 2B) and a noise- only distribution (e.g., the noise-only distribution 236 of FIG. 2B), and each datapoint of the set of noisy data has a hidden label as coming from the noiseless distribution (P(m)) or noise-only distribution At block 426, an estimate is generated for each of the hybrid distribution, the noiseless distribution and the noise-only distribution, and a circuit fidelity parameter (F) that indicates the noise associated with the quantum circuit, based on prior knowledge of the structureof the quantum circuit and the noise model and / or by taking additional data on the QCS. At block 428, each datapoint of the set of noisy data is labeled with a label based on the eigenvalue associated with the datapoint, wherein the label for a datapoint is an estimate of the hidden label. At block 430, a target quantity (e.g., d = d[P]) is determined based on at least one of the estimates for the noiseless distribution or the noisy datapoints and associated labels.Additional Embodiments

[0061] Another non-limiting embodiment includes a method for operating a quantum computing system (QCS) (e.g., the QCS 100 of FIG. 1). The method includes generating a set of noisy data (e.g., the set of noisy data 238 of FIG. 2B). The set of noise data is generated by repeatably implementing a quantum circuit (e.g., the noisy circuit 206 of FIGs. 2A-2B) on the QCS. Each data point of the set of noisy data corresponds to a sample from a noisy distribution (e.g., the noisy distribution 232 of FIG. 2B). The noisy distribution (Q(m)) is associated with an observable of a quantum state (e.g., the noiseless output state 204 and / or the noisy output state 208 of FIG. 2A). The quantum state corresponds to the quantum circuit. The noisy distribution includes a weighted combination of a noiseless distribution (e.g., the noiseless distribution 234 of FIG. 2B) and a noise-only distribution (e.g., the noise-only distribution 236 of FIG. 2B). The method includes labeling each data point of the set of noisy data. Thus, the set of noisy data is a labeled set of noisy data. A label assigned to each data point indicates an estimate of whether the data point corresponds to a sample from the noiseless distribution (P(m)) or the noise-only distribution A target quantity (e.g., d = φ[P]) that is associated with the observable of the quantum state is estimated. The target quantity is estimated based on the labeled set of noisy data.

[0062] The quantum circuit may be a noisy quantum circuit that is subject to quantum noise. The QCS includes a set of qubits (e.g., the plurality of qubits 120 of FIG. 1) and the quantum circuit includes a set of noisy quantum logic gates that operates on the set of qubits. The set of qubits may be a noisy set of qubits.

[0063] Each implementation of the quantum circuit may include performing a noisy quantum experiment. Repeatably implementing the quantum circuit includes performing a set of noisy quantum experiments. The quantum state may be a noisy quantum state (e.g., the noisy output state 208 of FIG. 2B) Performing each noisy quantum experiment of the set of noisy quantum experiments includes preparing the noisy quantum state by employing the quantum circuit to operate on a set of the qubits of the QCS. The observable of the noisy quantum state ismeasured in one or more bases. A data point is generated based on measuring the observable of the noisy quantum state. The data point is included in the set of noisy data.

[0064] The weighted combination of the noiseless distribution and the noise-only distribution is weighted by a circuit fidelity' scalar quantity (F). The circuit fidelity scalar quantity characterizes a level of noise in the quantum circuit. Labeling each data point of the set of noisy data set may be based on prior knowledge of at least one of noiseless distribution or the noise-only distribution. The prior knowledge of at least one of noiseless distribution or the noise-only distribution may be based on performing a set of measurements on the QCS that characterizes noise of the QCS. The set of measurements may be performed on the quantum circuit. In other embodiments, the set of measurements is performed on another quantum circuit implemented on the QCS.

[0065] Each label of the labeled set of noisy data may be a probabilistic label. The probabilistic label indicates a likelihood of whether the data point corresponds to the noiseless distribution or corresponds to the noise-only distribution. Each label of the labeled set of noisy data may be learned from a set of parameters that characterizes the noiseless distribution.Estimating the target quantity may include performing a quantum phase estimation (QPE) algorithm. Estimating the target quantity may include estimating an expectation value of the observable of the quantum state.

[0066] The quantum state may be a noiseless quantum state (e.g., noiseless output state 204 of FIG. 2A) The noiseless distribution (P(m) may be based on a trace of a product of a density matrix characterizing the noiseless quantum state and a positive operator-valued measure (POVM) associated with the observable (e.g., see eq (1)). The quantum circuit may be a noisy quantum circuit ( / !) that prepares a noisy quantum state (p). The noisy distribution (Q(m)) may be based on a trace of a product of a density matrix characterizing the noisy quantum state and a positive operator-valued measure (POVM) associated with the observable (e.g., see eq (2)).

[0067] Another embodiment includes a quantum computing system (QCS). The QCS includes one or more processors and one or more memory devices. The one or more processors includes a quantum processor (e.g., the quantum system 110) that includes a set of qubits (e.g., the plurality of qubits 120 of FIG. 1). The one or more memory devices store computer-readable instructions that when executed by the one or more processors cause the one or more processors to perform operations for operating the QCS. The operations include generating a set of noisy data by repeatably operating a quantum circuit that is implemented by the QCS on the set of qubits, wherein each data point of the set of noisy data corresponds to a sample from a noisy distributionthat is associated with an observable of a quantum state that is prepared by the quantum circuit, and wherein the noisy distribution includes a weighted combination of a noiseless distribution and a noise-only distribution. The operations further include labeling each data point of the set of noisy data such that the set of noisy data is a labeled set of noisy data, wherein a label for each data point indicates an estimate of whether the data point corresponds to a sample from the noiseless distribution or the noise-only distribution. The operations further include estimating a target quantity associated with the observable of the quantum state based on the labeled set of noisy data.

[0068] Another example aspect of the present disclosure is directed to a method for operating a quantum computing system (QCS) that includes a set of qubits. The method includes generating a set of noisy data by repeatedly measuring a noisy observable of a quantum circuit operating on the set of qubits. The set of noisy data and the noisy observable are subject to noise associated with the quantum circuit and each datapoint of the set of noisy data is associated with an eigenvalue of the noisy observable. A hybrid distribution is generated based on the set of noisy data, wherein the hybrid distribution includes a combination of a noiseless distribution and a noise- only distribution. Each datapoint of the set of noisy data is assigned a label based on the eigenvalue associated with the datapoint. The label for a datapoint associates the datapoint with at least one of the noiseless distribution or the noise-only distribution. An estimate is generated for each of the noiseless distribution and the noise-only distribution based on the hybrid distribution, the label assigned to each datapoint of the set of noisy data, and a circuit fidelity parameter that indicates the noise associated with the quantum circuit. A target quantity is determined based on the estimate for the noiseless distribution and the noise-only distribution.

[0069] Another example aspect of the present disclosure is directed to another method for operating a quantum computing system (QCS) that includes a set of qubits. The method includes generating a set of noisy data by repeatedly measuring a noisy observable of a quantum circuit operating on the set of qubits. The set of noisy data and the noisy observable are subject to noise associated with the quantum circuit and each datapoint of the set of noisy data is associated with an eigenvalue of the noisy observable. The noisy data is modeled as a hybrid distribution, which includes a combination of a noiseless distribution and a noise-only distribution. Each datapoint of the set of noisy data is assumed to have a hidden label as coming from either the noiseless distribution or the noise-only distribution. This label is estimated for each datapoint, based on the eigenvalue associated with the datapoint, and a constructed model for any of the hybrid distribution, the noise-only distribution, and the noiseless distribution. This may require additional data to be taken from the QCS to model the noise-only distribution. A target quantity is determinedbased on either the model for the noiseless distribution or the datapoints combined with their attached labels.

[0070] Implementations of the digital, classical, and / or quantum subject matter and the digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-implemented digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The term “quantum computing systems” may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0071] 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, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, 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. 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.

[0072] The terms quantum information and quantum data refer to information or data that is earned 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 all 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.

[0073] The term ‘ ata processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors 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.

[0074] 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..

[0075] 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 cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data.

[0076] 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 quantum processors, 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.

[0077] 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.

[0078] 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

[0079] 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-opticaldisks, or optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0080] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non¬ volatile digital and / or quantum 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; magneto-optical 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.

[0081] 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.

[0082] 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 sub-combination.

[0083] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all 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 requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0084] 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 require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. WHAT IS CLAIMED IS:

1. A method for operating a quantum computing system (QCS) comprising: generating a set of noisy data by repeatably implementing a quantum circuit on the QCS, wherein each data point of the set of noisy data corresponds to a sample from a noisy distribution that, is associated with an observable of a quantum state that corresponds to the quantum circuit, and wherein the noisy distribution includes a weighted combination of a noiseless distribution and a noise-only distribution,labeling each data point of the set of noisy data such that the set of noisy data is a labeled set of noisy data, wherein a label assigned to each data point indicates an estimate of whether the data point corresponds to a sample from the noiseless distribution or the noise-only distribution; andestimating a target quantity associated with the observable of the quantum state based on the labeled set of noisy data.

2. The method of claim 1, wherein the quantum circuit is a noisy quantum circuit that is subject to quantum noise.

3. The method of claim 1, wherein the QCS includes a set of qubits and the quantum circuit includes a set of noisy quantum logic gates that operates on the set of qubits.

4. The method of claim 3, wherein the set of qubits is a noisy set of qubits.

5. The method of claim 1, wherein each implementation of the quantum circuit includes performing a noisy quantum experiment such that repeatably implementing the quantum circuit includes performing a set of noisy quantum experiments.

6. The method of claim 5, wherein the quantum state is a noisy quantum state and performing each noisy quantum experiment of the set of noisy quantum experiments comprises:preparing the noisy quantum state by employing the quantum circuit to operate on a set of qubits of the QCS,measuring the observable of the noisy quantum state in one or more bases; generating a data point based on measuring the observable of the noisy quantum state; andincluding the data point in the set of noisy data.

7. The method of claim 1, wherein the weighted combination of the noiseless distribution and the noise-only distribution is weighted by a circuit fidelity scalar quantity that characterizes a level of noise in the quantum circuit.

8. The method of claim 1, wherein labeling each data point of the set of noisy data set is based on prior knowledge of at least one of noiseless distribution or the noise-only distribution.

9. The method of claim 8, wherein the prior knowledge of at least one of noiseless distribution or the noise-only distribution is based on performing a set of measurements on the QCS that characterizes noise of the QCS.

10. The method of claim 9, wherein the set of measurements is performed on the quantum circuit.

11. The method of claim 9, wherein the set of measurements is performed on another quantum circuit implemented on the QCS.

12. The method of claim 1, wherein each label of the labeled set of noisy data is a probabilistic label that indicates a likelihood of whether the data point corresponds to the noiseless distribution or corresponds to the noise-only distribution.

13. The method of claim 1, wherein each label of the labeled set of noisy data is learned from a set of parameters that characterizes the noiseless distribution.

14. The method of claim 1, wherein estimating the target quantity includes performing a quantum phase estimation (QPE) algorithm.

15. The method of claim 1, wherein estimating the target quantity includes estimating an expectation value of the observable of the quantum state.

16. The method of claim 1, wherein the quantum state is a noiseless quantum state and the noiseless distribution is based on a trace of a product of a density matrix characterizing the noiseless quantum state and a positive operator-valued measure (POVM) associated with the observable.

17. The method of claim 1, wherein the quantum circuit is a noisy quantum circuit that prepares a noisy quantum state and the noisy distribution is based on a trace of a product of a density matrix characterizing the noisy quantum state and a positive operator-valued measure (POVM) associated with the observable.

18. A method for operating a quantum computing system (QCS) that includes a set of qubits, the method comprising:generating a set of noisy data by repeatably measuring a noisy observable of a quantum circuit operating on the set of qubits, wherein the set of noisy data and the noisy observable are subject to noise associated with the quantum circuit and each datapoint of the set of noisy data is associated with an eigenvalue of the noisy observable,generating a hybrid distribution based on the set of noisy data, wherein the hybrid distribution includes a combination of a noiseless distribution and a noise-only distribution;assigning each datapoint of the set of noisy data a label based on the eigenvalue associated with the datapoint, wherein the label for a datapoint associates the datapoint with at least one of the noiseless distribution or the noise-only distribution,generating an estimate for each of the noiseless distribution and the noise-only distribution based on the hybrid distribution, the label assigned to each datapoint of the set of noisy data, and a circuit fidelity parameter that indicates the noise associated with the quantum circuit; and determining a target quantity based on the estimate for the noiseless distribution and the noise-only distribution.

19. A method for operating a quantum computing system (QCS) that includes a set of qubits, the method comprising:generating a set of noisy data by repeatably measuring a noisy observable of a quantum circuit operating on the set of qubits, wherein the set of noisy data and the noisy observable are subject to noise associated with the quantum circuit and each datapoint of the set of noisy data is associated with an eigenvalue of the noisy observable;modeling the noisy data as coming from a hybrid distribution, wherein the hybrid distribution includes a combination of a noiseless distribution and a noise-only distribution, and each datapoint of the set of noisy data has a hidden label as coming from the noiseless or noise- only distribution;generating an estimate for each of the hybrid distribution, the noiseless distribution and the noise-only distribution, and a circuit fidelity parameter that indicates the noise associated with the quantum circuit, based on prior knowledge of a structure of the quantum circuit and the noise model and / or by taking additional data on the QCS;assigning each datapoint of the set of noisy data a label based on the eigenvalue associated with the datapoint, wherein the label for a datapoint is an estimate of the hidden label; and determining a target quantity based on at least one of the estimates for the noiseless distribution or the noisy datapoints and associated labels.

20. A quantum computing system (QCS) comprising:one or more processors that include a quantum processor that includes a set of qubits;one or more memory devices, the one or more memory devices storing computer- readable instructions that when executed by the one or more processors cause the one or more processors to perform operations for operating the QCS, the operations comprising:generating a set of noisy data by repeatably operating a quantum circuit that is implemented by the QCS on the set of qubits, wherein each data point of the set of noisy data corresponds to a sample from a noisy distribution that is associated with an observable of a quantum state that is prepared by the quantum circuit, and wherein the noisy distribution includes a weighted combination of a noiseless distribution and a noise-only distribution;labeling each data point of the set of noisy data such that the set of noisy data is a labeled set of noisy data, wherein a label for each data point indicates an estimate of whether the data point corresponds to a sample from the noiseless distribution or the noise-only distribution; andestimating a target quanti ty associated with the observable of the quantum state based on the labeled set of noisy data.

21. A method for operating a quantum computing system (QCS) that includes a set of qubits, the method comprising:generating a set of noisy data by repeatably measuring a noisy observable of a quantum circuit operating on the set of qubits, wherein the set of noisy data and the noisy observable are subject to noise associated with the quantum circuit and each datapoint of the set of noisy data is associated with an eigenvalue of the noisy observable,modeling the noisy data as coming from a hybrid distribution, wherein the hybrid distribution includes a combination of a noiseless distribution and a noise-only distribution, and each datapoint of the set of noisy data has a hidden label as coming from the noiseless or noise- only distribution;generating an estimate for each of the hybrid distribution, the noiseless distribution and the noise-only distribution, and a circuit fidelity parameter that indicates the noise associated with the quantum circuit, based on prior knowledge of a structure of the quantum circuit and the noise model and / or by taking additional data on the QCS;assigning each datapoint of the set of noisy data a label based on the eigenvalue associated with the datapoint, wherein the label for a datapoint is an estimate of the hidden label; and determining a target quantity based on at least one of the estimates for the noiseless distribution or the noisy datapoints and associated labels.

Images