Quantum expectation readout error reduction

Abstract

Techniques are presented for reducing readout error of quantum expectation. A calibration component applies a first random Pauli gate to a qubit at a first output of a first circuit prior to a first readout measurement of the qubit. An estimation component applies a second random Pauli gate to the qubit at a second output of a second circuit prior to a second readout measurement of the qubit, and generates an error-reduced readout determination based on the first random Pauli gate applied to the qubit at the first circuit output and the second random Pauli gate applied to the qubit at the second circuit output. The calibration component determines calibration data based on the first readout measurement. The estimation component determines estimation data based on the second readout measurement. The estimation component determines a normalization scalar value based on the calibration data, determines an estimation scalar value based on the estimation data, and determines an error-reduced readout determination associated with a circuit of interest based on the normalization scalar value and the estimation scalar value.

Description

BACKGROUND

[0001] The present disclosure relates to quantum computing using quantum circuits. Quantum computing employs quantum physics to encode and process information rather than transistor-based binary digital technology. Quantum computing devices can employ qubits (also referred to as qubits), which operate according to quantum physics laws and can exhibit phenomena such as superposition and entanglement. The superposition principle of quantum physics allows a qubit to be in a state that partially represents both a value of “1” and a value of “0” at the same time. The entanglement principle of quantum physics allows qubits to be correlated to each other such that a combined state of the qubits cannot be decomposed into individual qubit states. For example, a state of a first qubit can depend on a state of a second qubit. Thus, quantum circuits can employ qubits to encode and process information in a manner that is distinct from transistor-based binary digital technology.

[0002] Quantum programming can be performed with quantum computing. Quantum programming can involve a process of assembling sequences of instructions, which can be referred to as quantum programs, that are capable of running on a quantum computer. Each quantum program can be associated with a set of quantum circuits. When a quantum program is executed, a result can be produced by the quantum computer. The performance of the quantum computer can depend not only on the fidelity of the unitary gates of the quantum circuits, but also on the fidelity of the quantum readout of the result. In conventional quantum computers, there can often be an undesirable amount of error in the quantum readout.

[0003] One general approach of some conventional readout-error reduction methods can be to use quantum detector tomography to estimate the transfer matrix A and apply the inverse to obtain an estimate of the ideal probability vector. Assuming that the readout error is independent for each qubit, one can determine where each 2-by-2 matrix A i may represent a classical bit-flip channel, and where certain elements of the matrix A i may represent probabilities of measuring 1 instead of 0, while for certain other elements the opposite is true. Although this conventional approach can be relatively easy to implement for practical applications, this conventional approach can fail to capture crosstalk and other dependencies between qubits, which is undesirable.

[0004] In some conventional approaches, a cumulatively extended-based representation can be employed to capture correlations between variables. However, such conventional approaches do not include or provide algorithms for using such representations in the context of error reduction.

[0005] In yet other conventional approaches, crosstalk can be incorporated in the model to some extent by considering pairs of qubit interactions. To capture crosstalk, traditional approaches consider a correlated noise model based on continuous-time Markov processes, and propose a technique to avoid explicit computation of the inverse transition matrix. The noise model in this traditional approach can be represented using only 2n 2 The general difficulties and drawbacks associated with matrix inversion across different conventional approaches can be that the resulting probability vectors may be non-physical: the vectors can contain negative entries or sum to a value other than one. While there can be some ways to ensure that the estimated probability vectors are physical, such as by estimating the probability vectors based on constrained optimization, some drawbacks of conventional techniques to estimate the probability vectors based on constrained optimization can be that they do not scale well with system size.

[0006] These and other drawbacks of conventional approaches to estimate quantum computation readout and attempt to reduce readout error can result in inefficiencies, ineffectiveness, and / or inaccuracies in estimating quantum computation readout and reducing readout error. SUMMARY

[0007] The following presents a summary to provide a basic understanding of one or more embodiments of the disclosed subject matter. This summary is not intended to identify key or important elements, or delineate any scope of certain embodiments or any scope of any claims. Its sole purpose is to present concepts in a simplified form as a prelude to the more detailed description that is presented later. In one or more embodiments described herein, systems, apparatuses, structures, computer-implemented methods, devices, and / or computer program products are provided that can reduce readout error of quantum expectation readout.

[0008] One embodiment is directed to a computer-implemented method comprising: applying, by a system operatively coupled to a processor, a first random Pauli gate to a qubit at a first output of a first circuit prior to a first readout measurement of the qubit. The computer-implemented method can further comprise: applying, by the system, a second random Pauli gate to the qubit at a second output of a second circuit prior to a second readout measurement of the qubit. This embodiment of the method can provide a number of advantages, including that the method can more efficiently and accurately estimate quantum computation readout.

[0009] In some embodiments, the computer-implemented method can further include determining, by the system, calibration information based on a first definition function and the first readout measurement measured at the first output of the first circuit; and determining, by the system, estimation information based on the first definition function and the second readout measurement measured at the second output of the second circuit. In particular embodiments, the computer-implemented method can further include determining, by the system, a normalization scalar value based on the calibration information and a second definition function; determining, by the system, an estimation scalar value based on the estimation information and the second definition function; and determining, by the system, an error-reduced readout determination associated with the circuit of interest based on the normalization scalar value and the estimation scalar value. These embodiments of the method can provide a number of advantages, including that the method can more efficiently and accurately estimate quantum computing readout results, and can perform operations that can be performed in an efficient and less complex manner.

[0010] In some embodiments, elements of the techniques described in connection with the disclosed methods can be embodied in various forms, such as a system, a computer program product, or another form.

[0011] According to another embodiment, a system includes a memory that stores computer executable components; and a processor that is operatively coupled to the memory, the processor to execute the computer executable components. The computer executable components can include a calibration component to apply a first random Pauli gate to a qubit at a first output of a first circuit prior to a first readout measurement of the qubit. The computer executable components can also include an estimation component to apply a plurality of pairs of random Pauli gates to the qubit prior to a second readout measurement of the qubit associated with a second circuit, including applying a second random Pauli gate to the qubit at a second output of the second circuit. Such embodiments of the system can provide a number of advantages, including that the method can more efficiently and accurately estimate quantum computing readout results.

[0012] In particular embodiments, the system can further include that the calibration component is capable of determining calibration data based on a first defined function and the first readout measurements about the first random Pauli gate measured at the first output of the first circuit, and the estimation component is capable of determining estimation data based on the first defined function and the second readout measurements about the pairs of random Pauli gates measured at the second output of the second circuit. In some embodiments, the system can further include that the estimation component determines a normalization scalar value based on the calibration data and a second defined function, determines an estimation scalar value based on the estimation data and the second defined function, and generates an error-reduced readout determination associated with the circuit of interest based on the normalization scalar value and the estimation scalar value. Such embodiments of the system can provide a number of advantages, including that the system can more efficiently and accurately estimate quantum computing readout results, and can perform operations that can be performed in an efficient and less complex manner.

[0013] In some embodiments, elements described in connection with the disclosed systems can be embodied in various forms, such as a computer-implemented method, a computer program product, or another form.

[0014] These and other features will become apparent from the following detailed description of illustrative embodiments thereof, which is to be read in connection with the accompanying drawings. BRIEF DESCRIPTION OF DRAWINGS

[0015] Figure 1 A block diagram illustrating an example non-limiting system that can desirably reduce readout errors associated with readout results produced by a quantum computer, in accordance with various aspects and embodiments of the disclosed subject matter, is shown;

[0016] Figure 2 , Figure 3 and Figure 4 A block diagram depicting an example circuit that can be used to facilitate producing error-reduced readout results, in accordance with various aspects and embodiments of the disclosed subject matter, is shown;

[0017] Figure 5 A plot depicting example diagonalization masks for 12 qubits, in accordance with various aspects and embodiments of the disclosed subject matter, that are obtained by averaging the outer product of a respective number of random commutation vectors d q ;

[0018] Figure 6 and Figure 7a flowchart illustrating an example non-limiting method that can desirably reduce readout errors associated with readout results produced by a quantum computer, in accordance with various aspects and embodiments of the disclosed subject matter;

[0019] Figure 8 and Figure 9 a flowchart illustrating another example non-limiting method that can desirably reduce readout errors associated with readout results produced by a quantum computer, in accordance with various aspects and embodiments of the disclosed subject matter;

[0020] Figure 10 a block diagram illustrating an example non-limiting operating environment in which one or more embodiments described herein can be facilitated. DETAILED DESCRIPTION

[0021] The following detailed description is merely illustrative and is not intended to limit embodiments and / or the application or uses of embodiments. Furthermore, there is no intention to be bound by any expressed or implied information presented in preceding Background or Summary sections or in the Detailed Description section.

[0022] One or more embodiments will now be described, by way of example only, with reference to the accompanying drawings in which identical reference numerals are used across various drawings to refer to identical elements. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a more thorough understanding. It is apparent, however, that one or more embodiments can be practiced without these specific details.

[0023] Quantum programming can involve a process of assembling sequences of instructions, which can be referred to as quantum programs, that can be run on a quantum computer. Each quantum program can be associated with a set of quantum circuits. When a quantum program is executed, a result (e.g., an estimate) can be produced by the quantum computer. The performance of the quantum computer can depend, to a large extent, not only on the fidelity of the unitary gates of the quantum circuits, but also on the fidelity of the quantum readout of the result. In conventional quantum computers, there can often be an undesirable amount of error in the quantum readout and / or it can be inefficient to estimate or produce the readout result.

[0024] Some conventional readout-error reduction methods can use quantum detector tomography to estimate the transfer matrix A and apply the inverse to obtain an estimate of the ideal probability vector. Assuming that the readout errors are independent for each qubit, one can determine where each 2-by-2 matrix A i may represent a classical bit-flip channel, and where the matrix A iCertain elements of the can represent probabilities of measuring 1 instead of 0, while for certain other elements the opposite is true. While such conventional approaches can be relatively easy to implement for practical applications, such conventional approaches can fail to capture crosstalk and other dependencies between qubits, which is undesirable.

[0025] In some conventional approaches, a cumulatively extended representation can be employed to capture correlations between variables. However, such conventional approaches do not include or provide algorithms for using such representations in the context of error reduction.

[0026] In yet other conventional approaches, crosstalk can be incorporated in the model to some extent by considering pairwise qubit interactions. To capture crosstalk, conventional approaches consider a correlated noise model based on continuous-time Markov processes, and propose a technique to avoid explicit computation of the inverse transition matrix. The noise model in this conventional approach can be represented using only 2n 2 General difficulties and drawbacks associated with matrix inversion across different conventional approaches can be that the resulting probability vectors may be non-physical: the vectors can contain negative entries or sum to values other than 1. While there can be some ways to ensure that the estimated probability vectors are physical, such as by estimating the probability vectors based on constrained optimization, some drawbacks of conventional techniques to estimate the probability vectors based on constrained optimization can be that they do not scale well with system size.

[0027] It can be desirable to be able to reduce the impact of readout errors, including multi-qubit correlated and state-dependent errors, particularly in a practical and efficient manner. The disclosed subject matter can be implemented to produce solutions to all or at least some of these problems of conventional quantum computing and readout of quantum computing results, including introducing robust, practical, ideally implementable protocols that can ideally reduce readout errors and the impact of readout errors in state and process tomography using circuit randomization (e.g., partial state tomography and partial process tomography). The disclosed subject matter can provide unbiased estimates of readout results, and can eliminate asymmetries in readout errors. The techniques and protocols of the disclosed subject matter can be dynamically adjusted to make them resilient to temporal variations in noise associated with quantum computing, which can be quite useful for the success of any reduction scheme on near-term devices.

[0028] To this end, various aspects and embodiments herein relate to techniques for reducing readout errors of quantum expectation. The disclosed subject matter can include a readout management component (RMC) that is capable of reducing readout errors of quantum expectation associated with quantum computing. The RMC can include a calibration component that can apply a first random Pauli gate (or a corresponding first Pauli operator) to a qubit component (also referred to herein as a qubit) at a first output of a first circuit prior to a first readout measurement of the qubit or the first circuit. As to a second circuit, which can include a third circuit that can be a circuit of interest, the RMC can further include an estimation component that can apply a second random Pauli gate (or a corresponding second Pauli operator) to the qubit at a second output of the second circuit prior to a second readout measurement of the qubit or the second circuit, and can generate an error-reduced readout determination (e.g., a readout result that can have an ideally reduced error) associated with the circuit of interest based on the first random Pauli gate applied to the qubit at the first output of the first circuit and the second random Pauli gate applied to the qubit at the second output of the second circuit. For example, as to state tomography (e.g., partial state tomography), such an estimation process can be utilized. To facilitate such an error-reduced readout determination, the calibration component can determine calibration data based on the first readout measurement and a first definition function, and the estimation component can determine estimation data based on the second readout measurement and the first definition function. The estimation component can determine a normalization scalar value based on the calibration data and a second definition function, and can determine an estimation scalar value based on the estimation data and the second definition function. The estimation component can determine the error-reduced readout determination (e.g., the error-reduced readout result) associated with the circuit of interest based on (e.g., as a function of) the normalization scalar value and the estimation scalar value.

[0029] In other embodiments, for example, with respect to process tomography (e.g., partial process tomography), during estimation of a process, instead of utilizing only a random Pauli gate applied at an output of a circuit, the RMC can utilize multiple pairs of random Pauli gates and apply the multiple pairs of random Pauli gates to the circuit, as more fully described herein. For example, the RMC employing a calibration component can apply a first random Pauli gate (or corresponding Pauli operator) to a qubit at a first output of a first circuit prior to a first readout measurement of the qubit or the first circuit. With respect to a second circuit, which can include a third circuit that can be a circuit of interest, the estimation component can apply multiple pairs of random Pauli gates (or corresponding Pauli operators) to a qubit associated with the second circuit prior to a second readout measurement of the qubit or the second circuit, including applying a second random Pauli gate to the qubit or the circuit of interest at a second output of the second circuit and applying a third random Pauli gate to the qubit or the circuit of interest at an input of the circuit of interest. The estimation component can generate an error-reduced readout determination associated with the circuit of interest based on the first random Pauli gate applied to the qubit or the first circuit and based on the multiple pairs of random Pauli gates applied to the qubit or the second circuit, as more fully described herein.

[0030] These and other aspects and embodiments of the disclosed subject matter will now be described with reference to the drawings.

[0031] Figure 1 A block diagram of an example, non-limiting system 100 that can desirably reduce readout errors associated with readout results produced by a quantum computer is shown in accordance with various aspects and embodiments of the disclosed subject matter. The system 100 can include a quantum computer component 102, which can include various quantum devices, quantum circuits, and / or other components. The quantum devices can include, for example, qubit components (also referred to herein as qubits). The quantum computer component 102 can be programmed and a desired quantum circuit 104 (including qubits and other quantum devices, circuits, and components) can be formed, for example, based on a set of instructions (e.g., an assembled sequence of instructions) that can be input to and run (e.g., executed) on the quantum computer component 102 to create and operate the desired quantum circuit 104, where the structure of the quantum circuit 104, and operations (e.g., quantum operations) performed by the quantum circuit 104, can be based on the set of instructions. In response to execution of a quantum program (that includes or is associated with the set of instructions and / or that includes input data or parameter data) and based on operation of the quantum circuit 104 of such a quantum program, the quantum computer component 102 can produce results (e.g., data results), which can also be referred to as readout results or readout determinations. The quantum computer component 102 can present (e.g., communicate or send) these results as output.

[0032] Traditionally, there can be undesirable readout errors associated with readout results output by quantum computers, as described herein. It can be desirable to be able to reduce the impact of readout errors, including multi-qubit dependent and state dependent errors, particularly in a practical manner. The disclosed subject matter can provide solutions to these and / or other problems with traditional quantum computing and quantum computing result readout.

[0033] To facilitate desirably reducing errors in readout results, and to do so in a fast and efficient manner, system 100 can include a readout management component (RMC) 106 that can be associated with (e.g., communicatively connected to) quantum computer component 102. RMC 106 can desirably (e.g., efficiently, quickly, and optimally) manage the production of readout results to reduce (e.g., decrease or minimize) readout errors. For example, RMC 106 can manage the production of readout results to reduce readout errors in partial state tomography and partial process tomography using randomization associated with a circuit according to defined readout management standards. In performing this, RMC 106 can provide unbiased estimates of readout results and can eliminate asymmetry in readout errors. The error reduction techniques employed by RMC 106 can enable and can implement dynamic adjustments to make the techniques resilient to temporal variations in noise that would otherwise cause readout errors, which can be useful and desirable in achieving success of reduction schemes on near-term devices. The error reduction techniques employed by RMC 106 are also desirably not dependent on a particular noise model.

[0034] According to various embodiments, the RMC 106 can utilize various estimation protocols to facilitate reducing readout errors associated with readout results produced by the quantum computer component 102. The estimation protocols can include, for example: an acquire data protocol (also referred to as Protocol AcquireData), which can specify a process for sampling and acquiring data and performing measurements of outputs (e.g., measurement responses) of circuits under various conditions; a first protocol (also referred to as Protocol 1 or a partial state tomography related protocol) and a second protocol (also referred to as Protocol 2 or a partial process tomography related protocol), each of which can utilize the acquire data protocol to generate error-reduced readout results (e.g., error-reduced readout determinations or error-reduced readout estimates or averages) associated with a desired circuit (e.g., a circuit of interest), as more fully described herein. For example, for readout results of the quantum computer component 102 associated with partial state tomography, Protocol AcquireData and Protocol 1 can be utilized. For example, for readout results of the quantum computer component 102 associated with partial process tomography, Protocol AcquireData and Protocol 2 can be utilized. To facilitate implementation of the protocols, the RMC 106 can include a calibration component 108, which can perform a calibration process, the calibration process can produce calibration data that can be used to reduce readout errors and can provide a benchmark associated with a circuit, and an estimation component 110, which can perform an estimation process, the estimation process can produce estimation data that can be used with the calibration data to provide readout results that can be ideally reduced in error, as more fully described herein.

[0035] Referring to Figure 2 , Figure 3 and Figure 4 (along with Figure 1 ), Figure 2 , Figure 3 and Figure 4 depict block diagrams of example circuits that can be used to facilitate producing error-reduced readout results, according to various aspects and embodiments of the disclosed subject matter. Figure 2 FIGS. 1-3 illustrate block diagrams of example circuits that can be used to facilitate producing error-reduced readout results, according to various aspects and embodiments of the disclosed subject matter. p and P q) of an example circuit 200. Circuit 200 can include a circuit of interest (C) 202, which can be a circuit (e.g., a quantum circuit) that can be used to determine or generate a readout result (e.g., a readout determination or estimate) in response to input data. Circuit 200 can include or employ various desired components and circuits of quantum computer component 102 (e.g., quantum components and circuits) to perform desired operations (e.g., quantum operations) on data (e.g., input data or other data). Circuit 200 can also include Pauli operators that can perform Pauli operations on data (e.g., input data or other data), including a Pauli operator (P p ) 204 that can be associated with (e.g., located at and / or connected to) an input of circuit 202 and a Pauli operator (P q ) 206 that can be associated with (e.g., located at and / or connected to) an output of circuit 202. The respective Pauli operators can be associated with respective values p and q, and can perform respective (e.g., different or unique) Pauli operations on data. The respective Pauli operators can also be associated with and / or can correspond to respective Pauli gates (e.g., Pauli gates employed in circuits such as circuit 200) and / or respective Pauli matrices.

[0036] Circuit 200 can also include desired qubits 208 (e.g., of quantum computer component 102), which can be associated with (e.g., part of or connected to) circuit 202 (where a set of qubits including one or more qubits 208 can be represented in circuit 200 by a diagonal line (e.g., a slash mark) through a horizontal line, as illustrated in circuit 200). RMC 106 can measure a response of circuit 200 (e.g., a response to input data) at an output 210 of circuit 200.

[0037] Figure 3 depicts a block diagram of an example circuit 300 that can include a circuit of interest (C) and a Pauli operation indexed by an integer q, in accordance with various aspects and embodiments of the disclosed subject matter. Circuit 300 can include circuit of interest (C) 202, Pauli operator P q 206, qubits 208 (where a set of qubits including one or more qubits 208 can be represented in circuit 300 by a diagonal line through a horizontal line, as illustrated in circuit 300), and / or other components (e.g., quantum components). Because P p , p = 0 and P0= I, P pIt can actually not affect or change the response of circuit 300, therefore, P p Not shown or not part of circuit 300, or at least not explicitly shown or not part of circuit 300. Similar to Figure 2 The RMC 106 can measure the response of the circuit 200 (e.g., response to input data) at the output 210 of the circuit 300.

[0038] Figure 4 A block diagram of an example circuit 400 according to various aspects and embodiments of the disclosed subject matter is shown. Example circuit 400 may include Pauli operations as indexed, where p = 0 and P0 = 1, and where C = 1, which could cause circuit 400 to actually employ Pauli operations indexed by an integer q. Circuit 300 may include the Pauli operator (P... q )206, qubits 208 (wherein a group of qubits including one or more qubits 208 can be represented in circuit 400 by a diagonal line crossing a horizontal line, as shown in circuit 400), and / or other components (e.g., quantum components). Because for P p Since p = 0 and P0 = 1, and because C = 1, and C can actually have no effect on or change the response of circuit 400, therefore C is not shown or is not part of circuit 400, or at least is not explicitly shown or is not part of circuit 400. Similar to... Figure 2 The RMC 106 can measure the response of the circuit 200 (e.g., response to input data) at the output 210 of the circuit 400.

[0039] According to various embodiments, with respect to state tomography (e.g., partial state tomography), calibration component 108 may implement protocols AcquireData and Protocol 1 to facilitate the generation of desired readout results (e.g., error-reduced readout determinations or estimates). As part of protocols AcquireData and Protocol 1, calibration component 108 may initially utilize circuitry 400, where p = 0 and P0 = 1, and where C = 1, which may result in the circuitry effectively employing Pauli arithmetic indexed by integer q.

[0040] According to AcquireData and Protocol 1, as part of the calibration process, calibration component 108 may randomly sample a first Pauli operator (e.g., a first random Pauli operator) P from a set of available Pauli operators based on a corresponding random value (e.g., a randomly generated number) associated with the corresponding random Pauli operator (and / or the corresponding Pauli gate). q A subset of 206. In some embodiments, calibration component 108 may uniformly sample a first random Pauli operator P from the set of available Pauli operators. q206. In other embodiments, the calibration component 108 can sample the first random Pauli operator P q 206. According to the defined readout management standard, the Pauli operators P q 206 can be a desired number N, where N can be a desired integer value.

[0041] The RMC 106 can employ a random number generator (RNG) 112 to generate random numbers for any of the operations described herein that utilize random numbers. Depending on the desired RNG algorithm, the RNG 112 can be a true random number generator that can generate true random numbers or a pseudo-random number generator that can generate pseudo-random numbers. Respective values (e.g., numbers) can be associated with (e.g., linked or mapped to) respective Pauli operators of the set of available Pauli operators (and / or corresponding Pauli gates), and where information related to associating (e.g., linking or mapping) respective numbers with respective Pauli operators can be stored in and retrieved from the data store 114 to facilitate determining which Pauli operator is associated with which value.

[0042] The calibration component 108 can apply (e.g., in corresponding instances) the first random Pauli operator (or corresponding Pauli gate) P q 206 to the qubit 208 at the output of the circuit 400 prior to a first readout measurement of the qubit 208 or the circuit 400. The calibration component 108 can employ the measurement component 116 of the RMC 106 to measure a respective first response (e.g., applied to the initial state |0>) at the output of the circuit 400 based on the respective first random Pauli operator (e.g., applied in corresponding instances to the respective measurement) and input data applied to the circuit 400. The first response can be a first readout measurement. According to the defined readout management standard, the calibration component 108 can perform a desired number M of readout measurements (e.g., one or more first readout measurements) for each of the first random Pauli operators P q 206. The calibration component 108 can store the respective first readout measurements and associated respective first values (e.g., respective q-values) in the data store 114, where the respective first values can be associated with (e.g., linked, mapped, or appended to) the respective first readout measurements and the respective first random Pauli operators P q 206. The calibration component 108 can store the respective first readout measurements and associated respective first values (e.g., respective q-values) in the data store 114, where the respective first values can be associated with (e.g., linked, mapped, or appended to) the respective first readout measurements and the respective first random Pauli operators P q206). In some embodiments, the respective first value can correspond to a respective random number used to determine and select the respective first random Pauli operator.

[0043] According to protocol AcquireData and protocol 1, as part of the estimation process, estimation component 110 can utilize Figure 3 circuit 300, where circuit 300 can include circuit of interest 202 and Pauli operators P q 206. Estimation component 110 can randomly sample (e.g., uniformly sample or non-uniformly sample) a second Pauli operator (e.g., a second random Pauli operator) P q 206 from the set of available Pauli operators based on a respective random value associated with the respective random Pauli operator (and / or corresponding Pauli gate). Estimation component 110 can apply the respective Pauli operator in the second random Pauli operator (e.g., P q 206) to (e.g., in the respective instance) qubit 208 or circuit of interest 202 at the output of circuit 300 prior to a second readout measurement of qubit 208 or circuit of interest 202. Estimation component 110 can employ measurement component 116 to measure a respective second response at the output of circuit 300 based on the respective second random Pauli operator (e.g., applied in the respective instance to the respective measurement) and the input data applied to circuit 300. The respective second response can be a respective second readout measurement. Estimation component 110 can store the respective second readout measurement and an associated respective second value in data repository 114, where the respective second value can be associated with (e.g., linked to, mapped to, or appended to) the respective second readout measurement and the respective second random Pauli operator (e.g., P q 206). In some embodiments, the respective second value can correspond to a respective random number used to determine and select the respective second random Pauli operator.

[0044] According to protocol AcquireData and protocol 1, calibration component 108 or estimation component 110 can determine and / or generate calibration data based on (e.g., from) the first readout measurement measured at the output of circuit 400 and the first defined function, as more fully described herein. Estimation component 110 can determine and / or generate estimation data based on the second readout measurement measured at the output of circuit 300 and the first defined function, as more fully described herein. Estimation component 110 can determine and / or generate a normalized scalar value based on (e.g., from) the calibration data and the second defined function, as more fully described herein. Estimation component 110 can determine and / or generate an estimated scalar value based on the estimation data and the second defined function, as more fully described herein. RMC 106 can include a calculator component 118 that can be used by calibration component 108, estimation component 110, or other components of RMC 106 to perform various calculations in connection with various operations and various protocols, as described above and as more fully described herein.

[0045] Estimation component 110 can determine and / or generate a readout result (e.g., an error-reduced readout determination) associated with circuit of interest 202 based on (e.g., from) the normalized scalar value and the estimated scalar value, as more fully described herein. The readout result can be an estimated (e.g., unbiased estimated) or expected readout result (e.g., a readout result determined and generated by quantum computer component 102 in connection with RMC 106 and processed by RMC 106) in which any readout error is ideally reduced (e.g., reduced, minimized, or substantially eliminated). The readout result can be, for example, an expected error-free readout determination or estimate (e.g., a readout result that can have a value (e.g., a quantum expectation) that can be expected in the absence of error).

[0046] RMC 106 can provide (e.g., communicate or produce) the error-reduced readout result as an output, which can be presented or displayed by interface component 120 (e.g., a display component including a display screen and an interface, and / or an audio component including an audio interface). Interface component 120 can present or display the readout result. Interface component 120 can also receive input data, quantum program information (e.g., instructions), and / or other information that can be processed by RMC 106 and / or provided to quantum computer component 102 to facilitate execution of a quantum program and generation of a readout result.

[0047] According to various embodiments, with respect to process tomography (e.g., partial process tomography), calibration component 108 can implement protocol AcquireData and protocol 2 to facilitate generating desired readout results (e.g., error-reduced readout determinations or estimates). As part of protocol AcquireData and protocol 2, calibration component 108 performs the same or similar calibration processes as employed with respect to protocol 1 as more fully described herein, or can use calibration data and / or calibration results from the calibration processes performed in connection with protocol 1 (if protocol 1 has been performed prior thereto). For example, calibration component 108 can randomly sample a subset of first Pauli operators (e.g., first random Pauli operators) P q 206, can apply the first random Pauli operators to qubits 208 at the output of circuit 400 prior to a first readout measurement of qubits 208 or circuit 400, can measure respective first responses (e.g., first readout measurements) at the output of circuit 400 (e.g., applied to initial state |0>) based on the respective first random Pauli operators and input data applied to circuit 400, and can store the respective first readout measurements and associated respective first values in data repository 114 as more fully described herein.

[0048] According to protocol AcquireData and protocol 2, as part of an estimation process, estimation component 110 can utilize Figure 2 circuit 200, where circuit 200 can include circuit of interest 202 and Pauli operators P p 204 and P q 206, where Pauli operators P p 204 can be applied to the input of circuit 200, and where Pauli operators P q 206 can be applied to the output of circuit 200. Estimation component 110 can randomly sample (e.g., uniformly sample or non-uniformly sample) a plurality of pairs of Pauli operators (e.g., P p 204 and P q 206) from the set of available Pauli operators based on respective random values associated with respective random Pauli operators (and / or corresponding Pauli gates), including a subset (e.g., N) of second Pauli operators (e.g., second random Pauli operators) 206 and a subset (e.g., N) of third Pauli operators (e.g., third random Pauli operators) 204.

[0049] The estimation component 110 can apply (e.g., in corresponding instances) pairs of random Pauli operators to the qubits 208 or the circuit 202 of interest. For example, prior to a second readout measurement of the qubits 208 or the circuit 200, the estimation component 110 can apply a second random Pauli operator P q 206to the qubits 208 or the circuit 202 of interest (e.g., in corresponding instances) at an output of the circuit 200, and can apply a third random Pauli operator P p 204to the qubits 208 or the circuit 202 of interest (e.g., in corresponding instances) at an input of the circuit 200. The estimation component 110 can employ the measurement component 116 to measure a respective second response at the output of the circuit 200 based on the respective second random Pauli operator (P q 206)(e.g., applied to the respective measurement in corresponding instances), the respective third random Pauli operator (P p 204)(e.g., applied to the respective measurement in corresponding instances), and input data applied to the circuit 200. The respective second response can be a respective second readout measurement. The estimation component 110 can store the respective second readout measurement, an associated respective second value (e.g., a q-value) and a respective third value (e.g., a p-value) in the data repository 114. The respective second value and the respective third value can be associated with (e.g., linked to, mapped to, or appended to) the respective second readout measurement, where the respective second value (e.g., a q-value) can be associated with the second random Pauli operator (P q 206), and where the respective third value (e.g., a p-value) can be associated with the third random Pauli operator (P p 204).

[0050] According to the protocol AcquireData and the protocol 2, the calibration component 108 or the estimation component 110 can determine and / or generate calibration data based on (e.g., from) the first readout measurement measured at the output of the circuit 400 and the first defined function, as more fully described herein. The estimation component 110 can also determine and / or generate estimation data based on the second readout measurement measured at the output of the circuit 200 and the first defined function, as more fully described herein. The estimation component 110 can also determine and / or generate a normalization scalar value based on (e.g., from) the calibration data and a second defined function, as more fully described herein. The estimation component 110 can also determine and / or generate an estimation scalar value based on the estimation data and the second defined function, as more fully described herein.

[0051] The estimation component 110 can determine and / or generate a readout result (e.g., an error-reduced readout determination) associated with the circuit of interest 202 based on (e.g., as a function of) the normalized scalar value and the estimated scalar value, as more fully described herein. The readout result can be an estimated (e.g., unbiased estimated) or an expected readout result (e.g., a readout result determined and generated by the quantum computer component 102 in conjunction with the RMC 106 and processed by the RMC 106) in which any readout error is ideally reduced (e.g., decreased, minimized, or substantially eliminated). The readout result can be, for example, an expected error-free readout determination or estimate. The RMC 106 can provide (e.g., communicate or produce) the error-reduced readout result as an output. For example, the interface component can present or display the error-reduced readout result.

[0052] According to various embodiments, the RMC 106 can also include (as depicted) or be associated with a processor component 122 that can work in conjunction with other components (e.g., the calibration component 108, the estimation component 110, the RNG 112, the data repository 114, the measurement component 116, the calculator component 118, the interface component 120, or other components) to facilitate performance of various functions of the RMC 106. The processor component 122 can employ one or more processors, microprocessors, or controllers that can process data related to circuits (e.g., quantum circuits), qubits, quantum components or devices, Pauli operators, Pauli gates, Pauli matrices, calibration processes, estimation processes, functions, algorithms (e.g., algorithms as indicated or defined by processes, protocols, methods, and / or techniques as described herein; and / or quantum algorithms), quantum logic, defined readout management standards, traffic flows, policies, protocols, interfaces, tools, and / or other information to facilitate operation of the RMC 106, as more fully disclosed herein, and control data flow between the RMC 106 and other components (e.g., the quantum computer component 102, quantum programs, data storage devices, user equipment or endpoint devices, or other computing or communication devices) associated with (e.g., connected to) the RMC 106.

[0053] Further to the data repository 114, the data repository 114 can store data structures (e.g., user data, metadata), code structures (e.g., modules, objects, hashes, classes, procedures), or instructions, circuits (e.g., quantum circuits), qubits, quantum components or devices, Pauli operators, Pauli gates, Pauli matrices, calibration processes, estimation processes, functions, algorithms (e.g., algorithms as indicated or defined by the processes, protocols, methods, and / or techniques described herein; and / or quantum algorithms), quantum logic, defined readout management standards, traffic flows, policies, protocols, interface-related information, and / or other information to facilitate control of operations associated with the RMC 106. In an aspect, the processor component 122 can be functionally coupled (e.g., by a memory bus) to the data repository 114 for storage and retrieval of information desirable for the functionality of the calibration component 108, estimation component 110, RNG 112, data repository 114, measurement component 116, calculator component 118, interface component 120, or other components and / or substantially any other operational aspect of the RMC 106.

[0054] These and other aspects and embodiments of estimation and readout error reduction techniques and estimation protocols in relation to the disclosed subject matter will now be further described.

[0055] A primary component for successful execution of quantum algorithms can be the ability to access results through measurement. One of the important challenges of quantum computing can be dealing with readout errors. The disclosed subject matter can enable reduction of readout errors in the computation of expectation values of Pauli observables, which arise in a wide range of applications from partial tomography of quantum states and processes to electronic structure determination using a variational quantum eigensolver (VQE). In this setting, individual measurements do not have to be corrected as used, for example, in quantum error correction and random number generation.

[0056] The measurement output of a quantum circuit can be characterized by an ideal probability vector p. Noisy readout can typically be represented by a classical noisy graph. For an n-qubit system, this graph can be represented by a 2 n times 2 n matrix A with entries A i , j in the matrix can represent the probability of measuring i instead of j. The noisy probability vector can thus be given by a linear transformation Some general way of conventional readout-error reduction methods can be to estimate the transition matrix A using quantum detector tomography and apply the inverse to obtain an estimate of the ideal probability vector: Assuming that the readout error is independent for each qubit, one can determine where each 2 by 2 matrix A imay represent a classical bit-flip channel,

[0057]

[0058] where r i may represent the probability of measuring 1 instead of 0, while s i is the opposite. While this approach can be relatively easy to implement for practical applications, this approach can fail to capture crosstalk and other dependencies between qubits. In theory, crosstalk can be captured by using the full A matrix, but this is not practical for all but the smallest systems, considering the calibration circuitry and memory requirements of just estimating and storing the exponential number of matrices. For larger systems, a reduced representation or model of the transfer matrix can be desirable (e.g., wanted or needed). In some conventional approaches, a cumulatively extended based representation can be employed to capture correlations between variables. However, such conventional approaches do not include or provide algorithms for using such representations in the context of error reduction. In other conventional approaches, crosstalk can be incorporated in the model to some extent by considering pairwise qubit interactions. To capture crosstalk, conventional approaches consider a correlated noise model based on continuous-time Markov processes, and propose a technique to avoid explicit computation of the inverse transfer matrix. The noise model in this conventional approach can be represented using only 2n 2 parameters. A general difficulty and drawback associated with matrix inversion across different conventional approaches can be that the resulting probability vector may be non-physical: the vector can contain negative entries or sum to a value other than 1. There can be different ways to ensure that the estimated probability vector is physical. As an example, the noise process can be modeled by a doubly stochastic matrix that can be obtained using gate-set tomography. Alternatively, constrained least squares or maximum likelihood estimation, or iterative Bayesian unfolding can be used. However, some drawbacks of conventional techniques to estimate the probability vector based on constrained optimization can be that these conventional techniques do not scale well with system size.

[0059] To overcome these and other problems associated with traditional methods for error reduction, the disclosed subject matter can employ techniques for readout-error reduction of values determined or estimated by a quantum computer that can involve quantum benchmarking. Unlike traditional methods, the techniques of the disclosed subject matter do not estimate a probability vector. Rather, the techniques of the disclosed subject matter (e.g., employed by RMC 106) can diagonalize a Pauli readout transition matrix that expresses the transition between the Pauli-z components of a system state p and their measurement, which can allow RMC 106 to form unbiased estimates of these components to within statistical uncertainty. Rather than directly using the A matrix, the techniques of the disclosed subject matter can diagonalize the transition matrix under a Hadamard transform. Like all current methods, the disclosed subject matter can operate under the assumption of accurate state preparation of calibration.

[0060] With respect to the techniques and methods of the disclosed subject matter, consider a system with n qubits and order the set of Pauli operators so that for P q may represent the unique Pauli operator. The Pauli-z operator can be assigned the index so that when read from right to left, the Pauli string represents may be in dictionary order (e.g., the operator on the first qubit can vary the fastest). The identity operator can have the index 0, and the singleton can be defined as Finally, the disclosed subject matter can express the Pauli-x operator as may represent the unique Pauli operator. The Pauli-z operator can be assigned the index

[0061] Any unitary operator U can be expressed in terms of its Pauli transition matrix T U The elements of the transition matrix may be defined as:

[0062]

[0063] so that

[0064]

[0065] For any state p, given an initial state

[0066]

[0067] It can be of interest to measure two quantities. The first quantity can estimate the Pauli-z component P i of the final state , i.e.

[0068]

[0069] The second quantity can take into account individual elements of the Pauli transfer matrix T U (i,j) throughout. It can be assumed that all measurements are done on a computational basis, and the initial state can be as given in equation (4). This can mean that only the Pauli-z component can be accessed in equations (2) and (5). To access the other components, the unitary U can be augmented with an appropriate basis change if desired (e.g., wanted or needed).

[0070] Further regarding basis changes, it should be noted that a basis change can be performed by applying a particular gate to a circuit, where for a first setting, a last part of the circuit can be (1) a basis change B, (2) a Pauli gate P q applied, and (3) a readout value (e.g., a readout estimate). When the basis change B is a Clifford operator (which can be a typical case), applying the basis change B and following with the Pauli gate P q can be equivalent to applying some Pauli gate P s and following with the basis change B. Given the Pauli gate P q , the Pauli gate P s can be effectively determined. This can mean that the following second setting can be equivalent to the first setting, where the second setting is: (1) applying a Pauli P s , (2) a basis change B, and (3) a readout value. Given the Pauli gate P q or the Pauli gate P s , the corresponding Pauli gate P s or Pauli gate P q can be determined or computed based on the basis change B, respectively. In the same or similar manner, this can also apply to the initial state, such that a Pauli gate (e.g., a random Pauli gate) following a circuit of interest can be equivalent to a first part (e.g., a Clifford operator) of some circuit of interest following a Pauli gate and a second part of the circuit of interest. The disclosed subject matter can include the indicated or applied settings (e.g., circuit settings or arrangements) involving basis changes and any and all equivalent types of settings involving such basis changes.

[0071] Further regarding the estimation protocol of the disclosed subject matter, to estimate a quantity of interest, the RMC 106 can run (e.g., execute) various instances of a circuit 200 of Figure 2 The circuit 200 can be parameterized by the Pauli indices p and q and the operator C (or a circuit C that can implement the operator C). It can be assumed that the identity operator can be made less complex, which can result in a circuit 300 of Figure 3 and a circuit 400 of Figure 4 ​circuit 400, as more fully described herein. The general procedure for obtaining data (e.g., protocol AcquireData) can be as follows:

[0072] Protocol AcquireData(S p , C, N) q

[0073] 1 : Initialize empty data set D

[0074] 2: For i = 1,..., N, perform

[0075] 3: Uniformly sample p p from S

[0076] 4: Uniformly sample q q from S

[0077] 5: Run circuit Figure 2 with parameters p, q, and operator C

[0078] 6: Record measurement m and add (p, q, m) to D

[0079] 7: Return D

[0080] Each measurement can be represented as an integer m, such that the least significant bit in the binary representation can correspond to the first qubit and the most significant bit can correspond to the last qubit. For classical post-processing of the data, a function can be defined as:

[0081]

[0082] where μ(a, b) can have a value of 1 if the Pauli P a and P b commute, and -1 if the Pauli P a and P b do not commute. The scalar value H m，i may represent the element at row m and column i of the unnormalized Hadamard matrix, which can be given by:

[0083]

[0084] The RMC 106 (e.g., employing the calibration component 108 and the estimation component 110) can utilize Protocol 1 for estimating Tr(P i ρ) for the Pauli-z operator in Equation (5), where Protocol 1 can include the following operations:

[0085] Protocol 1 ​

[0086] 1.

[0087] 2.

[0088] 3. Return Estimate

[0089] Note that the RMC 106 can reuse data obtained by the RMC 106 in operations 1 and 2 of protocol 1 (e.g., calibration data determined by the calibration component 108 and estimation data determined by the estimation component 110 ) to evaluate quantities in operation 3 of protocol 1 (e.g., the return estimate operation) for different values of the parameter In some embodiments, to reduce complexity, the RMC 106 can set the number of samples (e.g., sampling of the Pauli operators) in each of the two data sets to N, where N can be a desired integer. In other embodiments, the RMC 106 can use or select to use a different number of samples for each or some of these operations of protocol 1. The estimate T U (i,j) (e.g., as determined by the estimation component 110) in equation (2) can follow a similar approach and can be given by protocol 2 as follows:

[0090] Protocol 2

[0091] 1.

[0092] 2.

[0093] 3. Return Estimate

[0094] As before, data from operations 1 and 2 (e.g., calibration data determined by the calibration component 108 and estimation data determined by the estimation component 110 ) can be reused (e.g., by the estimation component 110 or other components of the RMC 106) to evaluate T U (i,j) for various in operation 3 (e.g., the return estimate operation of protocol 2). Because the data acquired in operation 1 can be independent of the choice of U, this data can be shared for use in both protocols (e.g., protocol 1 and protocol 2), and can be reused (e.g., by the RMC 106) for different operators U. As described more fully herein, in some cases, the parameter is replaced with in the data acquisition.It can or may be beneficial.

[0095] The derivation of the various aspects and characteristics of the disclosed subject begins with a certain notation, where 1 can represent a vector of suitable size containing only 1s, e i The i-th column of the identity matrix I can be represented by h, and correspondingly, the i-th column of the Hadamard matrix H can be represented by h. i =He i Also note that H can have inverse H. -1 =2 -n H is a symmetric real matrix. Based on the revealed order of Pauli operators, the Pauli transition matrix T is... U Top left 2 n Multiply by 2 n The block can contain the transition coefficients between Pauli-z operators, and this matrix can be represented by τU. Next, the function Z of the mapping density operator ρ can be defined as containing the weights of each Pauli-z operator of length 2. n vector:

[0096]

[0097] The disclosed topic can use an unscaled trace, meaning that for an initial state ρ0, Z(ρ0) = 1. Applying Z to the conjugate of ρ0 and operator U satisfies... The selected Pauli sort of the disclosed topic also allows the measurement probability vector corresponding to state ρ to be concisely written as p = H -1 Z(ρ). The disclosed subject (e.g., RMC 106 or other components of the disclosed subject) can be modeled by applying a transition matrix A, which gives the effective probability. The topics disclosed will also be described in this paper on how to handle more general readout errors.

[0098] To see the effect of adding the random Pauli operator, define a vector of commutative values. and the corresponding diagonal matrix D g =diag(d q Z(P) g ρP q ) = D g Z(ρ) holds, and based on the properties of the Pauli-x subset, for any

[0099]

[0100] Therefore, Pauli P p The combination of terms with the scalar μ(i, p) on average allows all terms in ρ0 except for one of the Pauli-z terms to be zeroed. Figure 2Circuit 200 (C=U) and Pauli matrix P p and P q It can have a noise measurement probability vector, which can be written as:

[0101]

[0102] Let M = HAH -1 And define the function:

[0103]

[0104] Based on the disclosed topic, p = 0 can be fixed, which can give P p =°, and define

[0105]

[0106] Alternatively, it can be The expected results were achieved:

[0107]

[0108] The transition matrix of the identity operator I can be obtained from τ I (i, j) = δ i，j Given, and from which f can be derived I (i)=M i，i The disclosed subject (e.g., RMC 106 or another component) can be obtained (e.g., determined or calculated) by scalar division (e.g., using equation (9) below) accordingly, yielding (e.g., determining or calculating) the value of the reduction in readout error for equations (5) and (2):

[0109]

[0110] As long as M i，i ≠0.

[0111] For a more general approach, Z can be replaced by a function C from which all Pauli coefficients can be extracted. The restricted transition matrix τ U It can be obtained by the full transition matrix T U Replaced. Can be applied by 2 n Multiply by 4 n A linear graph (e.g., from the RMC 106 or other components of the disclosed subject) yields an ideal noise-free measurement, wherein the first 2 n Each column contains a Hadamard matrix, and the remaining terms can be zero. In a typical setting, it can be envisioned that for some... Tr(P k ρ0) ≠ 0. Similarly, measurements can or may potentially be affected by terms not in the Pauli-z group. This can be achieved by adding terms from the Pauli subgroup (e.g., RMC 106). of the disclosed subject matter can filter out all terms that do not exchange with all elements in In the particular case corresponding to the Pauli-z group, this can mean that the disclosed subject matter (e.g., RMC 106) can filter out all terms outside of this group because it is the largest exchange subgroup. For the initial state, RMC 106 can thus have P p multiplied by a random Pauli-z matrix. This can be equivalent to sampling p from instead of from Likewise, the disclosed subject matter (e.g., RMC 106) can filter out all terms not in the Pauli-z group by sampling q from -1 A, where Λ is a quantum noise channel. This can be equivalent to (AH -1 ΛH)H -1 = A' H -1 Thus, the disclosed subject matter (e.g., RMC 106) can allow quantum noise to be modeled as a classical noise channel.

[0112] The goal of Protocols 1 and 2 can be to estimate the quantities in Equation (9). If these quantities are written in the form x / y, it can be seen that the protocol employed by RMC 106 can work by generating estimates and and returning Now the sample complexity of the protocol can be considered. That is, what value of N can the disclosed subject matter (e.g., RMC 106) choose so that with probability at least 1 - δ, the final estimate can be off the exact value by at most ε. Before doing this, the accuracy of the estimates can be considered in the case that x and y can be estimated to within an error of at most a.

[0113] Lemma IV.1. Let x, y be such that 0 ≤ |x| ≤ |y| ≤ 1. Given estimates where and so that 0 ≤ a ≤ |t| / 2. Then,

[0114]

[0115] Proof. Without loss of generality, assume x, y ≥ 0. For a small enough a, Taylor series expansion around zero can show that in the worst case:

[0116]

[0117] ​​In the last inequality, the disclosed subject matter can use the fact that x / y≤1 and l-α / y≥1 / 2. The disclosed subject matter (e.g., RMC 106 or other component) can similarly derive a lower bound to obtain a given result.

[0118] Thus, the disclosed subject matter can obtain the following sample complexity:

[0119] Theorem IV.2. With probability at least 1-δ, protocols 1 and 2 (e.g., RMC 106 employing protocols 1 and 2) can estimate equations (5) and (2), respectively, when the number of samples N satisfies the following for a fixed error at most ∈:

[0120]

[0121] Proof. Protocols 1 and 2 can obtain data and use the function in equation (6) to estimate different quantities. For a fixed i and j, each term in the sum can be viewed as an independent ±1 sample from a particular distribution depending on U, which can be marginalized over the Pauli indices p and q. For the error in the estimated quantities, the disclosed subject matter can apply Hoeffding’s inequality, which states that given independent random variables X i , deviating from the expected value satisfies

[0122]

[0123] It can be desirable (e.g., wanted) to ensure that the probability of deviating from expectation by more than or equal to α is bounded by δ / 2. Using the union bound can lead to the conclusion that the enumerator and the denominator are at least 1-δ in probability that they are within α of their expectations. Bounding the failure probability in equation (10) above by δ / 2 gives the following sufficient condition:

[0124]

[0125] The disclosed subject matter can choose (e.g., pick) α so that the final estimate can be ∈ accurate. From Lemma IV.1, it can be seen that taking 4α / y≤∈ can be sufficient, where y=f I (i) = M i，i . Substituting α=∈M i，i / 4 into equation (11) can give the desired result.

[0126] Regarding the number of circuit instances, for a given q, the term D q MD q can be written as M and the outer product The outer product has the remarkable property that the diagonal elements are always 1, independent of the signs in d q For a random sampling or Each off-diagonal value has the same chance of being positive or negative 1, and thus can have an expected value of zero. When the number of qubits n is small, the disclosed subject matter (e.g., RMC 106 or another component) can iterate over all possible q values and obtain an exact diagonalization of M. For larger values of n, this becomes intractable, and the disclosed subject matter can therefore only approximately diagonalize M, as shown in Figure 5 Figure 5 is a plot of example diagonalization masks 500 for 12 qubits, obtained by averaging outer products of random swap vectors d q according to various aspects and embodiments of the disclosed subject matter. Diagonalization masks 500 can include: diagonalization mask 502 for 12 qubits, obtained by averaging outer products of 30 random swap vectors d q ; diagonalization mask 504 for 12 qubits, obtained by averaging outer products of 100 random swap vectors d q ; diagonalization mask 506 for 12 qubits, obtained by averaging outer products of 1000 random swap vectors d q ; and diagonalization mask 508 for 12 qubits, obtained by averaging outer products of 3000 random swap vectors d q . For calibration, the disclosed subject matter (e.g., RMC 106) can estimate e i M1= M i，i + e i M(1 - e i ). To suppress the second term, it can be desirable for RMC 106 to sample over enough circuit instances. For an actual estimate of M i，i itself, it can be desirable for RMC 106 to sample enough times without considering circuit instances. At this point, it can be noted that the actual process can depend on the probability vector ​measurements, and thus, it can be desirable (e.g., useful or suitable) for the RMc 106 to sample each circuit a sufficient number of times (e.g., a sufficient number of times that can satisfy a defined readout management criterion, as indicated or described herein by the disclosed subject matter).

[0127] The disclosed subject matter can give bounds on the number of circuit instances for or desiring to i，i Estimating the number of circuit instances to a given accuracy gives bounds. Given the disclosed subject matter can multiply by an approximate diagonal mask, the bounds can depend in part on the largest off-diagonal element in M. The disclosed subject matter can show how this corresponds to properties of the transition matrix A and can investigate these properties for different types of transition matrices. When sampling individual elements of U It can be desirable (e.g., useful or suitable) to approximate to zero all coefficients except one of the coefficients in Z(p0) when sampling individual elements of Z(p0). Properties of the Pauli transition matrix can be investigated to determine to what extent it can be desirable for the disclosed subject matter to reduce these elements. The final estimate (e.g., a final estimate of a readout result) can be given by a ratio of two quantities, and thus it can be considered how estimation errors in these quantities can affect the result. Note that in this part of the disclosed subject matter, for clarity, the disclosed subject matter can work with the full matrix representation; and as shown in the disclosed subject matter and described in other parts, the processing itself can be performed based on individual elements.

[0128] For the disclosed subject matter can expect the number of circuit instances to approximate diagonalize M, the disclosed subject matter can have the following results:

[0129] Theorem IV.3. Given k random sample values and an index set Define

[0130] and Then, as long as simultaneously satisfy with probability at least 1 - δ for all and

[0131] Proof. Let i be any element in Given independent random variables X i From Hoeffding’s inequality, it follows that deviation from the expected value satisfies

[0132]

[0133] For the scaling of off-diagonal elements, the disclosed subject matter (e.g., RMC 106) can uniformly sample X from {-1, 1}. If one can ensure that each element is scaled by at most a factor of ∈b, then there can be an additive term of at most ∈b i，i in the estimation of M b . For the estimation of M i，i itself, the disclosed subject matter can apply equation (13) where X follows a suitable distribution over [-1, 1] and the maximum bias is ∈ a . Using the joint constraint on the off-diagonal elements in a row, the disclosed subject matter can obtain the condition:

[0134]

[0135] If β = 0, the disclosed subject matter can choose ∈ a = ∈ and let ∈ b → ∞. Using the joint constraint on a row in can give a sufficient number of circuit instances of .

[0136] For the more general case where β ≠ 0, the disclosed subject matter can choose ∈ a = ∈ b , which can simplify the condition of equation (14) to

[0137]

[0138] To satisfy ∈ a + ∈ b β ≤ ∈, the disclosed subject matter can choose ∈ a ≤ ∈ / (1 + β). In combination with the joint constraint obtained by multiplying the left-hand side of equation (14) by the number of elements in a set or group of , this can give the sample complexity stated in equation (12).

[0139] On the other hand, it is noted that diagonalizing a quantum noise channel using Pauli twirls can follow exactly the same principles as the disclosed subject matter for diagonalizing M. The disclosed subject matter (e.g., RMC 106 or other component) can utilize the desired modification to Theorem IV.3 to determine the number of desired (e.g., wanted) circuits to ensure that all off-diagonal noise terms are bounded by ∈.

[0140] With respect to example transition matrices, for a given transition matrix A, the disclosed subject matter can define a corresponding transformed matrix M = HAH -1 as the readout transition matrix of the Pauli-z operator. It can be seen that M -1 = HA as long as the inverse of A exists.-1 H -1 For convex combinations of two error channels, i.e., A = μΑ1+ (1 - μ)Α2and μ ∈ [0, 1], the disclosed subject matter can make M = μΜ1+ (1 - μ)Μ2. This directly generalizes to convex combinations of any number of transition matrices.

[0141] As a basic example of a transition matrix, consider the case where the outcome of each qubit is independently flipped with some probability r. The transition matrix for a single qubit can be given by equation (1) with r = s, and combined into a global transition matrix The corresponding Pauli readout transition matrix can have a particularly basic structure:

[0142]

[0143] In this case, since M is already diagonal, the disclosed subject matter does not have to shrink the off-diagonal elements. Thus, the disclosed subject matter is sufficient to select arbitrary but fixed values for g instead of sampling them. Selecting g = 0 can make the resulting circuit less complex. For simplicity, assume that all probabilities r l are equal to r, from equation (16) it follows that the diagonal elements M i，i can directly relate to the weights of the Pauli-z operator P i . For each σ i term in P z , the disclosed subject matter can have a multiplication term (1 - 2r). The diagonal term of P i with k non-identity terms can be given by (1 - 2r) k . The term (1 - 2r) k can have a lower bound of 1 - 2kr, which can mean that for 30 qubits measured with a 1% flip probability, the diagonal elements in M are still at least 0.4. In the noiseless case, the bit flip probability can be zero, and the disclosed subject matter can obtain A = M = I.

[0144] The transition matrix for the disclosed subject matter measuring only zeros can be given by A = e0e T and the corresponding matrix In the case of measuring each outcome with equal probability regardless of the state, the disclosed subject matter can make A = 2 -n ee T and While they are not realistic by themselves, the disclosed subject matter can use these matrices, e.g., in convex combinations with other transition matrices. A good but not very realistic example of a transition matrix that can be completely invertible but can provide some difficulty can be the following permutation matrix:

[0145]

[0146] If the disclosed subject matter can have approximately inverse then the disclosed subject matter can adjust the disclosed scheme to work with instead of This form of pre-conditioning (e.g., by the RMC 106 or other component of the disclosed subject matter) can help increase the number of diagonal elements in M or reduce the number of off-diagonal elements, but potentially can be computationally expensive.

[0147] With respect to certain practical considerations, in most of the discussion thus far, ideal state preparation has been assumed. Instead of p0= |0><0|, only For calibration, this can mean that after diagonalizing M, the disclosed subject matter can obtain the vector

[0148]

[0149] instead of m. If it is assumed that the state preparation of the qubits is independent and each qubit / is initialized to the state The disclosed subject matter can make

[0150] Under this assumption, this can mean that if the disclosed subject matter can estimate a l value, the disclosed subject matter (e.g., the RMC 106 or other component) can incorporate this information in equation (17) to better estimate m. In addition to improving calibration, knowledge of p can also allow the RMC 106 to better estimate individual elements in the Pauli transmon matrix T U However, for Pauli observables, due to the mixing of terms in T U , the disclosed subject matter can at most be able to obtain a readout error estimate of If T U is diagonal, e.g., in a benchmarking setting, there is no mixing, and readout error correction can simultaneously account for state preparation errors without knowledge of

[0151] Once the RMC 106 (e.g., the calibration component 108 of the RMC 106) has acquired a calibration dataset, the RMC 106 can utilize the calibration data to reduce readout errors for circuits with different U (potentially with underlying variations). In a real system, gradual variations in system gates and readout errors can be expected. This can mean that the calibration data can have a limited lifetime. For error reduction in the disclosed methods, e.g., whenever the RMC 106 is to be used to compute correction factors for individual Pauli-z operators, the RMC 106 can iterate over the calibration data. This approach of the RMC 106 can update the calibration dataset, which is ideally lightweight (e.g., very lightweight): the calibration component 108 can easily timestamp the calibration data and periodically add some new data points (e.g., calibration data points) while retiring (e.g., removing, discarding, deleting, or discontinuing use of) data that falls outside of a desired time window (e.g., a current time window) according to defined readout management standards. For methods based on explicit inversion of the transfer matrix, any such update by the calibration component 108 can amount to a regeneration of the entire matrix and its inverse. The computational complexity of the calibration component 108 to update the computation of correction factors can be linear in the size of the dataset. Both the evaluation of elements in the Hadamard matrix and the exchange between two n-qubit Pauli operators can take time

[0152] As disclosed herein, the disclosed subject matter typically only accesses information about M by sampling from In practice, the disclosed subject matter can thus trade off between the number of circuit instances and the number of samples per circuit according to defined readout management standards.

[0153] By employing the techniques and estimation protocols described herein to facilitate estimating readout results, the disclosed subject matter (e.g., RMC 106) can have a number of advantages over conventional estimation techniques for quantum readout results. The techniques and protocols of the disclosed subject matter can be more effective and accurate in estimating readout results, and can more desirably reduce (e.g., reduce or minimize) errors in the estimation of Pauli observables. Unlike conventional techniques, the techniques and protocols of the disclosed subject matter do not necessarily have any a priori assumptions or models of the readout error process. The techniques and protocols of the disclosed subject matter can be based on augmenting a quantum circuit with randomly selected Pauli operators and evaluating a scalar function from measurements obtained using a series of random instances. The disclosed subject matter can desirably reduce readout errors by dividing the function value of a quantum circuit of interest by the function value of a reference circuit (e.g., a calibration circuit). The techniques and protocols of the disclosed subject matter (e.g., utilized by RMC 106) can work by diagonalizing the readout-error transfer matrix in the Hadamard domain, which can make inverting relatively trivial. In comparison to many conventional algorithms, the techniques and protocols of the disclosed subject matter (e.g., utilized by RMC 106) can directly estimate the weights of the Pauli-z components (or elements of the Pauli transfer matrix) in the state, rather than the distribution of measurement values. Simulations of implementations of the techniques and protocols of the disclosed subject matter indicate that such techniques and protocols can reduce relevant readout errors in a 12-qubit system with relatively few (e.g., very few) measurements and circuit instances.

[0154] Systems and / or devices have been (or will be) described herein with respect to interactions among a number of components. It should be understood that such systems and components can include those components or sub-components specifically identified, some of those components or sub-components, and / or additional components. Sub-components can also be implemented as components communicatively coupled to other components rather than included within parent components. Further, one or more components and / or sub-components can be combined into a single component to provide aggregate functionality. Components can also interact with one or more other components not specifically described herein for the sake of brevity, but known by those of skill in the art.

[0155] Figure 6 and Figure 7A flowchart of an example non-limiting method 600, capable of ideally reducing readout errors associated with readout results generated by a quantum computer, is shown according to various aspects and embodiments of the disclosed subject matter. In some embodiments, method 600 may be performed by, for example, an RMC and / or processor component that may be associated with a data repository. The RMC may include calibration components, estimation components, and / or other components (e.g., other constituent components) as described herein. The RMC may be associated with a quantum computer and may receive readout determinations (e.g., readout results) from circuits (e.g., quantum circuits) that may be formed using quantum computer components including qubits (e.g., quantum components) and circuits (e.g., quantum circuits). For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted or may be omitted.

[0156] In 602, based on the corresponding random number (e.g., a randomly generated number) associated with the corresponding random Pauli gate, a first random Pauli gate (P) can be randomly sampled from a set of available Pauli gates. q RMC can employ RNG to generate random numbers, where corresponding numbers can be associated with corresponding Pauli gates (e.g., linked or mapped to corresponding Pauli gates), and where information related to associating corresponding numbers with corresponding Pauli gates (e.g., linking or mapping) can be stored in and retrieved from the data repository to facilitate determining which Pauli gate is associated with which number. In 604, in the qubit or first circuit (e.g., Figure 4 Before the first readout measurement of the circuit 400, a first random Pauli gate can be applied to the qubit at the first output of the first circuit. At 606, based on the corresponding first random Pauli gate applied to the first circuit and the input data, a corresponding first response can be measured at the first output of the first circuit (e.g., applied to the initial state |0>). The first response can be a first readout measurement. A calibration component can perform one or more readout measurements (e.g., one or more first readout measurements) on each of the first random Pauli gates. At 608, the corresponding first response (e.g., the first readout measurement) and the associated corresponding first value can be stored in a data repository, wherein the corresponding first value can be associated with the corresponding first readout measurement (e.g., the corresponding first readout measurement value) and the corresponding first random Pauli gate. The corresponding first value can correspond to a corresponding random number used to determine and select the corresponding first random Pauli gate.

[0157] In 610, based on the corresponding random number associated with the corresponding random Pauli gate, a second random Pauli gate (P) can be randomly sampled from the set of available Pauli gates. q In 612, prior to the second readout measurement of the qubit, it can be done in the second circuit (e.g.,Figure 3 a second random Pauli gate to a qubit or circuit of interest (e.g., C) at a second output end of the second circuit 300), where the second circuit can include a third circuit that can be the circuit of interest. At this point, the method 600 can proceed to reference point A, where, as shown, the method 600 can proceed from reference point A to reference label 614. Figure 7

[0158] At 614, a respective second response at a second output end of the second circuit can be measured based on the respective second random Pauli gate applied to the second circuit and the input data. The respective second response can be a respective second readout measurement. At 616, the respective second response (e.g., the second readout measurement) and an associated respective second value can be stored in a data repository, where the respective second value can be associated with the respective second readout measurement (e.g., a respective second readout measurement value) and the respective second random Pauli gate. The respective second value can correspond to the respective random number used to determine and select the respective second random Pauli gate.

[0159] At 618, calibration data can be determined based on (e.g., from) the first readout measurement measured at the first output end of the first circuit and a first defined function, as more fully described herein. At 620, estimation data can be determined based on the second readout measurement measured at the second output end of the second circuit and the first defined function, as more fully described herein.

[0160] At 622, a normalization scalar value can be determined based on (e.g., from) the calibration data and a second defined function. At 624, an estimation scalar value can be determined based on the estimation data and the second defined function, as more fully described herein. At 626, an error-reduced readout determination associated with the circuit of interest can be generated based on (e.g., from) the normalization scalar value and the estimation scalar value (e.g., the estimation scalar value divided by the normalization scalar value), as more fully described herein. The error-reduced readout determination can be an estimated (e.g., unbiased estimated) or expected readout result (e.g., a readout result determined and generated by the quantum computer and processed by the RMC), where readout errors are ideally reduced (e.g., reduced or minimized). At 628, the error-reduced readout determination can be provided (e.g., communicated or produced) as an output (e.g., an output from the RMC associated with the quantum computer).

[0161] Figure 8 and Figure 9 ​A flow diagram of another example non-limiting method 800 that can desirably reduce readout errors associated with readout results produced by a quantum computer in accordance with various aspects and embodiments of the disclosed subject matter is depicted. The method 800 can be performed by, for example, an RMC and / or a processor component that can be associated with a data store. The RMC can include a calibration component, an estimation component, and / or other components (e.g., other constituent components) as described herein. The RMC can be associated with a quantum computer and can receive readout determinations (e.g., readout results) from circuits (e.g., quantum circuits) that can be formed using components (e.g., quantum components) of the quantum computer including qubits and circuits (e.g., quantum circuits). Repetitive description of like elements employed in other embodiments described herein can be omitted or can be omitted in the interest of brevity.

[0162] At 802, a first random Pauli gate (P q ) can be randomly sampled from a set of available Pauli gates based on a respective random number (e.g., a randomly generated number) associated with the respective random Pauli gate. The RMC can employ an RNG to generate the random numbers, where respective numbers can be associated (e.g., linked or mapped to) respective Pauli gates, and where information related to associating (e.g., linking or mapping) respective numbers with respective Pauli gates can be stored in and retrieved from a data store to facilitate determining which Pauli gate is associated with which number. At 804, the first random Pauli gate can be applied to the qubit at a first output of the first circuit (e.g., circuit 400) prior to a first readout measurement of the qubit or the first circuit. At 806, a respective first response at the first output of the first circuit (e.g., applied to the initial state |0>) can be measured based on the respective first random Pauli gate applied to the first circuit and the input data. The first response can be a first readout measurement. The calibration component can perform one or more readout measurements (e.g., one or more first readout measurements) for each of the first random Pauli gates. At 808, the respective first responses (e.g., first readout measurements) and associated respective first values can be stored in the data store, where the respective first values can be associated with the respective first readout measurements and the respective first random Pauli gates. The respective first values can correspond to the respective random numbers used to determine or select the respective first random Pauli gates.

[0163] At 810, a second random Pauli gate (P q ) and a third random Pauli gate (P ppairs of random Pauli gates. At 812, prior to a second readout measurement of the qubits, the pairs of random Pauli gates can be applied to the qubits or the circuit of interest (C), including applying a second random Pauli gate to the qubits or the circuit of interest at a second output of a second circuit (e.g., circuit 200), where the second circuit can include a third circuit that can be the circuit of interest, and where the third random Pauli gate can be associated with an input of the circuit of interest. At this point, the method 800 can proceed to reference point B, where, as shown in Figure 9

[0164] At 814, based on the pairs of random Pauli gates applied to the second circuit and the input data applied to the second circuit, a respective second response at the second output of the second circuit can be measured. The respective second response can be a respective second readout measurement. At 816, the respective second response (e.g., the second readout measurement) and associated respective second and third values can be stored in a data repository, where the respective second and third values can be associated with (e.g., linked to, mapped to, or appended to) the respective second readout measurement, where the respective second value can be associated with the second random Pauli gate, and where the respective third value can be associated with the third random Pauli gate.

[0165] At 818, calibration data can be determined based on (e.g., from) the first readout measurement measured at the first output of the first circuit and the first defined function, as more fully described herein. At 820, estimation data can be determined based on the second readout measurement measured at the second output of the second circuit and the first defined function, as more fully described herein.

[0166] At 822, a normalized scalar value can be determined based on (e.g., from) the calibration data and a second defined function, as more fully described herein. At 824, an estimation scalar value can be determined based on the estimation data and the second defined function, as more fully described herein. At 826, an error-reduced readout determination associated with the circuit of interest can be generated based on (e.g., from) the normalized scalar value and the estimation scalar value (e.g., the estimation scalar value divided by the normalized scalar value). The error-reduced readout determination can be an estimated (e.g., unbiased estimate) or an expected readout result (e.g., a readout result determined and generated by the quantum computer and processed by the RMC), where readout error is ideally reduced (e.g., reduced or minimized). At 828, the error-reduced readout determination can be provided (e.g., communicated or produced) as an output (e.g., an output from the RMC associated with the quantum computer).

[0167] ​For simplicity of explanation, the methods and / or computer-implemented methods are depicted and described as a series of acts. It is to be understood and appreciated that the disclosed subject matter is not limited by the acts illustrated and / or by the order of acts, for example acts can occur in different orders and / or concurrently, and with other acts not presented and described herein. Furthermore, not all illustrated acts can be required to implement the computer-implemented methods in accordance with the disclosed subject matter. In addition, those skilled in the art will understand and appreciate that the computer-implemented methods could alternatively be represented as a series of interrelated states via a state diagram or events. Additionally, it should be further appreciated that the computer-implemented methods disclosed hereinafter and throughout this specification are capable of being stored on an article of manufacture to facilitate transporting and transferring such computer- implemented methods to computing devices.

[0168] To provide context for aspects of the disclosed subject matter, Figure 10 and the following discussion is intended to provide a general description of a suitable environment in which aspects of the disclosed subject matter can be implemented. Figure 10 A block diagram illustrating an example, non-limiting operating environment that can facilitate one or more embodiments described herein is shown. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity. Reference is made to Figure 10A suitable operating environment 1000 for implementing various aspects of this disclosure can also include the computer 1012. The computer 1012 can also include a processing unit 1014, a system memory 1016, and a system bus 1018. The system bus 1018 couples system components including, but not limited to, the system memory 1016 to the processing unit 1014. The processing unit 1014 can be any of various available processors. Dual microprocessors and other multiprocessor architectures can also be employed as the processing unit 1014. The system bus 1018 can be any of several types of bus structures including memory buses or memory controllers, peripheral buses or external buses, and / or a local bus using any of a variety of bus architectures including, but not limited to, Industrial Standard Architecture (ISA), Micro Channel Architecture (MSA), Extended ISA (EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB), Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus (USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and Small Computer Systems Interface (SCSI). The system memory 1016 can further include volatile memory 1020 and nonvolatile memory 1022. The basic input / output system (BIOS), containing the basic routines to transfer information between elements within the computer 1012, such as during start-up, is stored in nonvolatile memory 1022. By way of illustration, and not limitation, nonvolatile memory 1022 can include read only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). Volatile memory 1020 can also include random access memory (RAM), which acts as external cache memory. By way of illustration and not limitation, RAM can be available in many forms such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), direct Rambus RAM (DRRAM), and Rambus dynamic RAM (RDRAM).

[0169] The computer 1012 can also include removable / non-removable, volatile / non-volatile computer storage media. Figure 10A disk storage device 1024 is shown. Disk storage device 1024 can also include, but is not limited to, devices such as a magnetic disk drive, a soft disk drive, a tape drive, a Jaz drive, a Zip drive, an LS-100 drive, a flash memory card, or a Figure 10 Software is also depicted as being used to mediate between the user and the basic computer resources described in the suitable operating environment 1000. Such software can also include, for example, an operating system 1028. Operating system 1028, which can be stored on disk storage device 1024, acts to control and allocate resources of the computer 1012. System applications 1030 take advantage of the management of the resources by operating system 1028 through program modules 1032 and program data 1034, e.g., stored both on the system memory 1016 and on the disk storage device 1024. It is to be appreciated that this disclosure can be implemented with various operating systems or combinations of operating systems. A user enters commands or information into the computer 1012 through input device(s) 1036. Input devices 1036 include, but are not limited to, a pointing device such as a mouse, trackball, stylus, touchpad, keyboard, microphone, joystick, game pad, satellite dish, scanner, TV tuner card, digital camera, digital video camera, web camera, and the like. These and other input devices connect to the processing unit 1014 through the system bus 1018 via interface port(s) 1038. Interface port(s) 1038 include, for example, a serial port, a parallel port, a game port, and a universal serial bus (USB). Output device(s) 1040 use some of the same type of ports as input device(s) 1036. Thus, for example, a USB port can be used to provide input to computer 1012, and to output information from computer 1012 to an output device 1040. Output adapter 1042 is provided to illustrate that there are some output devices 1040 like monitors, speakers, and printers, among other output devices 1040 that require special adapters. The output adapters 1042 include, by way of illustration and not limitation, video and sound cards that provide a method of connection between the output device 1040 and the system bus 1018. It should be noted that other devices and / or systems of devices provide both input and output capabilities such as remote computer(s) 1044.

[0170] The computer 1012 can operate in a networked environment using logical connections to one or more remote computers, such as a remote computer 1044. The remote computer 1044 can be a computer, a server, a router, a network PC, a workstation, a microprocessor-based appliance, a peer device or other common network node, and typically includes many or all of the elements described relative to the computer 1012, although, for purposes of brevity, not all of the elements are

[0171] One or more embodiments can be a system, a method, an apparatus, and / or a computer program product at any possible technical detail level of integration. The computer program product can include a computer readable storage medium (or media) having computer readable program instructions thereon for causing a processor to carry out aspects of one or more embodiments. The computer readable storage medium can be a tangible device that can retain and store instructions for use by an instruction execution device. The computer readable storage medium can be, for example, but is not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. A non-exhaustive list of more specific examples of the computer readable storage medium can also include the following: a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), a static random access memory (SRAM), a portable compact disc read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanically encoded device such as punch-cards or raised structures in a groove having instructions recorded thereon, and any suitable combination of the foregoing. A computer readable storage medium, as used herein, is not to be construed as being transitory signals per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission media (e.g., light pulses passing through a fiber-optic cable), or electrical signals transmitted through a wire.

[0172] The computer readable program instructions described herein can be downloaded to respective computing / processing devices from a computer readable storage medium or to external computers or external storage devices via a network, for example, the Internet, a local area network, a wide area network and / or a wireless network. The network can comprise copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers and / or edge servers. A network adapter card or network interface in each computing / processing device receives computer readable program instructions from the network and forwards the computer readable program instructions to storage media within the respective computing / processing device for execution by a processor. Computer readable program instructions for carrying out operations of the disclosed subject matter can be assembler instructions, instruction-set-architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state-setting data, configuration data for integrated circuitry, or either source code or object code written in any combination of one or more programming languages, including an object oriented programming language such as Smalltalk, C++ or the like, and a procedural programming language such as the "C" programming language or similar programming languages. The computer readable program instructions can execute entirely on the user's computer, partly on the user's computer, as a stand-alone software package, partly on the user's computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer can be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection can be made to an external computer (for example, through the Internet using an Internet Service Provider). In some embodiments, electronic circuitry including, for example, programmable logic circuitry, field-programmable gate arrays (FPGA), or programmable logic arrays (PLA) can execute the computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry, in order to perform aspects of the disclosed subject matter.

[0173] The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational elements to be performed on the computer, other programmable apparatus or other device to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0174] The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational elements to be performed on the computer, other programmable apparatus or other device to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0175] While the subject matter has been described above in the general context of computer-executable instructions of a computer program product that runs on a computer and / or computers, those skilled in the art will recognize that the disclosure also can be implemented in combination with other program modules. Generally, program modules include routines, programs, components, data structures, that perform particular tasks and / or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the computer-implemented methods disclosed herein can be practiced with other computer system configurations, including single-processor or multiprocessor computer systems, minicomputers, mainframe computers, as well as computers, hand-held computing devices (e.g., PDA, phone), microprocessor-based or programmable consumer or industry electronics, and the like. The illustrated aspects can also be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. However, some, if not all, aspects of this disclosure can be practiced on stand-alone computers. In a distributed computing environment, program modules can be located in local and remote memory storage devices.

[0176] As used in this application, the terms “component,” “system,” “platform,” “interface,” and the like can refer to and / or can include a computer-related entity or an entity that is related to an operational machine with one or more specific functionalities. The entities disclosed herein can be either hardware, a combination of hardware and software, software, or software in execution. For example, a component can be, but is not limited to being, a process running on a processor, a processor, an object, an executable, a thread of execution, a program, and / or a computer. By way of illustration, both an application running on a server and the server can be a component. One or more components can reside within a process and / or thread of execution, and a component can be localized, co-resident, and / or distributed amongst one computer and / or across multiple computers. In another example, a component can execute from various computer readable media having various data structures stored thereon. The components can communicate via local and / or remote processes such as in accordance with a signal having one or more data packets (e.g., data from one component interacting with another component in a local system, distributed system, and / or across a network such as the Internet with other systems via the signal). As another example, a component can be an apparatus with specific functionality provided by mechanical parts operated by an electric or electronic circuit, which is operated by a software or firmware application executed by a processor. In such a case, the processor can be internal or external to the apparatus and can execute at least a part of the software or firmware application. As yet another example, a component can be an apparatus that provides specific functionality through electronic components without mechanical parts, wherein the electronic components can include a processor or other method for executing software or firmware that conveys at least portions of the functionality of the electronic components. In an aspect, a component can emulate an electronic component via, for example, a virtual machine within a cloud computing system.

[0177] Moreover, the term “or” is intended to mean an inclusive “or” rather than an exclusive “or.” That is, unless specified otherwise, or clear from context, “X employs A or B” is intended to mean any of the natural inclusive permutations. That is, if X employs A; X employs B; or X employs both A and B, then “X employs A or B” is satisfied under any of the foregoing instances. Moreover, articles “a” and “an” as used in the subject specification and annexed drawings should generally be construed to mean “one or more” unless specified otherwise or clear from context to be directed to a singular form. As used herein, the terms “example” and / or “exemplary” are utilized to mean serving as an example, instance, or illustration. For the avoidance of doubt, the subject matter disclosed herein is not limited by such examples. In addition, any aspect or design described herein as an “example” and / or “exemplary” is not necessarily to be construed as preferred or advantageous over other aspects or designs, nor is it meant to preclude equivalent exemplary structures and techniques for performing the same functions.

[0178] As employed in this specification, the term "processor" can refer to substantially any computing processing unit or device comprising virtually any physical hardware asset configured to compute, such as, without limitation, a single-core processor; single processor with software multithread execution capability; multi-core processors; multi-core processors with software multithread execution capability; multi-core processors with hardware multithread technology; parallel platforms; and parallel platforms with distributed shared memory. Additionally, a processor can refer to an integrated circuit, an application-specific integrated circuit (ASIC), a digital signal processor (DSP), a field- programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. Further, a processor can utilize nano- scale architectures such as, but not limited to, molecular and quantum-dot based transistors, switches and gates, in order to optimize space usage or enhance the performance of user devices. A processor can also be implemented as a combination of computing processing units. In the present disclosure, terms such as "store," "storage," "data store," data storage," "database," and substantially any other information storage component relevant to operation and functionality of a component are utilized to refer to "memory components," entities embodied in a "memory," or components comprising a memory. It is to be appreciated that memory and / or memory components described herein can be volatile memory or nonvolatile memory, or can include both volatile and nonvolatile memory. By way of illustration, and not limitation, nonvolatile memory can include read only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable ROM (EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). Volatile memory can include, for example, RAM that can act as external cache memory. By way of illustration and not limitation, RAM can be available in many forms such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), direct Rambus RAM (DRRAM), and Rambus dynamic RAM (RDRAM). Additionally, memory components of systems or computer-implemented methods disclosed herein are intended to comprise, without being limited to, these and any other suitable types of memory.

[0179] The above-described embodiments include only examples of systems and computer-implemented methods. Of course, not every combination of components or computer-implemented method can be described for purposes of describing the present disclosure, but one of ordinary skill in the art can recognize many other combinations and permutations of the present disclosure are possible. Moreover, with respect to the use of the term "comprise", "have", "hold", and the like, these terms are used in their inclusive, open sense, in the same way the term "comprising" is used in a claim as a transition word, and are therefore to be construed as specifying the noted component or elements after which the term is used, but not precluding the presence of additional components or elements. The description of the different embodiments has been presented for purposes of illustration, but is not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application, or technical improvement over technology found in the marketplace, or to enable others skilled in the art to understand the embodiments disclosed herein.

Claims

1. A system for quantum computing, comprising: a memory that stores computer executable components; and a processor that is operatively coupled to the memory, the processor executes computer executable components that perform: generating a set of calibration circuits, where each calibration circuit comprises a set of qubits, and a respective first subset of Pauli operators that is randomly selected for the calibration circuit from a set of Pauli operators, where the respective first subset of Pauli operators is at an output of the calibration circuit; generating a set of estimation circuits, where each estimation circuit comprises a set of qubits, and a respective second subset of Pauli operators that is randomly selected for the estimation circuit from a set of Pauli operators, where the respective second subset of Pauli operators is at an output of the estimation circuit; measuring, at a respective output of each calibration circuit, a respective calibration readout measurement of the set of qubits based on a respective application of each calibration circuit in the set of calibration circuits to the set of qubits; measuring, at a respective output of each estimation circuit, a respective estimation readout measurement of the set of qubits based on a respective application of each estimation circuit in the set of estimation circuits to the set of qubits; and generating an error-reduced readout result related to a target circuit based on a defined function of the respective calibration readout measurements and the respective estimation readout measurements.

2. The system of claim 1, wherein, An amount of error that is reduced in association with the error-reduced readout result is based on the respective first subset of Pauli operators and the respective second subset of Pauli operators.

3. The system of any one of claims 1-2, wherein, Generating the error-reduced readout result includes generating calibration data based on the respective calibration readout measurements and a first defined function.

4. The system of claim 3, wherein, Generating the error-reduced readout result includes generating estimation data based on the respective estimation readout measurements and a first defined function.

5. The system of claim 4, wherein, Generating the error-reduced readout result further includes: generating a normalization scalar value based on the calibration data and a second defined function; generating an estimation scalar value based on the estimation data and the second defined function; and generating the error-reduced readout result based on the normalization scalar value and the estimation scalar value.

6. The system of claim 5, wherein, The at least one computer executable component further performs: updating the calibration data based on generating one or more new calibration circuits.

7. The system of any one of claims 5-6, wherein, Generating the error-reduced readout result further includes: modeling quantum noise as a classical noise channel based on the calibration data and the estimation data.

8. A computer-implemented method, comprising: generating, by a system that is operatively coupled to a processor, a set of calibration circuits, where each calibration circuit comprises a set of qubits, and a respective first subset of Pauli operators that is randomly selected for the calibration circuit from a set of Pauli operators, where the respective first subset of Pauli operators is at an output of the calibration circuit; generating, by the system, a set of estimation circuits, where each estimation circuit comprises a set of qubits, and a respective second subset of Pauli operators that is randomly selected for the estimation circuit from a set of Pauli operators, where the respective second subset of Pauli operators is at an output of the estimation circuit; measuring, by the system, a respective calibration readout measurement of the set of qubits at the respective output of each calibration circuit based on respective application of each calibration circuit in the set of calibration circuits to the set of qubits; measuring, by the system, a respective estimation readout measurement of the set of qubits at the respective output of each estimation circuit based on respective application of each estimation circuit in the set of estimation circuits to the set of qubits; and generating, by the system, an error-reduced readout result related to the target circuit based on a defined function of the respective calibration readout measurement and the respective estimation readout measurement.

9. The computer-implemented method of claim 8, wherein, an amount of error reduced associated with the error-reduced readout result is based on the respective first subset of Pauli operators and the respective second subset of Pauli operators.

10. The computer-implemented method of any one of claims 8 to 9, wherein, generating the error-reduced readout result includes generating calibration data based on the respective calibration readout measurement and a first defined function.

11. The computer-implemented method of claim 10, wherein, generating the error-reduced readout result includes generating estimation data based on the respective estimation readout measurement and a first defined function.

12. The computer-implemented method of claim 11, wherein, generating the error-reduced readout result further includes: generating a normalization scalar value based on the calibration data and a second defined function; generating an estimation scalar value based on the estimation data and the second defined function; generating the error-reduced readout result based on the normalization scalar value and the estimation scalar value.

13. The computer-implemented method of claim 12, further comprising: updating, by the system, the calibration data based on generating one or more new calibration circuits.

14. A computer program product that facilitates reducing readout errors associated with quantum circuits, the computer program product comprising program instructions executable by a processor to cause the processor to: generate a set of calibration circuits, where each calibration circuit includes a set of qubits, followed by a circuit of interest, and followed by a respective first subset of Pauli operators randomly selected for the calibration circuit from a set of Pauli operators, where the respective first subset of Pauli operators is at an output of the calibration circuit; generate a set of estimation circuits, where each estimation circuit includes a set of qubits, followed by a circuit of interest, and followed by a respective second subset of Pauli operators randomly selected for the estimation circuit from a set of Pauli operators, where the respective second subset of Pauli operators is at an output of the estimation circuit; measure, by the system, a respective calibration readout measurement of the set of qubits at the respective output of each calibration circuit based on respective application of each calibration circuit in the set of calibration circuits to the set of qubits; measure, by the system, a respective estimation readout measurement of the set of qubits at the respective output of each estimation circuit based on respective application of each estimation circuit in the set of estimation circuits to the set of qubits; and generate, by the system, an error-reduced readout result related to the target circuit based on a defined function of the respective calibration readout measurement and the respective estimation readout measurement.

15. The computer program product of claim 14, wherein, generating the error-reduced readout result includes: generating calibration data based on the respective calibration readout measurement and a first defined function; generating estimation data based on the respective estimation readout measurement and a first defined function; generating a normalization scalar value based on the calibration data and a second definition function; generating an estimate scalar value based on the estimate data and the second definition function; and generating the error-reduced readout based on the normalization scalar value and the estimate scalar value.

16. A system for quantum computing, comprising: a memory that stores computer executable components; and a processor that is operatively coupled to the memory, the processor executes computer executable components that perform: generating a set of calibration circuits, where each calibration circuit includes a set of qubits and a respective first subset of Pauli operators that is randomly selected from a set of Pauli operators for the calibration circuit, where the respective first subset of Pauli operators is at an output of the calibration circuit; generating a set of estimate circuits, where each estimate circuit includes a set of qubits, a respective second subset of Pauli operators that is randomly selected from a set of Pauli operators for the estimate circuit, a circuit of interest, and a respective third subset of Pauli operators that is randomly selected from a set of Pauli operators for the estimate circuit, where the respective third subset of Pauli operators is at an output of the estimate circuit; measuring a respective calibration readout measurement of the set of qubits at the respective output of each calibration circuit based on respective application of each calibration circuit in the set of calibration circuits to the set of qubits; measuring a respective estimate readout measurement of the set of qubits at the respective output of each estimate circuit based on respective application of each estimate circuit in the set of estimate circuits to the set of qubits; and generating an error-reduced readout related to the target circuit based on a definition function of the respective calibration readout measurements and the respective estimate readout measurements.

17. The system of claim 16, wherein, Generating the error-reduced readout includes generating calibration data based on the respective calibration readout measurements and a first definition function.

18. The system of claim 17, wherein, Generating the error-reduced readout includes generating estimate data based on the respective estimate readout measurements and a first definition function.

19. The system of claim 18, wherein, Generating the error-reduced readout further includes: generating a normalization scalar value based on the calibration data and a second definition function; generating an estimate scalar value based on the estimate data and the second definition function; and generating the error-reduced readout based on the normalization scalar value and the estimate scalar value.

20. The system of claim 19, wherein, The at least one computer executable component further performs: updating the calibration data based on generating one or more new calibration circuits.

21. The system of claim 19 or 20, wherein, Generating the error-reduced readout further includes: modeling quantum noise as a classical noise channel based on the calibration data and the estimate data.

22. A computer-implemented method, comprising: generating, by a system operatively coupled to a processor, a set of calibration circuits, where each calibration circuit includes a set of qubits and a respective first subset of Pauli operators that is randomly selected from a set of Pauli operators for the calibration circuit, where the respective first subset of Pauli operators is at an output of the calibration circuit; generating, by the system, a set of estimation circuits, where each estimation circuit includes a set of qubits, followed by a respective second subset of Pauli operators randomly selected for the estimation circuit from a set of Pauli operators, followed by the circuit of interest, and followed by a respective third subset of Pauli operators randomly selected for the estimation circuit from a set of Pauli operators, where the respective third subset of Pauli operators is at an output of the estimation circuit; measuring, by the system, a respective calibration readout measurement of the set of qubits at a respective output of each calibration circuit based on the respective application of each calibration circuit to the set of qubits; measuring, by the system, a respective estimation readout measurement of the set of qubits at a respective output of each estimation circuit based on the respective application of each estimation circuit to the set of qubits; and generating, by the system, an error-reduced readout result related to the target circuit based on a defined function of the respective calibration readout measurements and the respective estimation readout measurements.

23. The computer-implemented method of claim 22, wherein, Generating the error-reduced readout result includes: generating calibration data based on the respective calibration readout measurements and a first defined function; and generating estimation data based on the respective estimation readout measurements and the first defined function.

24. The computer-implemented method of claim 23, wherein, Generating the error-reduced readout result further includes: generating a normalization scalar value based on the calibration data and a second defined function; generating an estimation scalar value based on the estimation data and the second defined function; generating the error-reduced readout result based on the normalization scalar value and the estimation scalar value.

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