Readout error mitigation for quantum forecasting

Abstract

A technique for mitigating readout errors in quantum conjectures is presented. A calibration component applies a first random Pauli gate to a qubit at a first output of a first circuit before 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 before a second readout measurement of the qubit, and generates an error-mitigated readout decision quantity 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 estimated data based on the second readout measurement. The estimation component determines a normalized scalar value based on the calibration data, determines an estimated scalar value based on the estimation data, and determines an error-mitigated readout decision quantity associated with the circuit of interest based on the normalized scalar value and the estimated scalar value.

Description

[Technical Field]

[0001] The subject disclosure relates to quantum computing using quantum circuits. Quantum computing uses quantum physics, rather than transistor-based binary digital technology, to encode and process information. Quantum computing devices can use quantum bits (also called qubits), which operate according to the laws of quantum physics and can exhibit phenomena such as superposition and entanglement. The quantum physics principle of superposition allows a qubit to assume a state that partially represents both a value of "1" and a value of "0" simultaneously. The quantum physics principle of entanglement allows qubits to correlate with each other in such a way that their combined state cannot contain the individual qubit states as components. For example, the state of a first qubit can depend on the state of a second qubit. As such, quantum circuits can use qubits to encode and process information in a manner significantly different from transistor-based binary digital technology. [Background technology]

[0002] Quantum computing can be used to perform quantum programming. Quantum programming can involve the process of assembling a sequence of instructions, which can be called a quantum program, that can be run on a quantum computer. Each quantum program can be associated with a collection of quantum circuits. When a quantum program is executed, the quantum computer can produce a result. The performance of a quantum computer depends, in large part, not only on the fidelity of the unitary gates in the quantum circuits, but also on the fidelity of the quantum readout of the result. Traditional quantum computers often have an undesirable amount of error in the quantum readout.

[0003] One common approach among several conventional readout error mitigation techniques is to use quantum detector tomography to estimate the transition matrix A and then apply the inverse matrix to obtain an estimate of the ideal probability vector. Assuming that the readout errors are independent for each qubit,

number

[0004] Some conventional approaches can capture interrelationships between variables using representations based on cumulant expansions, however, such conventional approaches do not include or provide algorithms for using such representations in the context of error mitigation.

[0005] Other conventional approaches can incorporate crosstalk to some extent by considering pairwise qubit interactions. To capture crosstalk, one conventional approach considers a correlated noise model based on a continuous-time Markov process and proposes techniques to avoid the explicit computation of the inverse transition matrix. The noise model in this conventional approach is a 2n 2 A common problem and drawback associated with matrix inversion, common to different conventional approaches, is that the resulting probability vector

number

[0006] These and other deficiencies in traditional approaches that attempt to estimate quantum computing readout results and mitigate readout errors may result in inefficient, ineffective, or inaccurate estimation of quantum computing readout results, or inefficient, ineffective, and inaccurate estimation and inefficient, ineffective, or inaccurate readout error mitigation, or inefficient, ineffective, and inaccurate readout error mitigation. Summary of the Invention

[0007] The following presents a summary in order to provide a basic understanding of one or more embodiments of the disclosed subject matter. This summary is not intended to identify key or critical elements or to limit the scope of particular embodiments or the claims. Its sole purpose is to present ideas in a simplified form as a prelude to the more detailed description that is presented later. One or more embodiments described herein provide a system, device, structure, computer-implemented method, apparatus, or computer program product, or a combination thereof, that can mitigate read errors in readout results for quantum prediction.

[0008] One embodiment relates to a computer-implemented method, the method including applying, by a system operatively coupled to a processor, a first random Pauli gate to the qubit at a first output of a first circuit before a first readout measurement of the qubit. The computer-implemented method may further include applying, by the system, a second random Pauli gate to the qubit at a second output of a second circuit before a second readout measurement of the qubit. Such an embodiment of the method may provide several advantages, including the method being able to more efficiently and accurately estimate quantum computing readout results.

[0009] In some embodiments, the computer-implemented method may further include determining, by the system, calibration information based on the defined first function and a first readout measurement measured at the first output of the first circuit, and determining, by the system, estimated information based on the defined first function and a second readout measurement measured at the second output of the second circuit. In certain embodiments, the computer-implemented method may further include determining, by the system, a normalized scalar value based on the calibration information and the defined second function, determining, by the system, an estimated scalar value based on the estimated information and the defined second function, and determining, by the system, an error-mitigated readout determination associated with the circuit of interest based on the normalized scalar value and the estimated scalar value. These embodiments of the method may provide several advantages, including the method being able to more efficiently and accurately estimate quantum computing readout results and to perform operations that can be performed in an efficient and less complex manner.

[0010] In some embodiments, the elements described in connection with these disclosed methods may be embodied in different forms, such as a system, a computer program product, or other forms.

[0011] According to another embodiment, a system includes a memory storing computer-executable components and a processor operatively coupled to the memory for executing the computer-executable components. The computer-executable components can include a calibration component that applies a first random Pauli gate to a qubit at a first output of a first circuit before a first readout measurement of the qubit. The computer-executable components can further include an estimation component that applies a pair of random Pauli gates to a qubit associated with a second circuit, the applying including applying a second random Pauli gate to the qubit at a second output of the second circuit before a second readout measurement of the qubit. Such an embodiment of the system can provide several advantages, including the system's ability to more efficiently and accurately estimate quantum computing readout results.

[0012] In certain embodiments, the system further includes a calibration component capable of determining calibration data based on the defined first function and a first readout measurement measured for a first random Pauli gate at a first output of the first circuit, and an estimation component capable of determining estimation data based on the defined first function and a second readout measurement measured for a second random Pauli gate at a second output of the second circuit. In some embodiments, the system further includes an estimation component that determines a normalized scalar value based on the calibration data and the defined second function, determines an estimated scalar value based on the estimation data and the defined second function, and determines an error-mitigated readout measurement associated with the circuit of interest based on the normalized scalar value and the estimated scalar value. Such an embodiment of the system can provide several advantages, including the system being able to more efficiently and accurately estimate quantum computing readout results and perform operations that can be performed in an efficient and less complex manner.

[0013] In some embodiments, the elements described in connection with the disclosed system may be embodied in different 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 should be read in conjunction with the accompanying drawings. [Brief explanation of the drawings]

[0015] [Figure 1] FIG. 1 is a block diagram of an exemplary, non-limiting system that can desirably mitigate readout errors associated with readout results generated by a quantum computer, in accordance with various aspects and embodiments of the disclosed subject matter. [Figure 2]FIG. 1 is a block diagram of an example circuit that can be utilized to facilitate generating error-mitigated read results in accordance with various aspects and embodiments of the disclosed subject matter. [Figure 3] FIG. 1 is a block diagram of an example circuit that can be utilized to facilitate generating error-mitigated read results in accordance with various aspects and embodiments of the disclosed subject matter. [Figure 4] FIG. 1 is a block diagram of an example circuit that can be utilized to facilitate generating error-mitigated read results in accordance with various aspects and embodiments of the disclosed subject matter. [Figure 5] FIG. 10 illustrates an example diagonalization mask for 12 qubits obtained by averaging the cross product of the exchange vector dq with a corresponding respective number of random q∈X, in accordance with various aspects and embodiments of the disclosed subject matter. [Figure 6] 1 is a flow diagram of an exemplary, non-limiting method by which read errors associated with read results generated by a quantum computer may be desirably mitigated, in accordance with various aspects and embodiments of the presented subject matter. [Figure 7] 1 is a flow diagram of an exemplary, non-limiting method by which read errors associated with read results generated by a quantum computer may be desirably mitigated, in accordance with various aspects and embodiments of the presented subject matter. [Figure 8] 1 is a flow diagram of another exemplary, non-limiting method by which read errors associated with read results generated by a quantum computer may be desirably mitigated, in accordance with various aspects and embodiments of the disclosed subject matter. [Figure 9] 1 is a flow diagram of another exemplary, non-limiting method by which read errors associated with read results generated by a quantum computer may be desirably mitigated, in accordance with various aspects and embodiments of the disclosed subject matter. [Figure 10] FIG. 1 is a block diagram of an exemplary non-limiting operating environment that can facilitate one or more embodiments described herein. DETAILED DESCRIPTION OF THE INVENTION

[0016] The following detailed description is for illustrative purposes only and is not intended to limit the embodiments or the application or uses of the embodiments, or both, nor is it intended to be bound by any express or implied information presented in the Background or Summary or Detailed Description above.

[0017] One or more embodiments will now be described with reference to the drawings. Like reference numerals are used throughout to refer to like elements. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a more thorough understanding of one or more embodiments. It will be apparent, however, that in various cases one or more embodiments may be practiced without these specific details.

[0018] Quantum programming can include the process of assembling a sequence of instructions, which can be called a quantum program, that can be run on a quantum computer. Each quantum program can be associated with a collection of quantum circuits. When a quantum program is executed, the quantum computer can generate a result (e.g., an estimate). The performance of a quantum computer can depend, in large part, not only on the fidelity of the unitary gates in the quantum circuits, but also on the fidelity of the quantum readout of the result. Traditional quantum computers often have an undesirable amount of error in the quantum readout, or can be inefficient in estimating or generating the readout result, or both.

[0019] Some conventional readout error mitigation techniques use quantum detector tomography to estimate the transition matrix A, and then apply the inverse matrix to obtain an estimate of the ideal probability vector. Assuming that the readout errors are independent for each qubit,

number

[0020] Certain conventional approaches can capture interrelationships between variables using representations based on cumulant expansions, however, such conventional approaches do not include or provide algorithms for using such representations in the context of error mitigation.

[0021] Other conventional approaches can incorporate crosstalk to some extent by considering pairwise qubit interactions. To capture crosstalk, one conventional approach considers a correlated noise model based on a continuous-time Markov process and proposes techniques to avoid the explicit computation of the inverse transition matrix. The noise model in this conventional approach is a 2n 2 A common problem and drawback associated with matrix inversion, common to different conventional approaches, is that the resulting probability vector

number

[0022] It may be desirable to be able to mitigate the effects of readout errors, including interrelated and state-dependent errors across multiple qubits, and in particular, it may be desirable to be able to mitigate the effects of readout errors in a practical and efficient manner. The disclosed subject matter can be implemented to provide solutions to all or at least part of these and / or other problems with traditional quantum computing and with readout of quantum computing results, including introducing robust, practical, and desirably implementable protocols that can desirably mitigate readout errors and the effects of readout errors in state tomography and process tomography (e.g., partial state tomography and partial process tomography) using circuit randomization. The disclosed subject matter can provide unbiased estimators 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 time variations in noise associated with quantum computing, which may be highly useful for the success of mitigation schemes on near-term devices.

[0023] To that end, various aspects and embodiments herein relate to techniques for mitigating readout errors for quantum conjectures. The disclosed subject matter can include a readout management component (RMC) capable of mitigating readout errors for quantum conjectures 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. With respect to the second circuit, which may include a third circuit, which may be the circuit of interest, the RMC may 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 generate an error-mitigated readout determination (e.g., a readout result that may have a desired mitigated 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. Such an estimation process may be utilized, for example, in connection with state tomography (e.g., partial state tomography). To facilitate such error-mitigated readout determination, the calibration component can determine calibration data based on the first readout measurement and the defined first function, and the estimation component can determine estimation data based on the second readout measurement and the defined first function. The estimation component can determine a normalized scalar value based on the calibration data and the defined second function, and can determine an estimated scalar value based on the estimation data and the defined second function.The estimation component can determine an error-mitigated read decision quantity (e.g., an error-mitigated read result) associated with the circuit of interest based on the normalized scalar value and the estimated scalar value (e.g., as a function of the normalized scalar value and the estimated scalar value).

[0024] In other embodiments, e.g., with respect to process tomography (e.g., partial process tomography), RMC can utilize pairs of random Pauli gates and apply pairs of random Pauli gates to a circuit during the estimation process, instead of just utilizing a random Pauli gate applied to the output of the circuit, as described more fully herein. For example, RMC can use the calibration component to 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 the second circuit, which may include a third circuit, which may be the circuit of interest, the estimation component may apply a pair of random Pauli gates (or corresponding Pauli operators) to a qubit associated with the second circuit, including applying a second random Pauli gate to the qubit or circuit of interest at a second output of the second circuit and applying a third random Pauli gate to the qubit or circuit of interest at an input of the circuit of interest prior to a second readout measurement of the qubit or second circuit. The estimation component may generate an error-mitigating readout determination quantity associated with the circuit of interest based on the first random Pauli gate applied to the qubit or first circuit and the pair of random Pauli gates applied to the qubit or second circuit, as more fully described herein.

[0025] These and other aspects and embodiments of the disclosed subject matter are now described with reference to the drawings.

[0026] FIG. 1 illustrates a block diagram of an exemplary, non-limiting system 100 that can desirably mitigate readout errors associated with readout results generated by a quantum computer, in accordance with various aspects and embodiments of the disclosed subject matter. System 100 can include a quantum computer component 102, which can include various quantum devices, quantum circuits, or other components, or combinations thereof. The quantum devices can include, for example, qubit components (also referred to herein as qubits). 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 based on, for example, a set of instructions (e.g., an assembled sequence of instructions) that can be input to quantum computer component 102 and run (e.g., executed) on quantum computer component 102 to generate and operate the desired quantum circuit 104, and the structure of quantum circuit 104 and the operations (e.g., quantum operations) performed by quantum circuit 104 can be based on this set of instructions. In response to execution of a quantum program that includes or is associated with this set of instructions, or that includes input data or parameter data, or a combination thereof, and operation of quantum circuit 104 based on such a quantum program, quantum computer component 102 can generate results (e.g., data results), which may also be referred to as readout results or readout decisions. Quantum computer component 102 can present (e.g., communicate or transmit) those results as output.

[0027] Traditionally, there may be undesirable read errors associated with readout results output by quantum computers such as those described herein. It may be desirable to be able to mitigate the effects of readout errors, including correlated and state-dependent errors across multiple qubits, and in particular, to be able to mitigate the effects of readout errors in a practical manner. The disclosed subject matter may provide solutions to these and / or other problems with traditional quantum computing and with reading out quantum computing results.

[0028] To facilitate the desired mitigation of errors in the readout results and to facilitate the desired mitigation of errors in the readout results in a fast and efficient manner, the system 100 may include a readout management component (RMC) 106, which may be associated with (e.g., communicatively connected to) the quantum computer component 102. The RMC 106 may desirably (e.g., efficiently, quickly, and optimally) manage the generation of the readout results to mitigate (e.g., reduce or minimize) the readout errors. For example, the RMC 106 may manage the generation of the readout results to mitigate readout errors in partial state tomography and partial process tomography using circuit-associated randomization according to defined readout management criteria. In doing so, the RMC 106 may provide an unbiased estimator of the readout results and remove asymmetries in the readout errors. The error mitigation techniques used by RMC 106 may allow for and implement dynamic adjustments to make the techniques resilient to time variations in noise that might otherwise cause read errors, which may be useful and desirable to enable the success of mitigation schemes on near-future devices. Desirably, the error mitigation techniques used by RMC 106 are also independent of a particular noise model.

[0029] According to various embodiments, the RMC 106 can utilize various estimation protocols to facilitate mitigating read errors associated with readout results generated by the quantum computing component 102. These estimation protocols can include, for example, a data acquisition protocol (also referred to as Protocol AcquireData), 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) that can specify a process for sampling and acquiring data and performing measurements of the circuit's output (e.g., measured response) under various conditions, as described more fully herein, where the first protocol and second protocol can each utilize the data acquisition protocol to generate error-mitigated readout results (e.g., error-mitigated readout decisions or error-mitigated readout estimates or averages of values) associated with a desired circuit (e.g., a circuit of interest). Protocol AcquireData and Protocol 1 can be utilized, for example, with respect to quantum computing component 102 readout results associated with partial-state tomography. Protocol AcquireData and Protocol 2 may be utilized, for example, with respect to readout results of quantum computer component 102 associated with partial process tomography. To facilitate implementation of these protocols, RMC 106 may include a calibration component 108 that may perform a calibration process that may generate calibration data that may be used to mitigate readout errors and provide a benchmark associated with the circuit, as described more fully herein, and an estimation component 110 that may perform an estimation process that may generate estimation data that may be used in conjunction with the calibration data to provide readout results that may have desirably mitigated errors.

[0030] Reference is now made to Figures 2, 3, and 4 (along with Figure 1), which illustrate block diagrams of example circuits that may be utilized to facilitate generating error-mitigated read results in accordance with various aspects and embodiments of the disclosed subject matter. Figure 2 illustrates a block diagram of an example circuit 200 in accordance with various aspects and embodiments of the disclosed subject matter, where the circuit 200 includes a circuit of interest (C), as well as a Pauli operator P that may be indexed (e.g., indexed by integers p and q). p and P q The circuit 200 may include a Pauli operation (using a Pauli operator P ) that may be associated with (e.g., located at or connected to) the input of the circuit 202. The circuit 200 may include a circuit of interest (C) 202, which may be a circuit (e.g., a quantum circuit) that can be utilized to determine or generate a readout result (e.g., a readout decision or estimate) in response to input data. To perform a desired operation (e.g., a quantum operation) on data (e.g., input data or other data), the circuit 200 may include or use various desired components and circuits (e.g., quantum components and circuits) of the quantum computer component 102. The circuit 200 may further include a Pauli operator (P ) that may be associated with (e.g., located at or connected to) the input of the circuit 202. p ) 204, and a Pauli operator (P q) 206, which can perform Pauli operations on data (e.g., input data or other data). Each corresponding Pauli operator can be associated with a corresponding respective value p and q and can perform a corresponding respective (e.g., different or unique) Pauli operation on the data. Each corresponding Pauli operator can further be associated with and / or correspond to a corresponding respective Pauli gate (e.g., a Pauli gate used in a circuit such as circuit 200) and / or a corresponding respective Pauli matrix.

[0031] Circuit 200 may further include desired qubits 208 (e.g., of quantum computer component 102) that may be associated with (e.g., be part of or connected to) circuit 202. (As shown in circuit 200, a qubit set that includes one or more qubits 208 may be represented in circuit 200 by a diagonal line (e.g., a slash mark) crossing a horizontal line.) RMC 106 may measure the response of circuit 200 (e.g., to input data) at output 210 of circuit 200.

[0032] 3 illustrates a block diagram of an exemplary circuit 300 according to various aspects and embodiments of the disclosed subject matter, which may include a circuit of interest (C) and a Pauli operation indexed with p=0 and P0=I, so that the Pauli operation can be effectively indexed by an integer q with the circuit of interest. q206, qubits 208 (as shown in circuit 300, a set of qubits containing one or more qubits 208 may be represented in circuit 300 by a diagonal line crossing a horizontal line), or other components (e.g., quantum components), or a combination thereof. p Since p=0 and P0=I for p can effectively not affect or change the response of circuit 300, and therefore P p is not shown or is not part of circuit 300, or at least is not explicitly shown or is at least not part of circuit 300. Similar to Figure 2, RMC 106 can measure the response of circuit 300 (e.g., to input data) at output 210 of circuit 300.

[0033] 4 illustrates a block diagram of an example circuit 400 in accordance with various aspects and embodiments of the disclosed subject matter, where the circuit 400 can include indexed Pauli operations where p=0 and P0=I and C=I, such that the circuit 400 can effectively use Pauli operations that can be indexed by an integer q. The circuit 400 can be implemented using the Pauli operator (P q ) 206, qubits 208 (as shown in circuit 400, a set of qubits including one or more qubits 208 may be represented in circuit 400 by a diagonal line crossing a horizontal line), or other components (e.g., quantum components), or a combination thereof. p Since p=0 and P0=I for C=I, P p and C cannot effectively affect or change the response of circuit 400, and therefore P pand C are not shown or are not part of circuit 400, or at least are not explicitly shown or are at least not part of circuit 400. Similar to Figure 2, RMC 106 can measure the response of circuit 200 (e.g., to input data) at output 210 of circuit 400.

[0034] According to various embodiments, for state tomography (e.g., partial state tomography), calibration component 108 can implement Protocol AcquireData and Protocol 1 to facilitate generating desired readout results (e.g., error-mitigated readout decisions or estimates). As part of Protocol AcquireData and Protocol 1, calibration component 108 can initially utilize circuit 400 where p=0 and P0=I and C=I, so that the circuit can effectively use Pauli operations indexed by the integer q.

[0035] According to Protocol AcquireData and Protocol 1, as part of the calibration process, calibration component 108 selects a first Pauli operator (e.g., a first random Pauli operator) P from the set of available Pauli operators based on a corresponding respective random value (e.g., a randomly generated number) associated with a corresponding respective random Pauli operator (and / or a corresponding Pauli gate). q 206. In some embodiments, the calibration component 108 may randomly sample a first random Pauli operator P from the set of available Pauli operators. q 206 may be uniformly sampled. In other embodiments, if and when desired, calibration component 108 may select a first random Pauli operator P from the set of available Pauli operators. q 206 can be sampled non-uniformly. The Pauli operator P qThe number 206 can be any desired number N according to a defined read management standard, where N can be any desired integer value.

[0036] The RMC 106 can use a random number generator (RNG) 112 to generate random numbers for any operation described herein that utilizes random numbers. The RNG 112 can be a true random number generator capable of generating true random numbers or a pseudo-random number generator capable of generating pseudo-random numbers according to a desired RNG algorithm. Each corresponding value (e.g., number) can be associated (e.g., linked or mapped) with a corresponding Pauli operator of a set of available Pauli operators (or corresponding Pauli gates, or both), and information regarding the association (e.g., linking or mapping) between each corresponding number and each corresponding Pauli operator can be stored in and retrieved from a data store 114 to facilitate determining which Pauli operator is associated with which value.

[0037] Calibration component 108 may (e.g., at each corresponding instance) calibrate a corresponding respective Pauli operator (or corresponding Pauli gate) P of the first random Pauli operator at the output of circuit 400 prior to a first readout measurement of qubit 208 or circuit 400. q 206 to the qubit 208. The calibration component 108 can use the measurement component 116 of the RMC 106 to measure a corresponding respective first response at the output of the circuit 400 (e.g., applied to an initial state) based on a corresponding respective first random Pauli operator (e.g., applied at a corresponding respective instance for a corresponding respective measurement) and the input data applied to the circuit 400. The first response can be a first readout measurand. The calibration component 108 can measure a corresponding respective first random Pauli operator P qFor each of the parameters 206, a desired number M (e.g., one or more) of readout measurements (e.g., one or more first readout measurements) may be performed according to a defined readout management standard, where M may be a desired integer. The calibration component 108 may store the corresponding respective first readout measurements and associated corresponding respective first values ​​(e.g., corresponding respective q values) in the data store 114, where the corresponding respective first values ​​are calculated based on the corresponding respective first readout measurements and the corresponding respective first random Pauli operators P q 206. In some embodiments, the corresponding respective first values ​​may correspond to corresponding respective random numbers used to determine and select the corresponding respective first Random Pauli operators.

[0038] According to Protocol AcquireData and Protocol 1, as part of the estimation process, the estimation component 110 may utilize circuit 300 of FIG. 3, which provides a Pauli operator P that may be applied to the circuit 202 of interest and the output of circuit 300. q The estimation component 110 may include a second Pauli operator (e.g., a second random Pauli operator) P from the set of available Pauli operators based on the corresponding respective random values ​​associated with the corresponding respective random Pauli operators (and / or corresponding Pauli gates). q 206. The estimation component 110 may randomly sample (e.g., uniformly or non-uniformly) a subset (e.g., N) of the qubits 208 or the circuit 202 of interest at the output of the circuit 300 (e.g., at each corresponding instance), prior to a second readout measurement of the qubit 208 or the circuit 202 of interest, the corresponding respective Pauli operator (e.g., P q206) to the qubit 208 or the circuit 202 of interest. The estimation component 110 can use the measurement component 116 to measure a corresponding respective second response at the output of the circuit 300 based on the corresponding respective second random Pauli operator (e.g., applied at a corresponding respective instance for the corresponding respective measurement) and the input data applied to the circuit 300. The corresponding respective second response can be a corresponding respective second readout measurement. The estimation component 110 can store the corresponding respective second readout measurement and the associated corresponding respective second value in the data store 114, where the corresponding respective second value is the value of the corresponding respective second readout measurement and the corresponding respective second random Pauli operator (e.g., P q 206), the corresponding respective second values ​​may correspond to corresponding respective random numbers used to determine and select the corresponding respective second Random Pauli operators.

[0039] According to Protocol AcquireData and Protocol 1, calibration component 108 or estimation component 110 can determine or generate, or determine and generate, calibration data based on a first readout measurement measured at the output of circuit 400 and a defined first function such as that described more fully herein (e.g., as a function of the first readout measurement measured at the output of circuit 400 and the defined first function). Estimation component 110 can determine or generate, or determine and generate, estimated data based on a second readout measurement measured at the output of circuit 300 and a defined first function such as that described more fully herein. Estimation component 110 can determine or generate, or determine and generate, a normalized scalar value based on calibration data and a defined second function such as that described more fully herein (e.g., as a function of the calibration data and the defined second function). Estimation component 110 can determine or generate, or determine and generate, an estimated scalar value based on estimation data and a defined second function such as that described more fully herein. The RMC 106 may include a calculator component 118 that may be utilized by the calibration component 108, the estimation component 110, or other components of the RMC 106 to perform various calculations for various operations and protocols, such as those described above and more fully herein.

[0040] The estimation component 110 can determine or generate, or determine and generate, a read result (e.g., an error-mitigated read decision) associated with the circuit 202 of interest based on (e.g., as a function of) the normalized scalar value and the estimated scalar value, such as those described more fully herein. The read result can be an estimated read result (e.g., an unbiased estimator of the read result) or an expected read result (e.g., a read result determined and generated by and processed by the quantum computer component 102 in conjunction with the RMC 106) in which read errors have been desirably mitigated (e.g., reduced, minimized, or substantially eliminated). The read result can be, for example, an error-free expected read decision or estimator (e.g., a read result that may have a value that may be expected (e.g., quantum predicted) in the absence of errors).

[0041] RMC 106 may provide (e.g., communicate or generate) the error-mitigated readout results as output, and may present or display the error-mitigated readout results via interface component 120 (e.g., a display component including a display screen and interface, or an audio component including an audio interface, or both). Interface component 120 may present or display the readout results. Interface component 120 may further receive input data, quantum program information (e.g., instructions), and / or other information that may be processed by RMC 106 and / or provided to quantum computer component 102 to facilitate execution of the quantum program and generation of the readout results.

[0042] According to various embodiments, for 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-mitigated readout determinations or estimates). As part of Protocol AcquireData and Protocol 2, calibration component 108 can perform the same or similar calibration process as used in connection with Protocol 1, described more fully herein, or (if Protocol 1 was previously performed) can use calibration data and / or calibration results of a calibration process performed in connection with Protocol 1. For example, calibration component 108 can perform a first Pauli operator (e.g., a first random Pauli operator) P q A subset of 206 may be randomly sampled, a first random Pauli operator may be applied to qubit 208 at the output of circuit 400 prior to a first readout measurement of qubit 208 or circuit 400, a corresponding respective first response (e.g., first readout measurement) at the output of circuit 400 (e.g., applied to an initial state) may be measured based on the corresponding respective first random Pauli operator and the input data applied to circuit 400, and the corresponding respective first readout measurement and associated corresponding respective first value may be stored in data store 114, such as that more fully described herein.

[0043] According to Protocol AcquireData and Protocol 2, as part of the estimation process, the estimation component 110 can utilize the circuit 200 of FIG. 2, which includes the circuit of interest 202 as well as the Pauli operator P p 204 and the Pauli operator P q 206, which can include the Pauli operator P p 204 can be applied to the input of the circuit 200, and the Pauli operator P q206 can be applied to the output of the circuit 200. The estimation component 110 can select a second Pauli operator (e.g., a second random Pauli operator) P from the set of available Pauli operators. q A subset of 206 (e.g., N second Pauli operators) and a third Pauli operator (e.g., a third random Pauli operator) P p A pair of Pauli operators (e.g., P p 204 and P q 206) can be randomly sampled (e.g., uniformly sampled or non-uniformly sampled) based on corresponding respective random values ​​associated with corresponding respective random Pauli operators (and / or corresponding Pauli gates).

[0044] Estimation component 110 can apply (e.g., at each corresponding instance) a pair of random Pauli operators to qubit 208 or circuit 202 of interest. For example, estimation component 110 can apply (e.g., at each corresponding instance) a second random Pauli operator P q 206 can be applied to qubit 208 or circuit 202 of interest, and (e.g., at each corresponding instance) at the input of circuit 200 prior to a second readout measurement of qubit 208 or circuit 200, a third random Pauli operator P p 204 to the qubit 208 or the circuit 202 of interest. The estimation component 110 uses the measurement component 116 to convert the corresponding respective second responses at the output of the circuit 200 into corresponding respective second random Pauli operators (P q 206), and the corresponding respective third random Pauli operators (P p204), and the input data applied to the circuit 200. The corresponding respective second responses may be corresponding respective second readout measurements. The estimation component 110 may store the corresponding respective second readout measurements, as well as the associated corresponding respective second values ​​(e.g., q-values) and corresponding respective third values ​​(e.g., p-values) in the data store 114. The corresponding respective second values ​​and the corresponding respective third values ​​may be associated (e.g., linked, mapped, or appended) with the corresponding respective second readout measurements, and the corresponding respective second values ​​(e.g., q-values) may be calculated by a second random Pauli operator (P q 206), and each corresponding third value (e.g., p value) can be associated with a third random Pauli operator (P p 204).

[0045] According to Protocol AcquireData and Protocol 2, calibration component 108 or estimation component 110 can determine or generate, or determine and generate, calibration data based on a first readout measurement measured at the output of circuit 400 and a defined first function such as that described more fully herein (e.g., as a function of the first readout measurement measured at the output of circuit 400 and the defined first function). Estimation component 110 can further determine or generate, or determine and generate, estimated data based on a second readout measurement measured at the output of circuit 200 and a defined first function such as that described more fully herein. Estimation component 110 can further determine or generate, or determine and generate, a normalized scalar value based on the calibration data and a defined second function such as that described more fully herein (e.g., as a function of the calibration data and the defined second function). Estimation component 110 can further determine or generate, or determine and generate, an estimated scalar value based on the estimation data and a defined second function such as that described more fully herein.

[0046] The estimation component 110 can determine or generate, or determine and generate, a read result (e.g., an error-mitigated read decision) associated with the circuit 202 of interest based on (e.g., as a function of) the normalized scalar value and the estimated scalar value, such as those described more fully herein. The read result can be an estimated read result (e.g., an unbiased estimator of the read result) or an expected read result (e.g., a read result determined and generated by the quantum computer component 102 in conjunction with the RMC 106 and processed by the RMC 106) in which read errors have been desirably mitigated (e.g., reduced, minimized, or substantially eliminated). The read result can be, for example, an error-free expected read decision or estimator. The RMC 106 can provide (e.g., communicate or generate) the error-mitigated read result as an output. For example, an interface component can present or display the error-mitigated read result.

[0047] According to various embodiments, to facilitate performing various functions of the RMC 106, the RMC 106 may further include or be associated with a processor component 122 that may function in conjunction with other components (e.g., calibration component 108, estimation component 110, RNG 112, data store 114, measurement component 116, calculator component 118, interface component 120, or other components) (as shown in the figures). The processor component 122 may use one or more processors, microprocessors, or controllers that can process data, such as information regarding 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 directed or defined by the processes, protocols, methods, and / or techniques described herein, and / or quantum algorithms), quantum logic, defined read control criteria, traffic flow, policies, protocols, interfaces, tools, or other information, or combinations thereof, to facilitate the operation of the RMC 106 as more fully disclosed herein, and to control the flow of data between the RMC 106 and other components associated with (e.g., connected to) the RMC 106 (e.g., quantum computer component 102, quantum programs, data storage devices, user devices or endpoint devices, or other computing or communication devices).

[0048] Further with respect to data store 114, to facilitate controlling operations associated with RMC 106, data store 114 may store information regarding data structures (e.g., user data, metadata), code structures (e.g., modules, objects, hashes, classes, procedures) or instructions, and 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 directed or defined by the processes, protocols, methods and / or techniques described herein, and / or quantum algorithms), quantum logic, defined readout management criteria, traffic flows, policies, protocols, interfaces, tools, or other information, or combinations thereof. In one aspect, the processor component 122 may be operatively coupled to the data store 114 (e.g., via a memory bus) to store and retrieve information desired to operate and / or provide functionality to the calibration component 108, the estimation component 110, the RNG 112, the data store 114, the measurement component 116, the calculator component 118, the interface component 120 or other components, and / or substantially any other operational aspect of the RMC 106, at least in part.

[0049] These and other aspects and embodiments of the estimation and read error mitigation techniques and estimation protocols of the disclosed subject matter are now further described.

[0050] A key component for the successful execution of a quantum algorithm can be the ability to access results through measurements. One of the significant challenges of quantum computing can be dealing with readout errors. The disclosed subject matter can enable the mitigation of readout errors in the calculation of expected values ​​of Pauli observables, which emerge in a wide range of applications, from partial tomography of quantum states and processes to electronic structure determination using variational quantum eigensolvers (VQEs). In this setting, there is no need to correct the individual measurements used, for example, in quantum error correction and random number generation.

[0051] The measurement output of a quantum circuit can be characterized by an ideal probability vector P. The noisy readout can usually be represented by a classical noise map. For an n-qubit system, this map can be written as n ×2 n The left probability matrix A can be expressed as i,j can denote the probability of measuring i instead of j. Thus, the noisy probability vector can be transformed into a linear

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[0052] To address these and other issues associated with traditional error mitigation approaches, the disclosed subject matter can employ readout error mitigation techniques for values ​​determined or estimated by a quantum computer, which can include quantum benchmarking. Unlike traditional methods, the techniques of the disclosed subject matter do not estimate probability vectors. Instead, the techniques of the disclosed subject matter (e.g., used by RMC 106) can diagonalize a Pauli readout transfer matrix that represents the transitions between the Pauli z components of the system state ρ and their measurands, which can enable RMC 106 to form unbiased estimators of these components up to their statistical uncertainty. The techniques of the disclosed subject matter do not use the A matrix directly; instead, they can diagonalize the transfer matrix under a Hadamard transform. Like all current methods, the disclosed subject matter can function assuming accurate state preparation for calibration.

[0053] With respect to the techniques and methods of the disclosed subject matter, consider a system of n qubits, P q But q∈P:=[0,4 n-1]. The Pauli string representation when read from right to left

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[0054] Any unitary operator U can be expressed as its Pauli transfer matrix T U The elements i,j∈P of the transfer matrix can be expressed as follows:

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[0055] Further, the basis change can be performed by applying certain gates to the circuit. The final part of the circuit is the first setting: (1) basis change B, (2) Pauli gate P. q Note that (3) the application of a basis change B, followed by the Pauli gate P, can be used to calculate the readout (e.g., the readout estimator). When the basis change B is a Clifford operator, which can be a typical case, q Applying the Pauli gate P s This can be equivalent to applying the Pauli gate P followed by a basis change B. q If a Pauli gate P is given, s This means that (1) Pauli P s (2) the application of the basis change B, and (3) the readout value, which means that the second setting can be equivalent to the first setting. q or Pauli Gate P s Given a Pauli gate P s or Pauli Gate P qcan each be determined or calculated based on a basis transformation B. Also or similarly, this can further apply to initial conditions such that a Pauli gate (e.g., a random Pauli gate) and subsequent circuit of interest can be equivalent to a first portion of a circuit of interest (e.g., a Clifford operator) and subsequent Pauli gate and second portion of the circuit of interest. The disclosed subject matter can include indicated or applied settings (e.g., circuit settings or configurations) that include basis transformations, and any type of equivalent setting that includes basis transformations.

[0056] Further with regard to the estimation protocols of the disclosed subject matter, the RMC 106 can run (e.g., execute) various instances of circuit 200 of FIG. 2 to estimate a quantity of interest. Circuit 200 can be parameterized by Pauli indices p and q and operator C (or circuit C that can implement operator C). It can be assumed that the identity operator can be made less complex, resulting in circuit 300 of FIG. 3 and circuit 400 of FIG. 4, which are more fully described herein. A general procedure for acquiring data (e.g., Protocol AcquireData) can be as follows: Protocol AcquireData(S p ,S q ,C,N) 1: Initialize an empty data set D 2: for i=1,...,N do 3: S p uniformly sample p from 4: S q uniformly sample q from 5: Run the circuit in Figure 2 with parameters p, q and operator C. 6: Record the measured quantity m and add (p,q,m) to D 7:Return D Each measurand can be expressed 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 to the last qubit. For classical post-processing of the data, we can write the function as

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[0057] The RMC 106 (e.g., using the calibration component 108 and the estimation component 110) calculates the Tr(P iρ ) can be estimated using Protocol 1, which can include the following operations: Protocol 1 1. D0 = AcquireData(T,X,I,N) 2. D1 = AcquireData(T,X,U,N) 3. Return the estimator f(D1,i,0) / f(D0,i,0)

[0058] It should be noted that data obtained by RMC 106 in operations 1 and 2 of Protocol 1 (e.g., determining calibration data D0 by calibration component 108 and determining estimated data D1 by estimation component 110) can be reused by RMC 106 to evaluate this quantity in operation 3 of Protocol 1 (e.g., the operation that returns an estimator) for different values ​​of i, j ∈ Z. In some embodiments, to reduce complexity, RMC 106 can set the number of samples (e.g., sampling of the Pauli operator) in each of the two data sets to N, where N can be any desired integer. In other embodiments, RMC 106 can use or choose to use a different number of samples for each or some of these operations of Protocol 1. The T in equation (2) for any i, j ∈ Z (e.g., as determined by estimation component 110) can be U The estimation of (i,j) can follow a similar approach and can be given by Protocol 2 as follows: Protocol 2 1. D0 = AcquireData(T,X,I,N) 2. D2=AcquireData(X,X,U,N) 3. Return the estimator f(D2,i,j) / f(D0,i,0)

[0059] As mentioned before, in operation 3 (e.g., the operation that returns the estimator of Protocol 2), we calculate T for various i, j∈Z. UTo evaluate (i,j), data (e.g.,) from operations 1 and 2 (e.g., determining calibration data D0 by calibration component 108 and determining estimation data D2 by estimation component 110) can be reused (e.g., by estimation component 110 or other components of RMC 106). Because the data obtained in operation 1 may be independent of the selection of U, the data can be shared with respect to both protocols (e.g., Protocol 1 and Protocol 2) and can be reused (e.g., by RMC 106) for different operators U. As described more fully herein, in some cases it can be or may be beneficial to replace the parameter X with P in data acquisition.

[0060] Regarding the derivation of various aspects and features of the disclosed subject matter, starting with notation, 1 can denote a vector among all vectors of appropriate size, e i can denote the i-th column of the identity matrix I, and therefore the i-th column of the Hadamard matrix H is h i =He i Furthermore, H can be expressed as the inverse H -1 =2 -n Note that the Pauli transfer matrix T can be a symmetric real matrix with H. Therefore, from the disclosed ordering of the Pauli operators, U 2 in the top left n ×2 n The block can contain the transfer coefficients between the Pauli z operators, and this matrix is Τ Next, we define the function Z that maps the density operator ρ as a function of length 2 containing the weights of each Pauli z operator. n can be defined for vectors of

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[0061] The disclosed subject matter can use an unscaled trace, which can mean that for an initial state ρ, Z(ρ) = 1. Applying Z to the conjugate of ρ with the operator U gives Z(UρU†) = ΤU Z(ρ0). The selected Pauli order of the disclosed subject matter further defines the measurement probability vector corresponding to the state ρ as p=H -1 Z(ρ) can be written succinctly as Z(ρ). The disclosed subject matter (e.g., RMC106 or other components of the disclosed subject matter) can model measurement errors by applying a transition matrix A, which can be expressed as a probability

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[0062] To see the effect of adding random Pauli operators, we consider the vector of exchange values ​​d q =Σ i∈Ζ μ(i,q)e i and the corresponding diagonal matrix D q =diag(d q ) is defined as Z(P q ρP q )=D q Z(ρ), and based on the properties of the Pauli x-subgroup, for any i∈Ζ

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[0063] M=HAH -1 indicates the function

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[0064] According to the disclosed subject matter, p=0 can be fixed, thereby P p = I can be given,

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[0065] For a more general treatment, Z can be replaced by a function C that extracts all the Pauli coefficients. U can be replaced by the complete transfer matrix TU. n2 columns contain the Hadamard matrix and the remaining elements can be zero n ×4 n By applying a linear map, one can obtain ideal noise-free measurements (e.g., by RMC106 or other components of the disclosed subject matter). In a general setting, for some k∈P\Z, T r (P kρ0 ) ≠ 0. Similarly, measurands can potentially be affected by terms that do not belong to the Pauli z group. By adding operators randomly sampled from the Pauli subgroup S (e.g., by RMC106), we can see that, on average, the disclosed subject matter can eliminate all Pauli terms that do not commute with all elements in S. In the particular case where S corresponds to a Pauli z group, this can mean that the disclosed subject matter (e.g., RMC106) can eliminate all terms outside this group, since it is a maximally commuting subgroup. Thus, for the initial state, RMC106 can add operators P to the random Pauli z matrix. p This can be equivalent to sampling p from P instead of X. Similarly, the disclosed subject matter (e.g., RMC106) can eliminate all terms that do not belong to the Pauli z group by sampling q from P. The disclosed subject matter then replaces the noisy measurement operator with AH -1 Λ, where Λ is the quantum noise channel. This can be expressed as (AH -1 ΛH)H -1 =A'H -1 and thus, this can enable the disclosed subject matter (e.g., RMC106) to model quantum noise as a classical noise channel.

[0066] The purpose of Protocols 1 and 2 can be to estimate the quantities in equation (9). If these quantities are written in the form x / y, the protocols used by RMC 106 estimate the quantities

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[0067] Lemma IV.1. Let x, y be 0≦|x|≦|y|≦1.

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[0068] Proof. Without loss of generality, assume x,y ≥ 0. Taking a Taylor series expansion around zero for a sufficiently small α, we find that in the worst case,

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[0069] In the last inequality, the disclosed subject matter can use that x / y≦1 and 1−α / y≧½. The disclosed subject matter (e.g., RMC 106 or other components) can derive a lower bound as well to obtain a given result.

[0070] This allows the disclosed subject matter to obtain the following sample complexity: Theorem IV.2. With probability at least 1-δ, Protocols 1 and 2 (e.g., RMC106 using Protocols 1 and 2), respectively, have a sample complexity N of

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[0071] Proof. Protocols 1 and 2 can take data and estimate different quantities using the function in equation (6). For fixed i and j, each term in the summation can be seen as an independent ±1 sample from some distribution that depends on U, which can be ignored over the Pauli indices p and q. For the error in the estimated quantity, the disclosed subject matter allows us to apply Höffding's inequality, which states that the independent random variable X from any distribution over [-1, 1] i Given, the expected value

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[0072] It is sometimes desirable (e.g., required) to ensure that the probability of deviation from the expectation by more than α is bounded by δ / 2. Using this union bound, the enumerator and denominator can be α close to their expectation with probability at least 1-δ. Bounding the failure probability in equation (10) by δ / 2 is a sufficient condition.

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[0073] The disclosed subject matter can choose (e.g., select) α such that the final estimator can be exactly ε. From Lemma IV.1, it follows that it is sufficient to take 4α / y≦ε, where y=f I (i)=M i,i In equation (11), α = εM i,i Substituting / 4 can give the desired result.

[0074] Regarding the number of circuit instances, for a given q, the term D q MD q cross product with M

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[0075] The disclosed subject matter is i,i This can provide a bound on the number of circuit instances utilized or desired to estimate T with a given accuracy. If the disclosed subject matter can multiply by an approximately diagonal mask, this bound can depend in part on the largest off-diagonal element of M. The disclosed subject matter can show how this corresponds to properties of the transfer matrix A, and these properties can be studied for different types of transfer matrices. T U When sampling individual elements of , it may be desirable (e.g., useful or appropriate) to have all but one of the coefficients of Z(ρ) approximately zero. To determine the extent to which the disclosed subject matter is desirably reduced to these elements, the properties of the Pauli transfer matrix can be studied. A final estimate (e.g., of a readout result) can be given by the ratio of two quantities, and thus consideration can be given to how estimation errors in these quantities may affect the results. In this section of the disclosed subject matter, for clarity, it is noted that the disclosed subject matter can work with a full matrix representation, but as shown and described elsewhere in the disclosed subject matter, the processing itself can be performed based on individual elements.

[0076] With respect to the number of circuit instances, the disclosed subject matter may wish to approximately diagonalize M, and the disclosed subject matter may have the following result: Theorem IV.3. Given k randomly sampled values ​​q1,...,q k ∈Χ and index set T⊆[2 n ] is given,

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[0077] Regarding scaling of off-diagonal elements, the disclosed subject matter (e.g., RMC106) can uniformly sample X from {-1, 1}. Each element has a maximum ε b If you can guarantee that the data is scaled by a factor of M i,i In the estimation of b There may be an additive term with magnitude β. M i,i For the estimation of σ, the disclosed subject matter can apply equation (13), where X is a suitable distribution in [-1, 1] and the maximum deviation ε α Using union bounds over the off-diagonal elements of the rows, the disclosed subject matter satisfies the condition

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[0078] If β=0, the disclosed subject matter is α = ε, and ε b →∞. Using the union bound over the rows of T,

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[0079] For the more general case where β ≠ 0, the disclosed subject matter is α =ε b can be selected, which satisfies the condition in Eq. (14).

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[0080] ε α +ε b Since β≦ε, the disclosed subject matter α≤ ε / (1 + β). Combined with the union bound obtained by multiplying the left-hand side of equation (14) by the number of elements in the set or group T, this gives the sample complexity shown in equation (12).

[0081] As an aside, diagonalizing quantum noise channels using Pauli rotation can follow exactly the same principles that the disclosed subject matter uses to diagonalize M. The disclosed subject matter (e.g., RMC106 or other components) can utilize any desired modification of Theorem IV.3 to determine the number of circuits desired (e.g., required) to ensure that all off-diagonal noise terms are bounded by ε.

[0082] With respect to example transition matrices, for a given transition matrix A, the disclosed subject matter provides a corresponding transformed matrix M=HAH -1 can be defined as the readout transition matrix for the Pauli z operator. Whenever there exists an inverse of A, M -1 =HA -1 H -1 For a convex combination of two error channels, i.e., A = μA1 + (1 - μ)A2, with μ∈|0,1|, the disclosed subject matter can have M = μM1 + (1 - μ)M2. This generalizes directly to a convex combination of any number of transition matrices.

[0083] As a basic example of a transition matrix, consider the case where the outcome of each qubit is flipped independently with some probability r. The transition matrix for a single qubit can be given by equation (1) where r = s, and the global transition matrix

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[0084] In this case, since M is already diagonal, the disclosed subject matter does not need to collapse the off-diagonal elements. Therefore, the disclosed subject matter may be sufficient to choose an arbitrary fixed value for q∈X for the circuit, rather than sampling. Choosing q=0 can make the resulting circuit less complex. For simplicity, let all probabilities r l Assuming that is equal to r, from equation (16), the diagonal element M i,i is the Pauli z operator P i can be directly related to the weight of P i Each term of σ z For P, the disclosed subject matter can have multiplicative terms (1-2r). i The diagonal terms for are (1-2r) k It can be given by the term (1-2r) k can be bounded by 1-2kr, which can mean that for 30 qubits with a 1% probability of measurement flip, the diagonal elements of M are still at least 0.4. In the noiseless case, the bit-flip probability can be forced to zero, and the disclosed subject matter can obtain A=M=I.

[0085] The transition matrix for the situation where the disclosed subject matter only measures zero is the corresponding matrix

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[0086] Regarding some practical considerations, much of the discussion so far has assumed ideal state preparation: instead of ρ0 = |0〉〈0|,

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[0087] After the RMC 106 (e.g., the calibration component 108 of the RMC 106) acquires a calibration data set, the RMC 106 can utilize the calibration data to mitigate read errors for the circuit using various Us and potentially using basis transformations. In practical systems, gradual changes in systemic gate and read errors can be expected. This may mean that the calibration data may have a limited lifetime. With regard to error mitigation in the disclosed techniques, the RMC 106 can, for example, traverse the calibration data and calculate correction coefficients for individual Pauli-z operators whenever the RMC 106 is used. This approach by RMC 106 makes updating the calibration data set desirably lightweight (e.g., very lightweight); calibration component 108 can easily augment the calibration data with time stamps and periodically add new data points (e.g., calibration data points), while erasing (e.g., removing, discarding, erasing, or deactivating) data outside a desired time window (e.g., the current time window) according to defined readout criteria. For approaches based on explicit inversion of the transfer matrix, such an update by calibration component 108 can amount to regenerating the entire matrix and its inverse. The computational complexity of calibration component 108 updating the correction coefficients using Equation (6) can be linear in the size of the data set. The evaluation of the elements of the Hadamard matrix and the exchange between two n-qubit Pauli operators by RMC 106 can both take time O(n).

[0088] As disclosed herein, the disclosed subject matter typically provides a

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[0089] The disclosed subject matter (e.g., RMC106) can have several advantages over conventional estimation techniques for quantum readout results by facilitating the estimation of readout results using the techniques and estimation protocols described herein. The techniques and protocols of the disclosed subject matter can be more efficient and accurate in estimating readout results and can more desirably mitigate (e.g., reduce or minimize) errors in estimating Pauli observables. Unlike conventional techniques, the techniques and protocols of the disclosed subject matter do not require a priori assumptions or models of the readout error process. The techniques and protocols of the disclosed subject matter can be based on expanding a quantum circuit with randomly selected Pauli operators and evaluating a scalar function based on measurements obtained using a series of random instances. The disclosed subject matter can desirably mitigate readout errors by dividing the function values ​​for the quantum circuit of interest by the function values ​​of a benchmark circuit (e.g., a calibration circuit). The techniques and protocols of the disclosed subject matter (e.g., as utilized by RMC106) can work by diagonalizing the readout error transfer matrix in the Hadamard domain, which makes inversion relatively trivial. In contrast to many conventional algorithms, the techniques and protocols of the disclosed subject matter (e.g., as utilized by RMC106) can directly estimate the weights of the Pauli z components of the states (or elements of the Pauli transfer matrix) rather than the distribution of measurements. Simulations of embodiments of the techniques and protocols of the disclosed subject matter show that such techniques and protocols can mitigate correlated readout errors in a 12-qubit system with a relatively small number (e.g., very few) of measurements and circuit instances.

[0090] Systems and / or devices have been described (or will be described) herein with respect to interactions between several components. It should be understood that such systems and components may include components or subcomponents designated therein, portions of designated components or subcomponents, or additional components, or combinations thereof. Subcomponents may be implemented as components communicatively coupled to other components rather than as components included within a parent component. Furthermore, one or more components and / or subcomponents may be combined into a single component that provides aggregate functionality. These components may further interact with one or more other components not specifically described herein for the sake of brevity but known to those skilled in the art.

[0091] 6 and 7 illustrate a flowchart of an exemplary, non-limiting method 600 that can desirably mitigate readout errors associated with readout results generated by a quantum computer, according to various aspects and embodiments of the disclosed subject matter. In some embodiments, method 600 can be performed by an RMC and / or processor component, which can be associated with a data store, for example. The RMC can include a calibration component, an estimation component, or other components (e.g., other configuration components), such as those described herein, or a combination thereof. The RMC can be associated with a quantum computer, and the RMC can receive a readout decision (e.g., a readout result) from a circuit (e.g., a quantum circuit) formed using components (e.g., quantum components) including qubits and the circuit (e.g., a quantum circuit) of the quantum computer. For the sake of brevity, repeated descriptions of identical elements used in other embodiments described herein have been omitted or may be omitted.

[0092] At 602, a first random Pauli gate (P) is selected from the set of available Pauli gates based on a corresponding respective random number (e.g., a randomly generated number) associated with the corresponding respective random Pauli gate. q ) can be randomly sampled. The RMC can generate random numbers using an RNG, and each corresponding number can be associated (e.g., linked or mapped) with a corresponding respective Pauli gate, and information regarding the association (e.g., linking or mapping) between each corresponding number and each corresponding Pauli gate can be stored in and retrieved from a data store to facilitate determining which Pauli gate is associated with which number. At 604, at a first output of the first circuit (e.g., circuit 400 of FIG. 4 ), a first random Pauli gate can be applied to the qubit prior to a first readout measurement of the qubit or the first circuit. At 606, each corresponding first response at the first output of the first circuit (e.g., applied to an initial state) can be measured based on the corresponding respective first random Pauli gate and the input data applied to the first circuit. The first response can be a first readout measurement. The calibration component may perform one or more readout measurements (e.g., one or more first readout measurements) for each of the first random Pauli gates. At 608, the corresponding respective first responses (e.g., first readout measurements) and associated corresponding respective first values ​​may be stored in a data store, where the corresponding respective first values ​​may be associated with the corresponding respective first readout measurements (e.g., corresponding respective first readout measurements) and the corresponding respective first random Pauli gates. The corresponding respective first values ​​may correspond to corresponding respective random numbers used to determine and select the corresponding respective first random Pauli gates.

[0093] At 610, a second random Pauli gate (P) is selected from the set of available Pauli gates based on the corresponding respective random numbers associated with the corresponding respective random Pauli gates. q ) can be randomly sampled. At 612, a second random Pauli gate can be applied to the qubit or circuit of interest (e.g., C) at a second output of a second circuit (e.g., circuit 300 of FIG. 3 ) prior to a second readout measurement of the qubit, which may include a third circuit, which may be the circuit of interest. At this point, method 600 can proceed to reference point A, from which method 600 can proceed to reference numeral 614, as shown in FIG. 7.

[0094] At 614, a corresponding respective second response at a second output of the second circuit may be measured based on the corresponding respective second random Pauli gate and the input data applied to the second circuit. The corresponding respective second response may be a corresponding respective second readout measurement. At 616, the corresponding respective second response (e.g., the second readout measurement) and an associated corresponding respective second value may be stored in a data store, where the corresponding respective second value may be associated with the corresponding respective second readout measurement (e.g., the corresponding respective second readout measurement) and the corresponding respective second random Pauli gate. The corresponding respective second value may correspond to a corresponding respective random number used to determine and select the corresponding respective second random Pauli gate.

[0095] At 618, calibration data can be determined based on the first readout measurement measured at the first output of the first circuit and a defined first function such as that described more fully herein (e.g., as a function of the first readout measurement measured at the first output of the first circuit and the defined first function). At 620, estimated data can be determined based on the second readout measurement measured at the second output of the second circuit and a defined first function such as that described more fully herein.

[0096] At 622, a normalized scalar value may be determined based on the calibration data and the defined second function (e.g., as a function of the calibration data and the defined second function). At 624, an estimated scalar value may be determined based on the estimated data and the defined second function, such as that described more fully herein. At 626, an error-mitigated read determination quantity associated with the circuit of interest may be generated based on the normalized scalar value and the estimated scalar value (e.g., the estimated scalar value divided by the normalized scalar value), such as that described more fully herein. The error-mitigated read determination quantity may be an estimated read result (e.g., an unbiased estimator of the read result) or an expected read result (e.g., a read result determined and generated by a quantum computer and processed by an RMC) that may have a desirably mitigated (e.g., reduced or minimized) read error. At 628, the error-mitigated readout determination may be provided (eg, communicated or generated) as an output (eg, an output from an RMC associated with a quantum computer).

[0097] 8 and 9 illustrate a flowchart of another exemplary, non-limiting method 800 that can desirably mitigate readout errors associated with readout results generated by a quantum computer, according to various aspects and embodiments of the disclosed subject matter. Method 800 can be performed by an RMC and / or processor component, which can be associated with a data store, for example. The RMC can include a calibration component, an estimation component, or other components (e.g., other configuration components), such as those described herein, or a combination thereof. The RMC can be associated with a quantum computer, and the RMC can receive a readout decision (e.g., a readout result) from a circuit (e.g., a quantum circuit) formed using components (e.g., quantum components) including qubits and the circuit (e.g., a quantum circuit) of the quantum computer. For the sake of brevity, repeated descriptions of identical elements used in other embodiments described herein have been omitted or may be omitted.

[0098] At 802, a first random Pauli gate (P) is selected from the set of available Pauli gates based on a corresponding respective random number (e.g., a randomly generated number) associated with the corresponding respective random Pauli gate. q) can be randomly sampled. The RMC can generate random numbers using an RNG, and each corresponding number can be associated (e.g., linked or mapped) with a corresponding respective Pauli gate, and information regarding the association (e.g., linking or mapping) between each corresponding number and each corresponding Pauli gate can be stored in and retrieved from a data store to facilitate determining which Pauli gate is associated with which number. At 804, at a first output of the first circuit (e.g., circuit 400), a first random Pauli gate can be applied to the qubit prior to a first readout measurement of the qubit or the first circuit. At 806, each corresponding first response at the first output of the first circuit (e.g., applied to an initial state) can be measured based on the corresponding respective first random Pauli gate and the input data applied to the first circuit. The first response can be a first readout measurement. The calibration component may 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 corresponding respective first responses (e.g., first readout measurements) and associated corresponding respective first values ​​may be stored in a data store, where the corresponding respective first values ​​may be associated with the corresponding respective first readout measurements and the corresponding respective first random Pauli gates. The corresponding respective first values ​​may correspond to corresponding respective random numbers used to determine and select the corresponding respective first random Pauli gates.

[0099] At 810, a second random Pauli gate (P q ) and the third random Pauli gate (P p) can be randomly sampled. At 812, a pair of random Pauli gates can be applied to a qubit or circuit of interest (C), which can include applying a second random Pauli gate to the qubit or circuit of interest at a second output of a second circuit (e.g., circuit 200) prior to a second readout measurement of the qubit, which can include a third circuit, which can be the circuit of interest, and the third random Pauli gate can be associated with an input of the circuit of interest. At this point, method 800 can proceed to reference point B, from which method 800 can proceed to reference numeral 814, as shown in FIG. 9 .

[0100] At 814, corresponding respective second responses at the second outputs of the second circuits may be measured based on the pairs of random Pauli gates applied to the second circuits and the input data applied to the second circuits. The corresponding respective second responses may be corresponding respective second readout measurements. At 816, the corresponding respective second responses (e.g., second readout measurements) and associated corresponding respective second values ​​and corresponding respective third values ​​may be stored in a data store, where the corresponding respective second values ​​and corresponding respective third values ​​may be associated (e.g., linked, mapped, or appended) with the corresponding respective second readout measurements, the corresponding respective second values ​​may be associated with the second random Pauli gates, and the corresponding respective third values ​​may be associated with the third random Pauli gates.

[0101] At 818, calibration data can be determined based on the first readout measurement measured at the first output of the first circuit and a defined first function such as that described more fully herein (e.g., as a function of the first readout measurement measured at the first output of the first circuit and the defined first function). At 820, estimated data can be determined based on the second readout measurement measured at the second output of the second circuit and a defined first function such as that described more fully herein.

[0102] At 822, a normalized scalar value can be determined based on the calibration data and a defined second function, such as one described more fully herein (e.g., as a function of the calibration data and the defined second function). At 824, an estimated scalar value can be determined based on the estimated data and a defined second function, such as one described more fully herein. At 826, an error-mitigated read determination quantity associated with the circuit of interest can be generated based on the normalized scalar value and the estimated scalar value (e.g., the estimated scalar value divided by the normalized scalar value) (e.g., as a function of the normalized scalar value and the estimated scalar value (e.g., the estimated scalar value divided by the normalized scalar value)). The error-mitigated read determination quantity can be an estimated read result (e.g., an unbiased estimator of the read result) or an expected read result (e.g., a read result determined and generated by a quantum computer and processed by an RMC) that may have a desirably mitigated (e.g., reduced or minimized) read error. At 828, the error-mitigated readout determination may be provided (eg, communicated or generated) as an output (eg, an output from an RMC associated with a quantum computer).

[0103] For simplicity of explanation, these 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 depicted acts or by the order of acts, or both; for example, acts can be performed in various orders or simultaneously, or with other acts not depicted and described herein. Moreover, not all depicted acts are required to implement a computer-implemented method in accordance with the disclosed subject matter. Furthermore, those skilled in the art will understand and appreciate that a computer-implemented method can alternatively be represented as a series of interrelated states via a state diagram or events. It is also to be appreciated that the computer-implemented methods disclosed below and throughout this specification can be stored on an article of manufacture to facilitate transporting and transferring such computer-implemented methods to a computer. As used herein, the term article of manufacture is intended to encompass a computer program accessible from a computer-readable device or storage medium.

[0104] To provide background for various aspects of the disclosed subject matter, FIG. 10 and the following discussion are intended to provide a general description of a suitable environment in which various aspects of the disclosed subject matter may be implemented. FIG. 10 illustrates a block diagram of an exemplary, non-limiting operating environment that may facilitate one or more embodiments described herein. For the sake of brevity, repeated descriptions of identical elements used in other embodiments described herein may be omitted or omitted. Referring to FIG. 10, a suitable operating environment 1000 for implementing various aspects of the present disclosure may further include a computer 1012. The computer 1012 may further 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 may be any of a variety of available processors. Dual microprocessors and other multiprocessor architectures may also be used as the processing unit 1014. The system bus 1018 may be any of several types of bus structures, including a memory bus or memory controller, a peripheral or external bus, or a local bus, or a combination thereof, using any of a variety of available bus architectures, including, but not limited to, Industrial Standard Architecture (ISA), MicroChannel Architecture (MCA), Enhanced 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 may further include volatile memory 1020 and nonvolatile memory 1022.The nonvolatile memory 1022 stores the basic input / output system (BIOS), which contains the basic routines for transferring information between elements within the computer 1012, such as during start-up. By way of example, and not limitation, the nonvolatile memory 1022 may include read-only memory (ROM), programmable ROM (PROM), erasable programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), flash memory, or nonvolatile random access memory (RAM), such as ferroelectric RAM (FeRAM). The volatile memory 1020 may also include random access memory (RAM), which acts as external cache memory. Many forms of RAM are available, including, by way of example and not limitation, 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), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM.

[0105] The computer 1012 may further include removable / non-removable, volatile / non-volatile computer storage media. FIG. 10 illustrates, for example, disk storage 1024. Disk storage 1024 may further include devices such as, but not limited to, a magnetic disk drive, a floppy disk drive, a tape drive, a Jaz drive, a Zip drive, an LS-100 drive, a flash memory card, or a memory stick. Disk storage 1024 may further include storage media separate from or combined with other storage media, including, but not limited to, an optical disk drive, such as a compact disc read-only memory (CD-ROM), a CD recordable drive (CD-R drive), a CD rewritable drive (CD-RW drive), or a digital versatile disc read-only memory (DVD-ROM). A removable or non-removable interface, such as interface 1026, is typically used to facilitate connection of the disk storage 1024 to the system bus 1018. FIG. 10 further illustrates software that acts as an intermediary between a user and the basic computer resources described in suitable operating environment 1000. Such software may further include, for example, an operating system 1028. The operating system 1028, which may be stored on disk storage 1024, functions to control and allocate resources of the computer 1012. System applications 1030 take advantage of the management of resources by the operating system 1028 through program modules 1032 and program data 1034 stored, for example, in system memory 1016 or on disk storage 1024. It should be appreciated that the present disclosure may be implemented with various operating systems or combinations of operating systems. A user enters commands or information into the computer 1012 through input devices 1036.The input devices 1036 include, but are not limited to, pointing devices such as mice, trackballs, styluses, touch pads, keyboards, microphones, joysticks, game pads, satellite dishes, scanners, TV tuner cards, digital cameras, digital video cameras, web cameras, etc. These and other input devices connect to the processing unit 1014 through the system bus 1018 via interface ports 1038. Interface ports 1038 include, for example, serial ports, parallel ports, game ports, and universal serial buses (USB). The output devices 1040 use some of the same types of ports as the input devices 1036. Thus, for example, a USB port can be used to provide input to the computer 1012 and to output information from the computer 1012 to the output device 1040. An output adapter 1042 is provided to illustrate that some output devices 1040, such as monitors, speakers, and printers, among others, require dedicated adapters. By way of example, output adapters 1042 include, but are not limited to, video cards and sound cards that provide a means of connection between output device(s) 1040 and system bus 1018. It should be noted that other devices and / or device systems, such as remote computer(s) 1044, provide both input and output capabilities.

[0106] 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, server, router, network PC, workstation, microprocessor-based device, peer device, or other common network node, and typically further includes many or all of the elements described relative to the computer 1012. For simplicity, only a memory storage device 1046 is shown for the remote computer 1044. The remote computer 1044 is logically connected to the computer 1012 through a network interface 1048 and then physically connected via a communication connection 1050. The network interface 1048 encompasses wired and / or wireless communication networks such as a local area network (LAN), a wide area network (WAN), and a cellular network. LAN technologies include Fiber Distributed Data Interface (FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ring, and the like. WAN technologies include, but are not limited to, point-to-point links, circuit-switched networks such as Integrated Services Digital Networks (ISDN) and variations thereon, packet-switched networks, and Digital Subscriber Lines (DSL). Communications connection 1050 refers to the hardware / software used to connect network interface 1048 to system bus 1018. For illustrative purposes, communications connection 1050 is shown internal to computer 1012, although communications connection 1050 can also be external to computer 1012. For illustrative purposes only, the hardware / software for connecting to network interface 1048 can further include internal and external technologies such as modems, including regular telephone-grade modems, cable modems, and DSL modems, ISDN adapters, and Ethernet cards.

[0107] One or more embodiments may be a system, method, apparatus, or computer program product, or combinations thereof, at any level of technical detail capable of integration. The computer program product may include a computer-readable storage medium having computer-readable program instructions thereon for causing a processor to execute aspects of one or more embodiments. The computer-readable storage medium may be a tangible device capable of holding and storing instructions for use by an instruction-execution device. The computer-readable storage medium may be, for example, but 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 thereof. A non-exhaustive list of more specific examples of computer-readable storage media may include portable computer diskettes, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disk read-only memory (CD-ROM), digital versatile disk (DVD), memory stick, floppy disk, mechanically encoded devices such as punch cards or raised structures in grooves on which instructions are recorded, and suitable combinations thereof. As used herein, computer-readable storage media should not be construed as being, per se, ephemeral signals, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating in waveguides or other transmission bodies (e.g., light pulses traveling in fiber optic cables), or electrical signals transmitted through electrical wires.

[0108] The computer-readable program instructions described herein can be downloaded from a computer-readable storage medium to each corresponding computing / processing device, or can be downloaded to an external computer or external storage device over a network, such as the Internet, a local area network, a wide area network, or a wireless network, or a combination thereof. This network can include copper transmission cables, optical fiber transmissions, wireless transmissions, routers, firewalls, switches, gateway computers, or edge servers, or a combination thereof. A network adapter card or network interface within each computing / processing device receives the computer-readable program instructions from the network and forwards the computer-readable program instructions to a computer-readable storage medium within each corresponding computing / processing device for storage. The computer-readable program instructions for carrying out the operations of the disclosed subject matter may be assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state setting data, or configuration data for an integrated circuit, or may be source or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk®, C++, and procedural programming languages ​​such as the "C" programming language or the like. The computer-readable program instructions may be executed entirely on a user's computer, partially on a user's computer, as a stand-alone software package, partially on a user's computer and partially on a remote computer, or entirely on a remote computer or remote server.In the last scenario above, the remote computer may be connected to the user's computer via any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (e.g., via the Internet using an Internet service provider). In some embodiments, to perform aspects of the disclosed subject matter, electronic circuitry, including, for example, a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA), may execute the computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry.

[0109] Aspects of the disclosed subject matter are described herein with reference to flowchart and / or block diagram illustrations of methods, apparatus (systems), and computer program products according to embodiments of the subject disclosure. It will be understood that each block of the flowchart and / or block diagram illustrations, and combinations of blocks in the flowchart and / or block diagram illustrations, can be embodied by computer-readable program instructions. These computer-readable program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, or other programmable data processing device forming a machine, such that the instructions, executed by the processor of the computer or other programmable data processing device, generate a method that implements the functions / operations specified in the blocks of the flowchart and / or block diagram illustrations. These computer-readable program instructions can also be stored on a computer-readable storage medium that can direct a computer, programmable data processing device, or other device, or combination thereof, to function in a particular manner, such that the computer-readable storage medium having instructions stored therein comprises a product containing instructions that implement aspects of the functions / operations specified in the blocks of the flowchart and / or block diagram illustrations. These computer-readable program instructions may then be loaded onto a computer, other programmable apparatus, or other device to cause the computer, other programmable apparatus, or other device to perform a series of operational steps to create a computer-implemented process, in a manner such that the instructions, executed on the computer, other programmable apparatus, or other device, perform the functions / acts specified in the blocks of these flowcharts and / or block diagrams.

[0110] The flowcharts and block diagrams in the accompanying figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the disclosed subject matter. In this regard, each block in the flowcharts or block diagrams may represent a module, segment, or portion of instructions, including one or more executable instructions for implementing specified logical functions. In some alternative implementations, the functions shown in the blocks may be performed out of the order shown in the figures. For example, two blocks shown in succession may be executed substantially concurrently, or the blocks may be executed in the reverse order, depending on the functionality involved. It should also be noted that each block of the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, may be implemented by a dedicated hardware-based system that performs the specified functions or operations or implements a combination of dedicated hardware and computer instructions.

[0111] While the subject matter has been described above in the general context of computer-executable instructions in a computer program product executing on one computer or multiple computers, or both, those skilled in the art will recognize that the disclosure can also be implemented in combination with other program modules. Generally, program modules include routines, programs, components, data structures, etc. that perform particular tasks and / or implement particular abstract data types. Furthermore, those skilled in the art will recognize that the computer-implemented methods disclosed herein can also be practiced with other computer system configurations, including single-processor or multiprocessor computer systems, minicomputing devices, mainframe computers, computers, handheld computing devices (e.g., PDAs, telephones), microprocessor-based or programmable consumer or industrial electronic devices, etc. The illustrated aspects can also be practiced in distributed computing environments where tasks are performed by remote processing devices linked through a communications network. However, some, if not all, aspects of the disclosure can also be practiced on stand-alone computers. In a distributed computing environment, program modules can be located in both local and remote memory storage devices.

[0112] As used in this application, the terms “component,” “system,” “platform,” “interface,” etc. may refer to or include a computer-related or operational machine-related entity having one or more specific functions, or both. The entities disclosed herein may be hardware, a combination of hardware and software, software, or software in execution. For example, a component may be, but is not limited to, a process running on a processor, a processor, an object, an executable, a thread of execution, a program, or a computer, or combinations thereof. By way of example, both an application running on a server and the server may be a component. One or more components may reside within a process or thread of execution, or both, and a component may be localized on one computer or distributed between two or more computers, or both. In another example, corresponding components may execute from various computer-readable media having various data structures stored thereon. Components may communicate via local or remote processes or both, e.g., according to signals having one or more data packets (e.g., data from one component interacting with another component via signals, with other systems, within a local system, within a distributed system, or across a network such as the Internet, or combinations thereof). As another example, a component may be a device having a specific function provided by mechanical parts operated by electrical or electronic circuits, which in turn are operated by software or firmware applications executed by a processor.In such cases, the processor may be located within or external to the device and may execute at least a portion of a software or firmware application. As another example, a component may be a device that provides certain functionality through electronic components that do not include mechanical parts, and those electronic components may include a processor or other means for executing software or firmware that provides at least a portion of the functionality of the electronic component. In one aspect, a component may emulate an electronic component via a virtual machine, for example, in a cloud computing system.

[0113] Furthermore, the term "or" is intended to mean an inclusive "or" rather than an exclusive "or." That is, unless otherwise stated or clear from the context, "X uses A or B" is intended to mean any of the natural inclusive permutations. That is, if X uses A, if X uses B, or if X uses both A and B, "X uses A or B" is satisfied under any of the above cases. Furthermore, unless otherwise stated or clear from the context that the singular form is indicated, the articles "a" and "an," as used in this specification and the accompanying drawings, should generally be construed to mean "one or more." 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. Furthermore, any aspect or design described herein as "example" and / or "exemplary" is not necessarily to be construed as preferred or advantageous over other aspects or designs, or to imply that such aspects or designs exclude equivalent exemplary structures and techniques known to those skilled in the art.

[0114] As used herein, the term "processor" may refer to virtually any computing processing unit or device, including, but not limited to, a single-core processor, a single-core processor with software multithreading execution capabilities, a multi-core processor, a multi-core processor with software multithreading execution capabilities, a multi-core processor with hardware multithreading techniques, a parallel platform, and a parallel platform with distributed shared memory. Furthermore, a processor may 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. Furthermore, a processor may utilize nanoscale architectures, such as, but not limited to, molecular-based and quantum-dot-based transistors, switches, and gates, to optimize space utilization or enhance performance of user equipment. A processor may also be implemented as a combination of computing processing units. In this disclosure, terms such as "store," "storage," "data store," "data storage," "database," and substantially any other information storage component associated with the operation and functionality of a component are utilized to refer to a "memory component," which is an entity embodied as a "memory" or a component that includes memory. It should be recognized that the memory and / or memory components described herein can be volatile or non-volatile memory, or can include both volatile and non-volatile memory.By way of example, non-volatile memory may include, but is not limited to, read-only memory (ROM), programmable ROM (PROM), erasable programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), flash memory, or non-volatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). Volatile memory may include RAM, which may function, for example, as external cache memory. Many forms of RAM are available, including, by way of example, but not limitation, 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), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM (RDRAM). Additionally, the disclosed memory components of the systems or computer-implemented methods herein are intended to include, but are not limited to, these and other suitable types of memory.

[0115] The foregoing description includes merely exemplary systems and computer-implemented methods. It is, of course, not possible to describe every conceivable combination of components or computer-implemented methods in order to describe the present disclosure, but those skilled in the art will recognize that many other combinations and permutations of the present disclosure are possible. Furthermore, to the extent that the terms "includes," "has," "possesses," and the like are used in the detailed description, claims, appendices, and drawings, such terms are intended to be inclusive, similar to the interpretation of the term "comprising" when used as a transitional word in the claims. The foregoing description of various embodiments has been provided for illustrative purposes and is not intended to be exhaustive or to limit the foregoing to only the disclosed embodiments. Many modifications and variations within the scope and spirit of the described embodiments will be apparent to those skilled in the art. The terminology used herein was selected to best explain the principles, practical applications, or technical improvements of the embodiments over commercially available technology, or to enable those skilled in the art to understand the embodiments disclosed herein.

Claims

1. 1. A system comprising: a memory storing computer-executable components; a processor operatively coupled to the memory for executing the computer-executable components; wherein the computer-executable components are: a calibration component at a first output of the first circuit that applies a first random Pauli gate to the qubit prior to a first readout measurement of the qubit; and an estimation component that applies a second random Pauli gate to the qubit at a second output of the second circuit, including a third circuit that is the circuit of interest, prior to a second readout measurement of the qubit. wherein the estimation component generates an error mitigation read decision associated with the circuit of interest based on a first read measurement at the first output of the first circuit and a second read measurement at the second output of the second circuit.

2. The system of claim 1, wherein the third circuit is associated with the second random Pauli gate, and the amount of error associated with the error-mitigated readout decision amount is mitigated 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.

3. 3. The system of claim 1, wherein the calibration component randomly samples, for each of a plurality of instances of the first circuit, a corresponding respective value of the first random Pauli gate, measures a corresponding respective response at the first output of the first circuit with respect to the first random Pauli gate, and stores the corresponding respective responses and the corresponding respective values ​​associated with the first random Pauli gate in a data store, wherein the corresponding respective responses are based on the first random Pauli gate and the corresponding respective values ​​are associated with the corresponding respective responses and the first random Pauli gate, and the corresponding respective responses are the first readout measurements.

4. 4. The system of claim 1, wherein the estimation component randomly samples, for each of a plurality of instances of the second circuit, a corresponding respective value of the second random Pauli gate, measures a corresponding respective response at the second output of the second circuit, and stores the corresponding respective response and the corresponding respective value associated with the second random Pauli gate in a data store, wherein the corresponding respective response is based on the second random Pauli gate and the corresponding respective value is associated with the corresponding respective response and the second random Pauli gate, and the corresponding respective response is the second readout measurement.

5. the calibration component determines calibration data based on a defined first function and the first readout measurement measured for the first random Pauli gate at the first output of the first circuit; the estimation component determines estimated data based on the defined first function and the second read measurement measured for the second random Pauli gate at the second output of the second circuit. A system according to any one of claims 1 to 4.

6. 6. The system of claim 5, wherein the third circuit is associated with the second random Pauli gate, and the estimation component determines a normalized scalar value based on the calibration data and a defined second function, determines an estimated scalar value based on the estimation data and the defined second function, and generates the error-mitigated readout determination quantity associated with the circuit of interest based on the normalized scalar value and the estimated scalar value.

7. 7. The system of claim 5, wherein the calibration component applies a third random Pauli gate to the qubit at a third output of the first circuit prior to a third readout measurement of the qubit or the first circuit, and determines an update to the calibration data based on a portion of the first readout measurement and based on a third readout measurement measured at the third output of the first circuit, the calibration component determines updated calibration data based on the update and stores the updated calibration data in a data store, and the third readout measurement is based on the third random Pauli gate.

8. The system of claim 5 , wherein the estimation component generates a quantum noise model that models quantum noise as a classical noise channel based on the calibration data and the estimation data.

9. 1. A computer-implemented method comprising: applying, by a system operatively coupled to a processor, a first random Pauli gate to the qubit at a first output of the first circuit prior to a first readout measurement of the qubit; applying, by the system, a second random Pauli gate to the qubit at a second output of a second circuit, including a third circuit that is a circuit of interest, prior to a second readout measurement of the qubit; generating, by the system, an error mitigation read decision associated with the circuit of interest based on a first read measurement at the first output of the first circuit and a second read measurement at the second output of the second circuit; 20. A computer-implemented method comprising:

10. randomly sampling, by the system, for each of a plurality of instances of the first circuit, a corresponding respective value of the first random Pauli gate; measuring, by the system, the first readout measurement at the first output of the first circuit, the first readout measurement being based on the first random Pauli gate; and storing, by the system, the first readout measurement and the corresponding respective values ​​in a data store, the corresponding respective values ​​being associated with the first readout measurement and the first random Pauli gate; The computer-implemented method of claim 9 further comprising:

11. randomly sampling, by the system, for each of a plurality of instances of the second circuit, a corresponding respective value of the second random Pauli gate; measuring, by the system, the second readout measurement at the second output of the second circuit, the second readout measurement being based on the second random Pauli gate; and storing, by the system, the second readout measurement and the corresponding respective values ​​in a data store, the corresponding respective values ​​being associated with the second readout measurement and the second random Pauli gate.

11. The computer-implemented method of claim 9, further comprising:

12. determining, by the system, calibration data based on a defined first function and the first readout measurement measured at the first output of the first circuit; and determining, by the system, estimated data based on the defined first function and the second readout measurement measured at the second output of the second circuit; 12. The computer-implemented method of claim 9, further comprising:

13. The method according to claim 12, wherein the third circuit is associated with the second random Pauli gate, and the method further comprises: determining, by the system, a normalized scalar value based on the calibration data and a defined second function; determining, by the system, an estimated scalar value based on the estimated data and the defined second function; generating, by the system, the error mitigation read decision quantity associated with the circuit of interest based on the normalized scalar value and the estimated scalar value; and and presenting, by said system, said error-mitigated read determination as an output. The computer-implemented method of claim 12 further comprising:

14. applying, by the system, at a third output of the first circuit, a third random Pauli gate to the qubit prior to a third readout measurement of the qubit or the first circuit; determining, by the system, updates to the calibration data based on a portion of the first readout measurement and based on a third readout measurement measured at the third output of the first circuit; determining, by the system, updated calibration data based on the update; and storing, by the system, the updated calibration data in a data store, the third readout measurement being based on the third random Pauli gate.

14. The computer-implemented method of claim 12, further comprising:

15. 1. A computer program that facilitates mitigating readout errors associated with a quantum circuit, comprising: applying a first random Pauli operator to the qubit at a first output of the first circuit prior to a first readout measurement of the qubit; applying a second random Pauli operator to the qubit prior to a second readout measurement of the qubit at a second output of a second circuit that includes a third circuit that is the circuit of interest; and generating an error mitigation read decision associated with the circuit of interest based on a first read measurement at the first output of the first circuit and a second read measurement at the second output of the second circuit; A computer program that causes a processor to execute the following.

16. The method of claim 15, wherein the third circuit is associated with the second random Pauli operator; determining calibration data based on the first readout measurement and a defined first function; determining estimated data based on the second readout measurement and the defined first function; determining a normalized scalar value based on the calibration data and a defined second function; determining an estimated scalar value based on the estimated data and the defined second function; generating the error mitigated read decision quantity associated with the circuit of interest based on the normalized scalar value and the estimated scalar value; and transmitting the error mitigated read determination as an output.

16. The computer program product of claim 15, which causes the processor to execute:

17. 1. A system comprising: a memory storing computer-executable components; a processor operatively coupled to the memory for executing the computer-executable components; wherein the computer-executable components are: a calibration component at a first output of the first circuit that applies a first random Pauli gate to the qubit prior to a first readout measurement of the qubit; and an estimation component that applies a pair of random Pauli gates to the qubit associated with a second circuit that includes a third circuit that is a circuit of interest, said applying including applying a second random Pauli gate to the qubit at a second output of the second circuit prior to a second readout measurement of the qubit; wherein the estimation component generates an error mitigation read decision associated with the circuit of interest based on a first read measurement at the first output of the first circuit and a second read measurement at the second output of the second circuit.

18. 18. The system of claim 17, wherein the calibration component randomly samples, for each of a plurality of instances of the first circuit, a corresponding respective value of the first random Pauli gate, measures the first readout measurement at the first output of the first circuit, and stores the first readout measurement and the corresponding respective value in a data store, the corresponding respective value being associated with the first readout measurement and the first random Pauli gate, and the first readout measurement is based on the first random Pauli gate.

19. 19. The system of claim 17, wherein the pair of Random Pauli gates includes the second Random Pauli gate and a third Random Pauli gate, the third Random Pauli gate associated with an input of the third circuit.

20. 20. The system of claim 19, wherein the estimation component randomly samples, for each of a plurality of instances of the second circuit, corresponding respective values ​​of the second random Pauli gate and the third random Pauli gate, measures the second readout measurement at the second output of the second circuit, and stores the second readout measurement and the corresponding respective values ​​in a data store, the second readout measurement being based on the second random Pauli gate and the third random Pauli gate, the corresponding respective values ​​being associated with the second readout measurement, a first portion of the corresponding respective values ​​being associated with the second random Pauli gate, and a second portion of the corresponding respective values ​​being associated with the third random Pauli gate.

21. the calibration component determines calibration data based on a defined first function and the first readout measurement measured for the first random Pauli gate at the first output of the first circuit; the estimation component determines estimated data based on the defined first function and the second read measurement measured for the second random Pauli gate at the second output of the second circuit.

21. A system according to any one of claims 17 to 20.

22. The system of claim 21, wherein the third circuit is associated with the pair of random Pauli gates, and the estimation component determines a normalized scalar value based on the calibration data and a defined second function, determines an estimated scalar value based on the estimation data and the defined second function, and generates the error-mitigated readout determination quantity associated with the circuit of interest based on the normalized scalar value and the estimated scalar value, and the error-mitigated readout determination quantity is provided as an output.

23. 1. A computer-implemented method comprising: applying, by a system operatively coupled to a processor, a first random Pauli operator to the qubit at a first output of the first circuit prior to a first readout measurement of the qubit; applying, by the system, a pair of random Pauli operators to the qubit associated with a second circuit, the third circuit being a circuit of interest, the applying including applying a second random Pauli operator to the qubit at a second output of the second circuit prior to a second readout measurement of the qubit; generating, by the system, an error mitigation read decision associated with the circuit of interest based on a first read measurement at the first output of the first circuit and a second read measurement at the second output of the second circuit; 20. A computer-implemented method comprising:

24. randomly sampling, by the system, for each of a plurality of instances of the first circuit, a corresponding respective first value of the first random Pauli operator; measuring, by the system, the first readout measurement at the first output of the first circuit, the first readout measurement being based on the first random Pauli operator; storing, by the system, the first readout measurements and the corresponding respective first values ​​in a data store, the corresponding respective first values ​​being associated with the first readout measurements and the first random Pauli operators; randomly sampling, by the system, for each of a plurality of instances of the second circuit, a corresponding respective second value of the second random Pauli operator and a corresponding respective third value of a third random Pauli operator; measuring, by the system, the second readout measurement at the second output of the second circuit, the second readout measurement being based on the second random Pauli operator and the third random Pauli operator; and storing, by the system, the second readout measurements and the corresponding respective second values ​​and the corresponding respective third values ​​in the data store, wherein the corresponding respective second values ​​are associated with the second readout measurements and the second random Pauli operators, and the corresponding respective third values ​​are associated with the second readout measurements and the third random Pauli operators.

24. The computer-implemented method of claim 23, further comprising:

25. The method according to claim 25, wherein the third circuit is associated with the second pair of random Pauli operators, and the method comprises: determining, by the system, calibration data based on the first readout measurement and a defined first function; determining, by the system, estimated data based on the second readout measurement and the defined first function; determining, by the system, a normalized scalar value based on the calibration data and a defined second function; determining, by the system, an estimated scalar value based on the estimated data and the defined second function; and generating, by the system, the error-mitigated read decision quantity associated with the circuit of interest based on the normalized scalar value and the estimated scalar value, the error-mitigated read decision quantity being presented as an output.

25. The computer-implemented method of any one of claims 23 and 24, further comprising:

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