Error Reduction Technology

Abstract

1. A method for mitigating errors in quantum computing, the method comprising: performing (S101) an operation on a state of a qubit in a group of qubits a plurality of times, the operation having a first error rate, each performance of the operation comprising: performing a first operation comprising a gate operation, a symmetry operation, and a first elementary operation; or performing a second operation comprising the gate operation, the symmetry operation, and a second elementary operation, wherein the first and second elementary operations are different elementary operations selected from a set of elementary operations; and further comprising measuring the state of the qubit; wherein a probability of performing the first operation is a first probability and a probability of performing the second operation is a second probability. performing the operation, where the error rate is a factor of 1; using the symmetry operation to obtain a symmetry measurement for the group of qubits after each performance of the operation (S102), where the group of qubits includes a plurality of qubits; wherein the symmetry measurement is a first symmetry result where the number of errors is even or a second symmetry result where the number of errors is odd; further obtaining a first state measurement by determining an average state of the qubits for the first symmetry result (S103); obtaining a second state measurement by determining an average state of the qubits for the second symmetry result (S104); [Equation 25] TIFF2023536794000026.tif12165 fitting (S105) the second state measurement to a first curve having the form: [Equation 26] TIFF2023536794000027.tif12165 and fitting a second curve having the form: where n is an error rate, and A and γ are fitting parameters; and further including using the fitted first and second curves to extrapolate the average state of the qubit at a second error rate (S106), where the second error rate is lower than the first error rate.

Description

Technical Field

[0001] The present invention relates to quantum computing, particularly error reduction techniques.

Background Art

[0002] A quantum computer can be used to calculate "observables", i.e., system properties. To measure an observable, it is possible to measure the output state of a qubit after performing a sequence of quantum operations on the qubit. Usually, the same sequence of quantum operations is repeated many times, and the average of the measured output states can be calculated to estimate the expected value of the observable.

[0003] However, the sequence of quantum operations performed on a qubit, i.e., the expected value to be estimated, is affected by errors. The goal of quantum computing is to reduce and even eliminate these errors. However, a more realistic approach for near-term quantum devices or quantum devices in the NISQ (Noisy Intermediate-Scale Quantum) era is to aim at reducing these errors using an analytical approach. In this method, it is possible to estimate the expected value of an observable without errors or noise.

[0004] Error reduction techniques use additional measurements to extract a noise-free expected value from noisy measurement results. Some existing error reduction techniques include symmetry verification, quasiprobability, and error extrapolation. Symmetry verification uses known properties of the system to determine whether errors are occurring without the need to measure (and thus collapse) the state of the qubit. Quasiprobability uses additional gates determined by modeling the errors associated with the components in the circuit. Error extrapolation involves physically changing the hardware to increase the noise level and predicting a noise-free expected value based on measurements at a higher noise level. Each of these error reduction techniques can be employed to reduce different types of noise.

[0005] Symmetry verification can be performed as is, but there may be associated errors, and the comprehensive error becomes undetectable using symmetry, so it cannot be assumed that there is no error in the operating circuit that passed the symmetry verification test.

[0006] Probability can eliminate errors. However, this elimination of errors is obtained at a very high cost and requires a large number of repetitions.

[0007] Error extrapolation requires an increase in the noise level by the experimenter. An accurate prediction of the noise-free observable according to the measured value of the increased noise level requires that the additional noise be generated from the same noise model as the original noise. Summary of the Invention Problems to be Solved by the Invention

[0008] The development of better error reduction techniques is desired. Means for Solving the Problems

[0009] One aspect of the present invention provides a method for reducing errors in quantum computing. The method includes executing an operation on the state of a qubit within a group of qubits multiple times. The operation has a first error rate, and each execution of the operation includes executing a first operation or a second operation and measuring the state of the qubit. The first operation includes a gate operation, a symmetry operation, and a first basic operation. The second operation includes the gate operation, the symmetry operation, and a second basic operation. The first and second basic operations are different basic operations selected from a set of basic operations. The probability of executing the first operation is a first probability, and the probability of executing the second operation is a second probability. The method further includes using the symmetry operation to obtain a symmetry measurement value for the group of qubits after each execution of the operation, wherein the group of qubits includes a plurality of qubits; the symmetry measurement value is a first symmetry result when the number of errors is even or a second symmetry result when the number of errors is odd. The method further includes obtaining a first state measurement value by determining an average state of the qubits for the first symmetry result and obtaining a second state measurement value by determining an average state of the qubits for the second symmetry result. The method further includes: fitting the first state measurement value to a first curve having the form of [Number] ; fitting the second state measurement value to a second curve having the form of [Number] ; and extrapolating the average state of the qubits at a second error rate using the fitted first and second curves, wherein the second error rate is lower than the first error rate.

[0010] Fortunately, the method of reducing this error results in an improved observable estimate at a reduced cost. The cost is given by the number of times the operation is performed. The method combines the quasi-probability, symmetry verification, and error extrapolation error reduction techniques in a multiplicative manner.

[0011] The first and second state measurement values are respectively fitted to the first and second curves. The combination of the first and second state measurement values can be modeled by an exponential decay curve. Since it has been shown that the exponential decay curve is a good model of the relationship between the observable and the error rate, it is advantageous to use it. Optionally, the exponential decay curve can be a multi-exponential decay curve that includes the sum of two or more exponential functions. The multi-exponential decay curve usually has a higher cost compared to a single exponential decay curve, but can usefully provide an improved model of the change in the expected value of the observable as a function of the error rate.

[0012] Typically, the gate operation can include any quantum logic gate operation such as, for example, a Pauli gate, a Hadamard gate, a SWAP (swap) gate, a controlled NOT (CNOT) gate, or a controlled Z gate. The gate operation can include a sequence of operations. The first basic operation and the second basic operation can be performed after the gate operation. Typically, the basic operation is selected from a set of basic operations. The set of basic operations can include 16 basic operations. Fortunately, any single qubit operation (which can be represented as a 4×4 matrix) can be represented as a linear combination of 16 basic operations. Generally speaking, to accommodate a larger number of qubits, the number of basic operations in the set of basic operations can be made larger. The set of basic operations can include Pauli's basic operations. Typically, for n qubits, the set of basic operations is at least 4n It can include the basic operations of Pauli. Therefore, each of the first and second basic operations can be one of those basic operations of Pauli.

[0013] Instead of the above, the execution of the operation can include the execution of a j-th operation that includes the gate operation, the symmetry operation, and the j-th basic operation. The number of operations that can be executed among the possible operations is preferably related to the number of basic operations within the set of basic operations. For example, if there is one gate operation, one qubit, and three basic operations in the set, the number of operations that can be executed among the possible operations can be three. The probability of executing the j-th operation can be the j-th probability. In this method, a random basic operation can be used to modify the gate operation. This has the advantage of being able to reduce the effective error rate of the operation. Optionally, the method can be repeated using different effective error rates. This can be achieved by changing the first probability and the second probability.

[0014] The symmetry measurement values are obtained for the group of qubits using the symmetry operation. The symmetry measurement values preferably have a known set of symmetry results that can depend on the symmetry operation and the system. The symmetry measurement values are a first symmetry result or a second symmetry result. Although stated as an option, using a k that can be greater than 2, the symmetry measurement values are a k-th symmetry result. In one example with two symmetry results, the first symmetry result is a pass and the symmetry measurement values match the expected symmetry. In this example, the second symmetry result is preferably a fail where the symmetry is broken. The first and second state measurement values are obtained by determining the average state of the qubits for both the pass and fail symmetry results. Although stated as an option, the k-th state measurement value is obtained by determining the average state of the qubits for the k-th symmetry result. This method has the advantage that the measurement values that failed the symmetry test are constructively used when estimating the average state of the qubits at the second error rate. This reduces the cost of performing the error reduction operation in a beneficial way.

[0015] Typically, the execution of the operation includes the execution of the symmetry operation. Although stated as an option, let the symmetry operation be a first symmetry operation. The execution of the operation can further include the execution of a second symmetry operation. The symmetry measurement values using the second symmetry operation can be a third symmetry result or a fourth symmetry result that can be a pass and a fail respectively. The first symmetry operation can be used for testing the first symmetry of the system, and the second symmetry operation can be used for testing the second symmetry of the system. In this method, multiple characteristics of the group of qubits can be measured simultaneously, providing additional data. Advantageously, the additional data can be used to provide a better estimate of the expected value of the observable.

[0016] Preferably, the symmetry operation and the basic operation are performed after the gate operation. Although mentioned as an option, the symmetry operation is performed before the basic operation. Alternatively, the basic operation can be performed before the symmetry operation.

[0017] Although mentioned as an option, the method can be performed at an additional error rate. The effective error rate can be determined by the selection of the basic operation and the probability that each basic operation within the set of basic operations is selected. In one example, the method further comprises: performing another operation multiple times on the state of the qubits within the group of qubits, wherein the another operation has a third error rate, and each execution of the another operation comprises: performing another first operation comprising the gate operation, the symmetry operation, and another first basic operation; or performing another second operation comprising the gate operation, the symmetry operation, and another second basic operation; measuring the state of the qubits; wherein the probability of performing the another first operation is another first probability and the probability of performing the another second operation is another second probability; performing the another operation; obtaining another symmetry measurement for the group of qubits using the symmetry operation; wherein the another symmetry measurement is another first symmetry result or another second symmetry result; further comprising obtaining another first state measurement by determining the average state of the qubits for the another first symmetry result; obtaining another second state measurement by determining the average state of the qubits for the another second symmetry result; fitting the another first state measurement to another first curve and the another second state measurement to another second curve; extrapolating the average state of the qubits at a fourth error rate using the another first fitted curve and the another second fitted curve; wherein the fourth error rate is lower than the third error rate.

[0018] Conveniently, this method attributes to a resulting different effective error rate, i.e., the third error rate, by modifying the gate operation using a random selection of basic operations. Usually, the first basic operation, the second basic operation, the another first basic operation, and the another second basic operation are selected from the same set of basic operations. Using this method, an arbitrary i-th error rate can be brought about by randomly selecting a basic operation from the set of basic operations using different associated probabilities. The probabilities for selecting the basic operations are preferably determined theoretically.

[0019] The qubits within the group of qubits can be the first qubits and, as part of the quantum computation, can operate on the states of further qubits. Preferably, the method further includes executing the operation multiple times on the states of the second qubits within the group of qubits; obtaining a third state measurement value by determining the average state of the second qubits for the first symmetry result; obtaining a fourth state measurement value by determining the average state of the second qubits for the second symmetry result; fitting the third state measurement value to a third curve and the fourth state measurement value to a fourth curve; and extrapolating the average state of the second qubits at the second error rate using the fitted third and fourth curves. Any number of qubits can be present within the group of qubits, and the states of each qubit can be operated on in a similar manner. Conveniently, this method can be executed for the quantum device as a whole.

[0020] As an option, one or more qubits within the group of qubits can be left idle while performing an operation on the state of one or more qubits within the group of qubits. The one or more idle qubits within the group of qubits can be operated on using a parity gate operation. The one or more idle qubits are typically affected by the decoherence errors associated with the execution of the parity gate. After each execution of the operation, a symmetry measurement for the group of qubits is obtained using the symmetry operation. The symmetry measurement preferably includes the one or more qubits within the group of qubits on which the operation is performed and the one or more qubits that are left idle.

[0021] The qubit (or qubits) can be an electron spin qubit. Preferably, when the qubit is an electron spin qubit, the state of the qubit is an electron spin. Advantageously, electron spin qubits can be easily manipulated and coupled to other electron spin qubits. Preferably, since electron spin qubits in a silicon-based device conveniently have a long coherence time and are compatible with existing technologies, the qubit is an electron spin qubit in a silicon-based device. Devices of that kind can be suitably adapted for use in the NISQ (Noisy Intermediate-Scale Quantum or Nisk) computing era.

[0022] Another aspect of the present invention provides a device for performing quantum computing calculations, including a selection module, a quantum processor, a quantum measurement device, a symmetry measurement device, and a classical processor. The selection module is configured to select a first basic operation from a set of basic operations using a first probability and to select a second basic operation from the set of basic operations using a second probability, wherein the first and second basic operations are different. The quantum processor is configured to perform operations on the state of certain qubits within a group of qubits multiple times, wherein the operations have a first error rate, and each execution of the operations includes performing a gate operation, a symmetry operation, and the selected basic operation. The quantum measurement device is configured to measure the state of the qubits. The symmetry measurement device is configured to measure the symmetry of the group of qubits after each execution of the operations using the symmetry operation, wherein the group of qubits includes a plurality of qubits; and the symmetry measurement value is a first symmetry result when the number of errors is even or a second symmetry result when the number of errors is odd. The classical processor is configured to obtain a first state measurement value by determining the average state of the qubits for the first symmetry result and to obtain a second state measurement value by determining the average state of the qubits for the second symmetry result; and to fit the first state measurement value to a first curve having the form of

Number

Number

[0023] This device can be conveniently used to reduce the effect of noise in the measured expected value of an observable. The selection module is configured to select a first or second elementary operation that is to be executed by a quantum processor following the execution of the gate operation, using corresponding probabilities. In this method, the measured value of the state of the qubit can be recombined to estimate the value of the observable with reduced error. The value of the observable estimated using this device is advantageously accurate and low-cost.

[0024] Preferably, when the total number of elementary operations in a set of elementary operations is J, preferably the selection module is further configured to select the j-th elementary operation for 3 ≤ j ≤ J. The set of elementary operations can be determined based on a theoretically determined noise model, benchmark experiments, and / or consideration of available experimental options.

[0025] A further aspect of the present invention provides a computer-readable memory medium including instructions that, when executed by a computer, cause the computer to: perform a plurality of times an operation on the state of a qubit within a group of qubits, the operation having a first error rate, and each execution of the operation including: performing a first operation including a gate operation, a symmetry operation, and a first basic operation; or performing a second operation including the gate operation, the symmetry operation, and a second basic operation, wherein the first and second basic operations are different basic operations selected from a set of basic operations; further including measuring the state of the qubit; wherein the probability of performing the first operation is a first probability and the probability of performing the second operation is a second probability, the step of performing the operation; obtaining a symmetry measurement value for the group of qubits after each execution of the operation using the symmetry operation, the group of qubits including a plurality of qubits; wherein the symmetry measurement value is a first symmetry result when the number of errors is even or a second symmetry result when the number of errors is odd; further including obtaining a first state measurement value by determining an average state of the qubits for the first symmetry result; obtaining a second state measurement value by determining an average state of the qubits for the second symmetry result; fitting the first state measurement value to a first curve having the form of

Number

Number

[0026] Conveniently, this computer-readable memory medium can be used to determine an error reduction value of an observable having a low estimation error.

[0027] One aspect of the present invention provides a method for reducing errors in quantum computing. The method includes performing an operation on the state of a qubit within a group of qubits multiple times. The operation has a first error rate, and each execution of the operation includes performing a first operation or a second operation and measuring the state of the qubit. The first operation includes a gate operation, a symmetry operation, and a first basic operation. The second operation includes the gate operation, the symmetry operation, and a second basic operation. The probability of performing the first operation is a first probability, and the probability of performing the second operation is a second probability. The method further includes obtaining a symmetry measurement value for the group of qubits using the symmetry operation, wherein the symmetry measurement value is a first symmetry result or a second symmetry result. The method further includes obtaining a first state measurement value by determining an average state of the qubits for the first symmetry result and obtaining a second state measurement value by determining an average state of the qubits for the second symmetry result. The method further includes combining the first state measurement value and the second state measurement value and estimating the average state of the qubit.

[0028] Conveniently, the method of reducing this error results in an improved observable estimate at reduced cost. The cost is given by the number of times the operation is performed. The method combines the quasi-probability and symmetry verification error reduction techniques in a multiplicative manner. The quasi-probability can be usefully employed to change the form of the error in such a way that it can remove errors undetectable by the use of the symmetry operation. Furthermore, using the knowledge of the relative probabilities of obtaining the first and second symmetry results respectively, it is possible to recombine the first and second state measurements to provide an improved estimate of the observable without the error.

[0029] Another aspect of the present invention provides a device for performing quantum computing calculations, including a selection module, a quantum processor, a quantum measurement device, a symmetry measurement device, and a classical processor. The selection module is configured to select a first basic operation from a set of basic operations using a first probability and to select a second basic operation from the set of basic operations using a second probability. The quantum processor is configured to perform operations on the state of certain qubits within a group of qubits over a plurality of times, where the operation has a first error rate, and each execution of the operation includes performing a gate operation, a symmetry operation, and the selected basic operation. The quantum measurement device is configured to measure the state of the qubits. The symmetry measurement device is configured to measure the symmetry of the group of qubits using the symmetry operation, where the symmetry measurement value is a first symmetry result or a second symmetry result. The classical processor is configured to obtain a first state measurement value by determining the average state of the qubits for the first symmetry result and to obtain a second state measurement value by determining the average state of the qubits for the second symmetry result, and to combine the first state measurement value and the second state measurement value to estimate the average state of the qubits.

[0030] This device can be conveniently used to reduce the effect of errors in the measured expected values of observables. The selection module is configured to select a first or second basic operation to be executed by a quantum processor following the execution of the gate operation, using corresponding probabilities. The measured values of the states of the qubits are classified according to the results of the symmetry measurements. In this method, the first and second state measurement values can be recombined using the knowledge of the probabilities of the first and second symmetry results to estimate the value of the observable with reduced error.

[0031] A further aspect of the present invention provides a computer-readable memory medium including instructions which, when executed by a computer, cause the computer to: execute, multiple times, an operation on the state of a qubit within a group of qubits, the operation having a first error rate, and each execution of the operation including: executing a first operation including a gate operation, a symmetry operation, and a first basic operation; or executing a second operation including the gate operation, the symmetry operation, and a second basic operation; measuring the state of the qubit; wherein the probability of executing the first operation is a first probability and the probability of executing the second operation is a second probability; obtaining, using the symmetry operation, a symmetry measurement value for the group of qubits, wherein the symmetry measurement value is a first symmetry result or a second symmetry result; further, obtaining a first state measurement value by determining an average state of the qubits for the first symmetry result; obtaining a second state measurement value by determining an average state of the qubits for the second symmetry result; combining the first state measurement value and the second state measurement value and estimating the average state of the qubits; thereby causing the execution of the steps to be performed on a quantum computer.

[0032] Advantageously, this computer-readable memory medium can be used to more accurately determine the observable error reduction value.

[0033] One aspect of the present invention provides a method for reducing errors in quantum computing. The method includes: performing a first operation on the state of a qubit, where the first operation has a first error rate; obtaining a first measurement value of the state of the qubit; performing a second operation on the state of the qubit, where the second operation has the first error rate; obtaining a second measurement value of the state of the qubit; calculating a first average value of the state of the qubit at the first error rate by averaging the first measurement value and the second measurement value; performing a third operation on the state of the qubit, where the third operation has a second error rate lower than the first error rate and the third operation includes the first operation and a first basic operation; obtaining a third measurement value of the state of the qubit; performing a fourth operation on the state of the qubit, where the fourth operation has the second error rate and the fourth operation includes the first operation and a second basic operation, where the first and second basic operations are different basic operations selected from a set of basic operations; obtaining a fourth measurement value of the state of the qubit; calculating a second average value of the state of the qubit at the second error rate by averaging the third measurement value and the fourth measurement value. The method further includes: fitting the first average value of the state of the qubit and the second average value of the state of the qubit to a curve; extrapolating the average stage of the qubit at a third error rate using the fitted curve, where the third error rate is lower than the first error rate and the second error rate.

[0034] Conveniently, the method of reducing this error results in an improved estimate of the observable at a reduced cost. The method combines the techniques of quasi-probability and error extrapolation for error reduction in a multiplicative manner.

[0035] The first basic operation and the second basic operation are selected from a set of basic operations. Preferably, the selection is made randomly using a weighting probability for selecting a particular basic operation. Although mentioned as an option, the probability of selecting the first basic operation is the first probability, and the probability of selecting the second basic operation is the second probability. The set of basic operations can further include additional basic operations each with an associated probability of selection. In this method, a random basic operation can be used to modify the first operation. This has the advantage that the effective error rate of the operation can be reduced from the first error rate to the second error rate. The second error rate may depend on the respective selection probabilities of the basic operations.

[0036] Both the first and second operations have the first error rate. The first average value is calculated by averaging the first measurement value and the second measurement value. Preferably, the second operation is made the same as the first operation. Furthermore, the first operation is usually repeated a plurality of times, and the first average value of the state of the qubit in the first error rate can be calculated by averaging each of the resulting measurement values. Each measurement value is either +1 or -1 corresponding to each of the two states of the qubit. Executing the first operation a plurality of times has the advantage that the uncertainty of the expected value of the observable in the first error rate is reduced.

[0037] The third and fourth operations each include the first and second basic operations in addition to the first operation. Optionally, the first and second basic operations are Pauli basic operations. The Pauli basic operations typically include an identity operation. Further modified operations can be performed using additional basic operations. Each of the first, second, and any additional basic operations can be randomly selected from a set of basic operations. In this way, the effective error rate can be beneficially reduced compared to the execution of the first operation without random sampling of modified operations.

[0038] Preferably, the qubit is one of a plurality of qubits within a group of qubits. The state of each qubit within the group of qubits can be transformed by performing one operation or a sequence of operations. Advantageously, this error reduction method can be applied to a quantum device that includes a plurality of qubits.

[0039] The qubit can be an electron spin qubit. Preferably, when the qubit is an electron spin qubit, the state of the qubit is an electron spin. The measurement of the state of the qubit typically returns either spin-up |↑> or spin-down |↓>. Advantageously, electron spin qubits can be easily manipulated and coupled to other electron spin qubits. Preferably, the qubit is an electron spin qubit within a silicon-based device that conveniently provides a long coherence time and is compatible with existing technology.

[0040] The first and second average values are fitted to a curve. Optionally, the curve is an exponentially decaying curve. Since an exponentially decaying curve is usually a good model for the relationship between an observable and an error rate, it is advantageous to use it. Optionally, the exponentially decaying curve is a multi-exponentially decaying curve that includes the sum of two or more exponential functions. Preferably, the multi-exponentially decaying curve is

Number

[0041] Another aspect of the present invention provides a device for performing quantum computing calculations that includes a selection module, a quantum processor, a quantum measurement device, and a classical processor. The selection module is configured to select a basic operation from a set of basic operations that includes a first basic operation and a second basic operation, where the first and second basic operations are different. The quantum processor is configured to perform a first operation having a first error rate on the state of a qubit, perform a second operation having the first error rate on the state of the qubit, perform a third operation having a second error rate lower than the first error rate and including the first operation and the first basic operation on the state of the qubit, and perform a fourth operation having the second error rate and including the first operation and the second basic operation on the state of the qubit. The quantum measurement device is configured to obtain first, second, third, and fourth measurement values after performing the first, second, third, and fourth operations, respectively. The classical processor is configured to calculate a first average value of the state of the qubit at the first error rate by averaging the first measurement value and the second measurement value, calculate a second average value of the state of the qubit at the second error rate by averaging the third measurement value and the fourth measurement value, fit the first average value of the state of the qubit and the second average value of the state of the qubit to a curve, and extrapolate the average state of the qubit at a third error rate lower than the first error rate and the second error rate using the fitted curve.

[0042] Advantageously, this device can be used to reduce the effect of errors in the measured expected value of an observable. The noise-free value of the observable estimated using this device is advantageously accurate and low-cost.

[0043] A further aspect of the present invention provides a computer-readable memory medium including instructions that, when executed by a computer, cause the computer to: execute a first operation on a state of a qubit, where the first operation has a first error rate; obtain a first measurement of the state of the qubit; execute a second operation on the state of the qubit, where the second operation has the first error rate; obtain a second measurement of the state of the qubit; calculate a first average value of the state of the qubit at the first error rate by averaging the first measurement and the second measurement; execute a third operation on the state of the qubit, where the third operation has a second error rate lower than the first error rate and the third operation includes the first operation and a first basic operation; obtain a third measurement of the state of the qubit; execute a fourth operation on the state of the qubit, where the fourth operation has the second error rate and the fourth operation includes the first operation and a second basic operation, and where the first and second basic operations are different basic operations selected from a set of basic operations; obtain a fourth measurement of the state of the qubit; calculate a second average value of the state of the qubit at the second error rate by averaging the third measurement and the fourth measurement; fit the first average value of the state of the qubit and the second average value of the state of the qubit to a curve; and extrapolate an average stage of the qubit at a third error rate lower than the first error rate and the second error rate using the fitted curve. The execution of the steps included results in a quantum computer.

[0044] Conveniently, this computer-readable memory medium can be used to determine an observable error reduction value having a low estimated error.

[0045] Another aspect of the present invention provides a method for reducing errors in quantum computing. The method includes executing an operation on the state of a qubit within a group of qubits multiple times. The operation has a first error rate, and each execution of the operation includes executing a first operation or a second operation and measuring the state of the qubit. The first operation includes a gate operation, a symmetry operation, and a first basic operation. The second operation includes the gate operation, the symmetry operation, and a second basic operation. The probability of executing the first operation is a first probability, and the probability of executing the second operation is a second probability. The method further includes obtaining a symmetry measurement value for the group of qubits using the symmetry operation, wherein the symmetry measurement value is a first symmetry result or a second symmetry result. The method further includes obtaining a first state measurement value by determining an average state of the qubits for the first symmetry result and obtaining a second state measurement value by determining an average state of the qubits for the second symmetry result. The method further includes fitting the first state measurement value to a first curve and the second state measurement value to a second curve; and extrapolating the average state of the qubits at a second error rate using the fitted first and second curves, wherein the second error rate is lower than the first error rate.

[0046] A further aspect of the present invention provides a device for performing quantum computing calculations, including a selection module; a quantum processor; a quantum measurement device; a symmetry measurement device; and a classical processor. The selection module is configured to select a first basic operation from a set of basic operations using a first probability and to select a second basic operation from the set of basic operations using a second probability. The quantum processor is configured to perform operations on the state of certain qubits within a group of qubits multiple times, wherein the operations have a first error rate, and each execution of the operations includes performing a gate operation, a symmetry operation, and the selected basic operation. The quantum measurement device is configured to measure the state of the qubits. The symmetry measurement device is configured to measure the symmetry of the group of qubits using the symmetry operation, wherein the symmetry measurement value is a first symmetry result or a second symmetry result. The classical processor is configured to obtain a first state measurement value by determining the average state of the qubits for the first symmetry result and to obtain a second state measurement value by determining the average state of the qubits for the second symmetry result; fitting the first state measurement value to a first curve and the second state measurement value to a second curve; and using the fitted first and second curves to extrapolate the average state of the qubits at a second error rate, wherein the second error rate is lower than the first error rate.

[0047] Another aspect of the present invention provides a computer-readable memory medium including instructions that, when executed by a computer, cause the computer to: perform an operation on the state of a certain qubit within a group of qubits multiple times, where the operation has a first error rate, and each execution of the operation includes: performing a first operation including a gate operation, a symmetry operation, and a first basic operation; or performing a second operation including the gate operation, the symmetry operation, and a second basic operation; measuring the state of the qubit; where the probability of performing the first operation is a first probability and the probability of performing the second operation is a second probability; performing the operation; obtaining a symmetry measurement value for the group of qubits using the symmetry operation, where the symmetry measurement value is a first symmetry result or a second symmetry result; further, obtaining a first state measurement value by determining the average state of the qubits for the first symmetry result; obtaining a second state measurement value by determining the average state of the qubits for the second symmetry result; fitting the first state measurement value to a first curve and the second state measurement value to a second curve; extrapolating the average state of the qubits at a second error rate using the fitted first and second curves, where the second error rate is lower than the first error rate, which results in the execution of the steps on a quantum computer.

[0048] One aspect of the present invention provides a method for reducing errors in quantum computing. The method includes: performing a first operation on the state of a qubit, where the first operation has a first error rate; obtaining a first measurement of the state of the qubit; performing a second operation on the state of the qubit, where the second operation has the first error rate; obtaining a second measurement of the state of the qubit; calculating a first average value of the state of the qubit at the first error rate by averaging the first measurement and the second measurement; performing a third operation on the state of the qubit, where the third operation has a second error rate lower than the first error rate and the third operation includes the first operation and a first basic operation; obtaining a third measurement of the state of the qubit; performing a fourth operation on the state of the qubit, where the fourth operation has the second error rate and the fourth operation includes the first operation and a second basic operation; obtaining a fourth measurement of the state of the qubit; calculating a second average value of the state of the qubit at the second error rate by averaging the third measurement and the fourth measurement. The method further includes: fitting the first average value of the state of the qubit and the second average value of the state of the qubit to a curve; extrapolating an average stage of the qubit at a third error rate using the fitted curve, where the third error rate is lower than the first error rate and the second error rate.

[0049] Another aspect of the present invention provides a device for performing quantum computing calculations that includes a selection module; a quantum processor; a quantum measurement device; and a classical processor. The selection module is configured to select a basic operation from a group of basic operations that includes a first basic operation and a second basic operation. The quantum processor is configured to: perform a first operation having a first error rate on the state of a qubit; perform a second operation having the first error rate on the state of the qubit; perform a third operation having a second error rate lower than the first error rate and including the first operation and the first basic operation on the state of the qubit; and perform a fourth operation having the second error rate and including the first operation and the second basic operation on the state of the qubit. The quantum measurement device is configured to obtain first, second, third, and fourth measurement values after performing the first, second, third, and fourth operations, respectively. The classical processor is configured to: calculate a first average value of the state of the qubit at the first error rate by averaging the first measurement value and the second measurement value; calculate a second average value of the state of the qubit at the second error rate by averaging the third measurement value and the fourth measurement value; fit the first average value of the state of the qubit and the second average value of the state of the qubit to a curve; and extrapolate the average state of the qubit at a third error rate lower than the first error rate and the second error rate using the fitted curve.

[0050] A further aspect of the present invention provides a computer-readable memory medium including instructions that, when executed by a computer, cause the computer to: execute a first operation on a state of a qubit, where the first operation has a first error rate; obtain a first measurement value of the state of the qubit; execute a second operation on the state of the qubit, where the second operation has the first error rate; obtain a second measurement value of the state of the qubit; calculate a first average value of the state of the qubit at the first error rate by averaging the first measurement value and the second measurement value; execute a third operation on the state of the qubit, where the third operation has a second error rate lower than the first error rate and the third operation includes the first operation and a first basic operation; obtain a third measurement value of the state of the qubit; execute a fourth operation on the state of the qubit, where the fourth operation has the second error rate and the fourth operation includes the first operation and a second basic operation; obtain a fourth measurement value of the state of the qubit; calculate a second average value of the state of the qubit at the second error rate by averaging the third measurement value and the fourth measurement value; fit the first average value of the state of the qubit and the second average value of the state of the qubit to a curve; and extrapolate an average stage of the qubit at a third error rate lower than the first error rate and the second error rate using the fitted curve. The execution of the steps included is brought about on a quantum computer.

[0051] Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings listed below.

Brief Description of the Drawings

[0052]

Figure 1

Figure 2

Figure 3

Figure 4

Best Mode for Carrying Out the Invention

[0053] FIG. 1 is a flowchart illustrating an error reduction method according to an embodiment. In this embodiment, a combination of quasiprobability, symmetry verification, and error extrapolation is used.

[0054] Quantum computation typically involves: initializing a group of qubits; executing a sequence of quantum operations on that group of qubits; and measuring the output state of each qubit. The measurement can also be performed on the group of qubits as a whole. The sequence of quantum operations has associated errors.

[0055] In step S101, an operation on the state of the qubits in a group of qubits is performed using a quantum processor. This operation has a first error rate and is performed multiple times. The first error rate n 1 is the number of errors expected to occur each time the operation is performed. The actual number of errors occurring each time varies, but the average number of errors over multiple operations is approximately n 1 . Errors including phase relaxation errors and depolarization errors can occur at a large number of possible error locations M. Assuming that the number of possible error locations is large and the first error rate is on the order of 1, i.e., M >> 1 and n 1 ~1.

[0056] Each time an operation is executed, the state of the qubit following that operation is measured using a quantum measurement device. After multiple executions of the operation, the expected value of the qubit state can be obtained by averaging the individual measurement values using a classical processor. This expected value can be mapped to a physical quantity of the system, such as position or momentum, often called an observable.

[0057] The operations executed in step S101 include gate operations and additional operations. The gate operations include one or more quantum logic gate operations. Any quantum gate or series of quantum gates can be selected according to the requirements of the experiment. The additional operations are executed after the gate operations and are used to reduce the errors resulting from the execution of the gate operations. These additional operations include symmetry operations used for error reduction using symmetry verification and basic operations used for error reduction using quasiprobability. Symmetry verification and quasiprobability error reduction techniques will be discussed below.

[0058] Without error reduction, it is possible to obtain measurements of noisy observables using gate operations. The gate operations are selected by the experimenter based on the observable to be measured. The aim of error reduction techniques is to estimate the value of the observable for which a noiseless or noise-free gate operation has been performed. A noisy gate operation U n can be represented as a noiseless operation U 0 followed by a noisy operation M, i.e., U n = MU 0 as represented.

[0059] In the quasiprobability error reduction technique, the inverse of the noise M -1 can be expressed as a function of each basic operation B j within the set of basic operations, where 1 ≤ j ≤ J and J is the number of basic operations in the set. In this embodiment, the Pauli principle is used, in which there are four different basic operations in the set (J = 4). The Pauli basic operation B acting on the state s of the qubitj is

Number

Number

Number

Number

[0060] To estimate the effect of the operation without noise, it is possible to perform a basic operation after the gate operation. The basic operation is randomly selected by the selection module. In this case, the possible operations for one gate operation and three basic operations are: B 1 U n , B 2 U n , B 3 U n , and B 4 U n . The j-th basic operation is selected by the selection module with a probability proportional to |a j |.

[0061] In an alternative embodiment, it is possible to perform additional gate operations and to select different basic operations from the set of basic operations. This increases the number of possible operations that can be selected by the selection module. Thus, reducing the effect of noiseless operations using quasiprobabilities usually requires the execution of a very large number of operations.

[0062] In this embodiment, quasiprobabilities are used to reduce the error rather than to eliminate it. The cost C of using quasiprobabilities to reduce the error rate from the natural error rate n to a first error rate n 1 is approximately

Number

[0063] The use of quasiprobabilities to reduce rather than eliminate the error conveniently makes the cost lower and thus requires fewer repetitions of the operations. A noisy gate operation U n can be converted into a gate operation U n1 with a first error rate, where the first error rate is lower than the error rate U n of the noisy gate operation (n 1 < n). The relationship between the original and the error-reduced operations can be expressed as U n = NU n1 .

[0064] The transformation N -1 is a function of the basic operations, N -1 = p 1 B 1 + p 2 B 2 + p 3 B 3 + p 4 B 4 and can be expressed as such, where each basic operation is |p jIt can be included in the execution of an operation using a probability proportional to |. For example, p 1 = 0.1, p 2 = -0.2, p 3 = 0.8 and p 4 = 0.3, and assuming that 140 operations are executed, the first operation B 1 U n will be executed approximately 10 times, the second operation B 2 U n will be executed approximately 20 times, the third operation B 3 U n will be executed approximately 80 times, and the fourth operation B 4 U n will be executed approximately 30 times.

[0065] Therefore, each operation executed in step S101 includes the j-th basic operation following the gate operation. The operation modified using the j-th basic operation can be called the j-th operation. The operation is executed multiple times, and each possible modified operation is executed multiple times according to the probability |p j |. Usually, step S101 includes a set of modified operations executed for the state of each qubit in a group of qubits.

[0066] In the symmetry verification error reduction technique, a symmetry operation is executed and a symmetry measurement is performed. In quantum computing, some properties of the system are known and can be verified. For example, regardless of the specific state of each electron, the number of electrons in the system needs to remain fixed. If an error occurs, it can appear as a decrease or increase in electrons.

[0067] However, when a symmetry operation is performed on a group of qubits as a whole, it is not possible to easily distinguish the occurrence of multiple errors using symmetry verification. In the above example, the symmetry operation may only be capable of determining whether the total number of electrons is odd or even. The result of the symmetry measurement following the execution of the symmetry operation will fail if the number of electrons changes by only ±1, ±3, ±5, etc. A failed symmetry test indicates that at least one error has occurred. The result will pass if the number of electrons does not change, or if it changes by only ±2, ±4, ±6, etc. Measured values that fail the symmetry verification test are usually discarded in existing error reduction techniques since it is already known that at least one error has occurred. However, it is not possible to conclude from a passed symmetry verification test that no error has occurred.

[0068] Each operation executed in step S101 includes a symmetry operation S. The noisy gate operation U n may include errors in a form that cannot be detected using the selected symmetry operation. Therefore, in this embodiment, a quasiprobability technique is used to remove error components that are not locally detectable using symmetry. From the above explanation of the symmetry verification technique, it is noted that the remaining locally detectable errors may combine again to form errors that are not globally detectable. The symmetry measurement is performed globally.

[0069] The symmetry operation and the basic operation executed after the gate operation in step S101 can be executed in any order following the gate operation. A first possible operation performed on a qubit among a group of qubits includes the execution of the gate operation U n followed by the symmetry operation S, and further followed by the first basic operation B 1 where the first basic operation is |p 1It is selected by a selection module using a first probability proportional to |. The first basic operation can change the symmetry of the system and, correspondingly, modify the pass / fail evaluation criteria for symmetry verification. The state of the qubit is measured by a quantum measurement device following the execution of the sequence of operations in the first operation.

[0070] A second possible operation performed on the qubits within the group of qubits is the gate operation U n followed by a second basic operation B 2 followed further by the execution of a symmetry operation S, where the second basic operation is selected by the selection module using a second probability proportional to |p 2 |. Similarly, the state of the qubit is measured by a quantum measurement device following the execution of the sequence of operations in the second operation.

[0071] The state of the qubit is generally a superposition of a first state |0> and a second state |1>. However, at the time of measurement, the state of the qubit is either the first state or the second state, i.e., either 0 or 1, which corresponds to -1 or +1 of the measurement result, respectively.

[0072] The first and second states differ depending on the type of qubit. Thus, the characteristics of the measured qubit depend on the type of that qubit. The quantum measurement device is selected corresponding to the type of qubit. For example, the first and second states of an electron spin qubit are spin-up and spin-down, where spin-up is recorded as +1 and spin-down is recorded as -1. Thus, the measurement of an electron spin qubit is obtained by measuring the electron spin, and the quantum measurement device is configured to measure the electron spin.

[0073] When the qubit is an electron charge qubit, the electron charge is measured, in which case the first and second states are no electron and one electron. When the qubit is a superconducting phase qubit, the excited state is measured, in which case the first and second states are the ground state and the first excited state. Any quantum system with first and second measurable states can be used as a qubit. An appropriate quantum measurement device that enables distinguishing between the first and second states is used to obtain the measurement value.

[0074] By executing the operation in step S101 multiple times, it is possible to determine the average state of the qubit. The original operation U n and the reduced error operation U n1 relationship between, that is, N -1 U n =U n1 is used (where N -1 =p 1 B 1 +p 2 B 2 +p 3 B 3 +p 4 B 4 ), to determine the effect of the reduced error operation on the state of the qubit, it is possible to computationally recombine the measurement values obtained following the execution of the modified operation in which different basic operations B j are sampled with probabilities proportional to |p j |. These measurement values are assigned a parity of ±1 according to the sign of the coefficient p j . For example, if p 2 =-0.2 and the state of the qubit following the execution of SB 2 U n is measured to be 1, then that measurement value is recorded as -1.

[0075] In step S102, a symmetry operation is used, and symmetry measurement values for a group of qubits are obtained using a symmetry measurement device. The group of qubits includes a plurality of qubits. The symmetry measurement values are obtained in S102 after each execution of operation S101. The symmetry measurement values are either a 'pass' of the first symmetry result when the number of errors is even or a 'fail' of the second symmetry result when the number of errors is odd. Examples of system characteristics that can be used for performing the symmetry verification test include parity, the number of particles, and energy. In existing symmetry verification techniques, measurement values that fail the symmetry test are discarded. However, in this embodiment, all measurement values are maintained and classified according to their symmetry results. This reduces the cost of the error reduction technique of this embodiment. Measurement values that fail the symmetry verification test can nevertheless be constructively used using the methods and analysis described below.

[0076] In this embodiment, a single symmetry operation is performed following the gate operation. In an alternative embodiment, more than one symmetry operation is used. Typically, each symmetry operation is used to verify the symmetry of different properties of the system, and a pass or fail result is returned.

[0077] In step S103, a first state measurement value E pass is obtained. Measurement values of the states of the qubits following the execution of the operation in S101, which are classified as 'pass' following the symmetry classification in S102, are averaged using a classical processor. The averaged measurement values are obtained from the set of the j-th operations executed. The weighted sampling of the possible elementary operations B j according to the determined coefficient p j experimentally reproduces the theoretically determined transformation N -1 .

[0078] In step S104, a second state measurement value E failis obtained, in which the measured values classified as 'Fail' in step S102 are averaged using a classical processor. These first state measurement values and second state measurement values will be different because the probability of an even error occurring is different from the probability of an odd error occurring. The first error rate n 1 in which the probability of an even error occurring is [Number] . The probability of an odd error occurring in the first error rate is [Number] . The measured values obtained in steps S102, S103, and S104 can be obtained simultaneously.

[0079] In an alternative embodiment where more than one symmetry operation is performed, additional state measurement values can be obtained. For example, if two symmetry operations are performed and each has a result of pass or fail, four state measurement values corresponding to pass-pass, pass-fail, fail-pass, and fail-fail will be measured.

[0080] In step S105, the first and second state measurement values are respectively fitted to the first and second curves using the classical processor of a conventional computer. This is an error reduction technique of extrapolation. The expected value of the observable, that is, the average of the individual measurement values of the qubit state, depends on the level of noise. Assuming that the relationship between the expected value of the observable and the noise level, or error rate, follows a trend line, it is possible to use the measured state measurement values to fit the trend line and predict the expected value of the observable at lower noise.

[0081] In this embodiment, it is assumed that the expected value of the observable decays exponentially as the error rate increases. The first and second state measurement values were brought about as a result with the first error rate from the operation performed in step S101, but since the probability of passing or failing the symmetry verification test is different, these two state measurement values will be attributed to different results of the measured expected value. The first curve used for fitting the first state measurement value is

Number

Number

[0082] At the first error rate, the expected value of the qubit state is expressed as E n1 =P even E pass +P odd E fail . In an alternative embodiment, this estimation of the expected value of the qubit state can be used to provide an error reduction value of the observable using the probabilities P even , P odd of the first and second symmetry results and the combination of the first and second state measurement values E pass , E fail .

[0083] In this embodiment, error extrapolation is used along with the assumption that the value of the observable decays approximately exponentially as the error rate increases, and the error is further reduced. Two categories of measurements obtained at a single error rate, namely pass and fail measurements, can be used to determine the noise-free expected value assuming that the exponential decay curve is a single exponential decay, i.e., E n =Ae -γn , where En is the average state of the qubit, n is the error rate, and A and γ are fitting parameters.

[0084] In an alternative embodiment, the curve is a multi-exponential decay curve, the sum of two or more exponential functions, i.e., for K > 1

Equation

[0085] In step S106, a classical processor is used to extrapolate the average state of the qubit at a second error rate that is lower than the first error rate. In this embodiment, the second error rate is selected to be the zero error rate so that an estimation of the noise-free expected value becomes possible.

[0086] Using the first and second fitting curves from step S105, the noise-free expected value E pass and E fail are respectively associated with the first and second state measurements E 0 to determine the noise-free expected value E

[0087]

Equation

[0088] Figure 2 is a flowchart illustrating an error reduction method according to another embodiment. In step S201, a first operation is performed on the state of the qubit using a quantum processor. The first operation has a first error rate n 1 which, in this embodiment, is typically the unmitigated error rate of the quantum computing system. Typically, the unmitigated error rate is the lowest experimentally achievable error rate. However, hardware variations and other environmental factors can affect that error rate.

[0089] The first operation is a gate operation selected by the experimenter. After the execution of the first operation, a first measurement value of the state of the qubit is obtained in step S202 using a quantum measurement device. The measurement result is either +1 or -1, as explained in connection with FIG. 1.

[0090] In step S203, a second operation is performed on the state of the qubit using a quantum processor. The second operation also has the first error rate n 1 The second operation in this embodiment is the same gate operation as that executed in step S201. After the execution of the second operation, a second measurement value of the state of the qubit is obtained in step S204 using a quantum measurement device.

[0091] The gate operation is typically repeated multiple times, and a measurement value of the state of the qubit is obtained after each execution of the gate operation. Then, in step S205, a first average value of the state of the qubit is calculated using a classical processor. The first average value is calculated by averaging the first measurement value, the second measurement value, and any additional measurement values obtained. The first average value represents the expected value of the observable at the first error rate.

[0092] In step S206, a third operation is performed on the state of the qubit using a quantum processor. The third operation has a first error rate n 1 lower than the second error rate n 2 , that is, n 2 < n 1 . The lower error rate is achieved using the probability technique as described in relation to FIG. 1. The third operation includes the first operation and the first basic operation. The first basic operation is one of the set of basic operations and is randomly selected using a selection module and the first probability. Following the execution of the third operation, in step S207, a third measurement value of the state of the qubit is obtained using a quantum measurement device.

[0093] In step S208, a fourth operation is performed on the state of the qubit. The fourth operation has a second error rate n 2 . The fourth operation includes the first operation and the second basic operation. The second basic operation is one of the set of basic operations and is randomly selected using a selection module and the second probability. The second basic operation is different from the first basic operation. Following the execution of the fourth operation, in step S208, a fourth measurement value of the state of the qubit is obtained.

[0094] The correction gate operation, that is, the first operation followed by a randomly selected basic operation, is typically repeated multiple times. The randomly selected basic operation is selected from a set of basic operations that includes the first basic operation, the second basic operation, and additional, different basic operations. In this embodiment, the set of basic operations used is the Pauli set, and there are four different basic operations for single qubit operations. Each selection of a basic operation is weighted according to a coefficient determined using probability as described in relation to FIG. 1. After each execution of the correction gate operation, a measurement value of the state of the qubit is obtained. In this method, it is possible to reduce the error rate by performing additional basic operations following the gate operation.

[0095] A gate operation is typically a sequence of operations that can include any typical quantum gate such as a Pauli or Hadamard gate, and a qubit is one of a group of qubits. The selection module is configured to modify each operation performed on the state of each qubit within the group of qubits. Optionally, some operations are not modified and are performed in their normal form. This random nature of the modification provides an unbiased statistical representation of the expected value of the qubit state. In this embodiment, for a group of n qubits, the selection module randomly selects a basic operation from a set of 4 n basic operations.

[0096] A second average value of the qubit state is calculated in step S210 by averaging a third measurement value, a fourth measurement value, and any additional measurement values obtained. The determined coefficient p j weighted sampling B of possible basic operations according to j theoretically determined transformation N -1 =Σ j p j B j is experimentally reproduced. In this embodiment, an operation having a first error rate and an operation having a second error rate are related as follows: U n1 =NU n2 , where n 2 <n 1 . The second average value represents the expected value of the observable at the second error rate. In this embodiment, the operator N is designed to change the form of the error so that the noise model is simplified.

[0097] In step S211, a classical processor is used to fit the first and second average values to a curve. The curve selected may depend on the theoretical understanding of the relationship between the expected value of the qubit state and the error rate. In this embodiment, the curve is an exponential decay curve that includes a single exponential function, i.e., E n =Ae -γn where En is the average state of the qubits, n is the error rate, and A and γ are fitting parameters.

[0098] In an alternative embodiment, the curve is a multi-exponential decay curve that includes at least two exponential functions, i.e., for K > 1

Equation

[0099] After the shape of the curve is determined by fitting the first and second average values, in step S212, it is possible to estimate the average state of the qubits at a third error rate using extrapolation. The third error rate is lower than the first and second error rates and is selected to be zero in this embodiment. In this method, a noise-free value of the observable can be estimated using a classical processor.

[0100] FIG. 3 is a schematic illustration of a first quantum computation according to an embodiment. In this embodiment, each qubit undergoes a sequence of four operations before being measured. Although the operations for a group of qubits including three qubits are illustrated in this illustration, typically there are dozens or hundreds of qubits within the group of qubits.

[0101] The first operation 311, the second operation 312, and the third operation 313 are respectively executed on the states of the first, second, and third qubits. In this embodiment, the first, second, and third operations 311-313 are executed simultaneously. Each of the first, second, and third operations 311-313 is a single-qubit operation. Subsequently, a fourth operation 314 and a fifth operation 315 are executed. In this embodiment, the fourth and fifth operations 314, 315 are executed simultaneously. The fourth operation 314 is a two-qubit operation executed on the first and second qubits, and the fifth operation 315 is a single-qubit operation executed on the third qubit. Subsequently, a sixth operation 316 and a seventh operation 317 are executed. In this embodiment, the sixth and seventh operations 316, 317 are executed simultaneously. The sixth operation 316 is a single-qubit operation executed on the first qubit, and the seventh operation 317 is a two-qubit operation involving the interaction of the second and third qubits.

[0102] The operations 311-317 from the first to the seventh are gate operations. The first, second, third, fifth, and sixth operations 311, 312, 313, 315, 316 are single-qubit gate operations executed on one of the qubits in a group of qubits. One or more of the first, second, third, fifth, and sixth operations 311, 312, 313, 315, 316 can be coincidence operations. The fourth and seventh operations 314, 317 are two-qubit gate operations executed on two of the qubits in a group of qubits. It is possible to select any single-qubit and / or two-qubit operation according to the requirements of the experiment.

[0103] Following the execution of the gate operations as described above, in this embodiment, for the states of the first, second, and third qubits, a first basic operation 321, a second basic operation 322, and a third basic operation 323 are respectively executed. Each of the first, second, and third basic operations 321 - 323 is randomly selected from a set of basic operations using a selection module. In this embodiment, the set of basic operations is the Pauli set, and there are three operations in the set. This random selection means that the first, second, and third basic operations 321 - 323 can all be different, or only two of them can be identical to each other, or all of them can be identical.

[0104] Following the execution of the sequence of operations, measurement values are obtained. In this embodiment, symmetry measurement values are obtained using a symmetry measurement device 330. The symmetry measurement device 330 measures the characteristics of the system as a whole. The symmetry measurement in this embodiment is designed not to affect the measurement of the state of each qubit within the group of qubits. For example, the symmetry measurement device 330 can measure the state of an auxiliary system qubit configured to change state when detecting a change in a specified state in any of the first, second, and third qubits. The measurement value of the state of the auxiliary system qubit can be obtained in a useful form without collapsing the state of the qubits within the group of qubits.

[0105] The quantum computation described can be executed multiple times. In this embodiment including symmetry verification, using the symmetry measurement device 330, symmetry measurement values are obtained after each execution of the quantum computation. The symmetry measurement value becomes a first symmetry result if the number of errors is even, or a second symmetry result if the number of errors is odd.

[0106] The first quantum measurement device 331 measures the state of the first qubit. The second quantum measurement device 332 measures the state of the second qubit. The third quantum measurement device 333 measures the state of the third qubit.

[0107] The use of symmetry measurements is optional, and in alternative embodiments, only the state of each qubit is measured following the operation. In further alternative embodiments, each of the operations performed on the groups of qubits, namely the first, second, third, fourth, fifth, sixth, and seventh operations 311-317, can be modified using randomly selected elementary operations.

[0108] FIG. 4 is an illustration of a fitting and extrapolation process according to an embodiment. The fitting and extrapolation are performed using a processor of a classical computer. The first measurement value 41 at the first error rate 42 and the second measurement value 43 at the second error rate 44 are obtained using the method described above. Since the first error rate 42 is greater than the second error rate 44, the first measurement value 41 is smaller than the second measurement value 43.

[0109] Using a classical processor, an exponential decay curve 45 of the form E = Ae -γn is fitted to the first measurement value 41 and the second measurement value 43. When the fitting parameters A and γ are determined, the curve is extrapolated using a classical processor to the third error rate 47. Here, the third error rate 47 is the zero error rate, i.e., n = 0. A value 46 of the observable without error is estimated by extrapolation to the zero error rate. In alternative embodiments, it is possible to perform additional measurements at additional error rates to improve the estimation of the fitting parameters.

[0110] As is recognized, an improved error reduction method is provided that significantly improves the estimation of observables without error. The combination of error reduction techniques as described results in an improved estimation of observables without error at a reduced cost.

Description of Signs

[0111] 41 First measurement value 42 First error rate 43 Second measurement value 44 Second error rate 45 Exponential decay curve 46 Value without error 47 Third error rate 311 - 317 Gate operation 321 - 323 Basic operation 330 Symmetry measurement device 331 - 333 Quantum measurement device

Claims

1. 1. A method for mitigating errors in quantum computing, comprising: performing an operation on a state of a qubit in a group of qubits a plurality of times, the operation having a first error rate that is a number of the errors expected to occur each time the operation is performed, each execution of the operation: performing a first operation including a gate operation, a symmetry operation, and a first elementary operation; or performing a second operation including the gate operation, the symmetry operation, and a second elementary operation; wherein the first and second elementary operations are different elementary operations selected from a set of elementary operations; further comprising measuring the state of the qubit; performing an operation, wherein a probability of performing the first operation is a first probability and a probability of performing the second operation is a second probability; obtaining a symmetry measurement for the group of qubits after each execution of the operation using the symmetry operation, the group of qubits comprising a plurality of qubits; wherein the symmetry measure is a first symmetry result when the number of errors is even or a second symmetry result when the number of errors is odd; and obtaining a first state measurement by determining an average state of the qubit for the first symmetry result. obtaining a second state measurement by determining an average state of the qubit for the second symmetry result; The first condition measurement [0020] fitting a first curve having the form: The second condition measurement ##EQU00021## and fitting a second curve having the form: where n is the error rate, A and γ are fitting parameters, and extrapolating the average state of the qubit at a second error rate using the fitted first and second curves. wherein the second error rate is lower than the first error rate; The method for reducing the error includes the steps of:

2. 2. The method of claim 1, wherein the first and second elementary operations are selected from a set of elementary operations that includes Pauli elementary operations.

3. The method of claim 1 or 2, wherein the first symmetry result is a pass and the second symmetry result is a fail.

4. The qubit is a first qubit, and the method further comprises: performing the operation multiple times on a state of a second qubit in the group of qubits; obtaining a third state measurement by determining an average state of the second qubit for the first symmetry result; and obtaining a fourth state measurement by determining an average state of the second qubit for the second symmetry result; and fitting the third condition measurements to a third curve and the fourth condition measurements to a fourth curve; extrapolating the average state of the second qubit at the second error rate using the fitted third and fourth curves; The method of reducing errors according to any one of claims 1 to 3, comprising:

5. 1. A device for performing quantum computing calculations, comprising: selecting a first elementary operation from the set of elementary operations using a first probability; and configured to select a second primitive operation from the set of primitive operations using a second probability; wherein the first and second basic operations are different. A selection module; The method is configured to perform an operation on a state of a qubit in a group of qubits a plurality of times, where the operation has a first error rate that is a number of errors expected to occur each time the operation is performed, and where each performance of the operation comprises: a quantum processor that includes performing gate operations, symmetry operations, and the selected elementary operations; a quantum measurement device configured to measure the state of the qubit; and a symmetry measurement device configured to measure the symmetry of the group of qubits after each execution of the operation using the symmetry operation, where the group of qubits comprises a plurality of qubits; a symmetry measurement device, in which the symmetry measurement value is a first symmetry result when the number of errors is even or a second symmetry result when the number of errors is odd; obtaining a first state measurement by determining an average state of the qubits for the first symmetry result, and obtaining a second state measurement by determining an average state of the qubits for the second symmetry result; When n is the error rate and A and γ are fitting parameters, [0022] fitting the first condition measurements to a first curve having the form: [0023] fitting the second condition measurements to a second curve having the form: a classical processor configured to use the fitted first and second curves to extrapolate the average state of the qubit at a second error rate that is lower than the first error rate; A device for performing quantum computing calculations, comprising:

6. 1. A method for mitigating errors in quantum computing, comprising: performing a first operation on a state of the qubit, the first operation having a first error rate that is a number of the errors expected to occur each time the operation is performed; obtaining a first measurement of the state of the qubit; performing a second operation on the state of the qubit, the second operation having the first error rate; obtaining a second measurement of the state of the qubit; and calculating a first average value of the state of the qubit at the first error rate by averaging the first measurement and the second measurement; performing a third operation on the state of the qubit, the third operation having a second error rate lower than the first error rate and the third operation comprising the first operation and a first elementary operation; obtaining a third measurement of the state of the qubit; and performing a fourth operation on the state of the qubit, the fourth operation having the second error rate and the fourth operation comprising the first operation and a second elementary operation; wherein the first and second elementary operations are different elementary operations selected from a set of elementary operations; obtaining a fourth measurement of the state of the qubit; and calculating a second average value of the state of the qubit at the second error rate by averaging the third measurement and the fourth measurement; and fitting the first average value of the states of the qubits and the second average value of the states of the qubits to a curve; using the fitted curve to extrapolate an average stage of the qubit at a third error rate, the third error rate being lower than the first error rate and the second error rate; The method for reducing the error includes the steps of:

7. 7. The method of claim 6, wherein the probability of selecting a first elementary operation is a first probability and the probability of selecting a second elementary operation is a second probability.

8. 8. A method for reducing errors as claimed in claim 6 or 7, wherein the set of elementary operations includes Pauli elementary operations.

9. The method of claim 6 , wherein the second operation is the same as the first operation.

10. 10. A method of mitigating errors as claimed in any one of claims 6 to 9, wherein the qubit is one of a plurality of qubits in a group of qubits.

11. A method for mitigating errors according to any one of claims 6 to 10, wherein the curve is an exponential decay curve.

12. The exponential decay curve is expressed as follows: E = E = the average state of the qubit, n = the error rate, A ｋ and γ ｋ When using as fitting parameters, ##EQU00024## 12. The method of claim 11, wherein the decay curve is a multi-exponential decay curve of the form:

13. 1. A device for performing quantum computing calculations, comprising: a selection module configured to select a base operation from a set of base operations including a first base operation and a second base operation, where the first and second base operations are different; performing a first operation on the state of the qubit having a first error rate, the first error rate being a number of errors expected to occur each time the operation is performed; performing a second operation on the state of the qubit having the first error rate; performing a third operation on the state of the qubit, the third operation having a second error rate lower than the first error rate and comprising the first operation and the first elementary operation; and a quantum processor configured to perform a fourth operation on the state of the qubit, the fourth operation having the second error rate and including the first operation and the second elementary operation; a quantum measurement device configured to obtain first, second, third, and fourth measurements after execution of the first, second, third, and fourth operations, respectively; calculating a first average value of the state of the qubit at the first error rate by averaging the first measurement and the second measurement; calculating a second average value of the state of the qubit at the second error rate by averaging the third measurement and the fourth measurement; fitting the first average value of the states of the qubits and the second average value of the states of the qubits to a curve; and a classical processor configured to use the fitted curve to extrapolate an average state of the qubit at the first error rate and a third error rate that is lower than the second error rate; and A device for performing quantum computing calculations, comprising:

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