Multi-exponential error extrapolation method

Abstract

A method of reducing errors when using a quantum computer, comprising: S101, performing a first operation 21 on a state of a qubit a plurality of times; wherein the first operation 21 has a first error rate 32; S102, obtaining a first measurement of an average state of the qubit; S103, modifying an error rate of the quantum computer from the first error rate 32 to a second error rate 34; S104, performing a second operation 23 on the state of the qubit a plurality of times; wherein the second operation 23 has the second error rate 34; S105, obtaining a second measurement of the average state of the qubit; S106, modifying the error rate of the quantum computer from the second error rate to a third error rate; S107, performing a third operation on the state of the qubit a plurality of times; wherein the third operation has the third error rate; S108, obtaining a third measurement of the average state of the qubit; S109, modifying the error rate of the quantum computer from the third error rate to a fourth error rate; S110, performing a fourth operation on the state of the qubit a plurality of times; wherein the fourth operation has the fourth error rate; S111, obtaining a fourth measurement of the average state of the qubit; S112, fitting the first, second, third, and fourth measurements to a multi-exponential decay curve 35; and S113, using the fitted curve 35 to extrapolate an average state of the qubit at a fifth error rate 37, wherein the fifth error rate 37 is lower than the first, second, third, and fourth error rates.

Description

Technical Field

[0001] This invention relates to error reduction techniques in quantum computing. Background Technology

[0002] Quantum computers can be used to compute "observables," which are properties of a system. To measure an observable, the output state of the qubit can be measured after performing a sequence of quantum operations on it. The same sequence of quantum operations is typically repeated multiple 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 qubits is limited by errors, and therefore the estimated expectation is also limited by errors. The goal of quantum computing is to reduce or even eliminate these errors. However, a more realistic approach for quantum devices in the near future, or quantum devices in the era of Noisy Medium-Sized Quantum (NISQ), is to aim at reducing these errors using analytical methods. In this way, error-free or noise-free expectation values ​​of observables can be estimated.

[0004] Error reduction techniques use additional measurements to extract a noise-free expected value from noisy measurements. One such technique is error extrapolation. In this technique, the noise level is artificially increased through physical control of hardware, and the expected value is measured and plotted as a function of noise. The expected value changes with increasing noise, following a trend. The noise-free value can be estimated by fitting the measurements to this trend and extrapolating to determine the expected value of error-free quantum computing.

[0005] The form of the trend is important because it will affect the estimation. Developing techniques that can improve the accuracy of the estimation is desirable. Summary of the Invention

[0006] One aspect of the present invention provides a method for reducing errors when using a quantum computer. The method includes: performing a first operation multiple times on the state of a qubit; wherein the first operation has a first error rate; obtaining a first measurement of the average state of the qubit; modifying the error rate of the quantum computer from the first error rate to a second error rate; performing a second operation multiple times on the state of the qubit; wherein the second operation has a second error rate; obtaining a second measurement of the average state of the qubit; modifying the error rate of the quantum computer from the second error rate to a third error rate; performing a third operation multiple times on the state of the qubit; wherein the third operation has a third error rate; obtaining a third measurement of the average state of the qubit; modifying the error rate of the quantum computer from the third error rate to a fourth error rate; performing a fourth operation multiple times on the state of the qubit; wherein the fourth operation has a fourth error rate; obtaining a fourth measurement of the average state of the qubit; fitting the first, second, third, and fourth measurements to a multi-exponential decay curve; and using the fitted curve to extrapolate the average state of the qubit at a fifth error rate, wherein the fifth error rate is lower than the first, second, third, and fourth error rates.

[0007] One advantage of this method is the improved error reduction. The use of multi-exponential decay curves provides an accurate estimate of noise-free observables. In particular, this estimate is generally more accurate than that obtained using alternative methods such as single-exponential decay curves. The fifth error rate is lower than each of the first, second, third, and fourth error rates.

[0008] Preferably, the multi-exponential decay curve is the sum of K exponential curves, where K ≥ 2. Each decay curve can have a different decay rate. Preferably, the multi-exponential decay curve used to fit the obtained measurements has the following form: Here, E is the average state of the qubit, n is the error rate, and A is the mean state of the qubit. k and γ k The fitting parameters are denoted as n. Preferably, the error rate is the number of errors expected to occur when the operation is performed. The form of the multi-exponential decay curve shows that as the error rate n increases, the expected value and the average state of the qubit decrease exponentially.

[0009] Entering K=1 into the above equation will return a single exponential decay curve. For multi-exponential decay curves, K is typically greater than or equal to 2. Alternatively, K=2 in the above equation. Therefore, a multi-exponential decay curve can be a double-exponential decay curve with the following form: One advantage of using the sum of only two exponents in this way is the reduced likelihood of overfitting. Overfitting can occur if the number of measurements is insufficient to determine the fitting parameters. The consequence of overfitting is a poor estimate of the expected value.

[0010] The number of exponents K selected to produce the best fit may be greater than two. To prevent overfitting of the data, a fit analysis can be performed to determine an appropriate value for K. For example, a threshold can be set for the fit loss function, where a curve with the lowest value of K that reaches the fit threshold can be selected. Advantageously, by setting a threshold, an improved fit can be determined. Furthermore, selecting a minimum acceptable number of exponents helps to avoid overfitting of the data.

[0011] A k and γ k This is the fitting parameter used in multi-exponential decay curves. Preferably, the fitting parameter γ... k Greater than or equal to 0 and less than or equal to 1, i.e., 0 ≤ γ k ≤1. Fitting parameter γ k This limitation can stem from the assumption that errors occur randomly. In this case, for a very large number of errors, the expected value of the observables may be expected to approach zero.

[0012] The error rate *n* is typically proportional to the number of possible error locations *M*. The number of possible error locations *M* can be assumed to be large, i.e., much greater than 1, and can depend on the specific design of the quantum circuit and the chosen implementation of the quantum operation. Therefore, the error rate *n* can also depend on the circuit design and implementation. The error can be a Pauli error, or can be converted to a Pauli error. The conversion from non-Pauli errors to Pauli errors can be achieved using Pauli rotations. Advantageously, multi-exponential decay curves have been found to be a good noise model for Pauli errors.

[0013] The first, second, third, and fourth error rates, performed respectively, can be different. This advantageously results in first, second, third, and fourth measurements at different error rates, which allows the first, second, third, and fourth measurements to fit to a curve. Preferably, the second error rate is higher than the first error rate. Optionally, the third error rate is higher than the second error rate, and the fourth error rate is higher than the third error rate. Alternatively, the first, second, third, and fourth error rates can be in any order. Preferably, the error rate is increased by adding noise from the same noise model. This advantageously improves the accuracy of the prediction of the average state of the qubit at a fifth error rate. The fifth error rate is lower than each of the first, second, third, and fourth error rates; the fifth error rate can be 0.

[0014] In one example, additional operations can be used to increase the error rate. Optionally, the second operation includes the first operation and the modification operation. Optionally, each of the second, third, and fourth operations includes both the first operation and the modification operation. One or more of the second, third, and fourth operations may include more than one modification operation. Preferably, each of the second, third, and fourth operations includes at least the first operation. For example, the third operation may include the first operation, the first modification operation, and the second modification operation. The second modification operation may be the same as the first modification operation or may be different from the first modification operation. In another example, the fourth operation may include the first operation as well as the first, second, and third modification operations. In this example, the first and third modification operations may be the same and the second modification operation may be different, or the second and third modification operations may be the same and the first modification operation may be different, or the first and second modification operations may be the same and the third modification operation may be different, or the first, second, and third modification operations may all be the same or all different. The use of "first," "second," and "third" in the above paragraphs is irrelevant to the order in which the modification operations are performed and is also irrelevant to any particular operation.

[0015] Typically, any of the second, third, and fourth operations may include one or more modification operations in addition to the first operation. These modification operations may be completely different, or in a scenario with two or more modification operations, any two or more operations may be identical.

[0016] The modification operations can be selected based on the predicted noise model. For example, each modification operation can include the Pauli operation, such as Pauli-X(σ). x ), Pauli-Y (σ y ) or Pauli-Z(σ z The operator () can be randomly selected. The second operation is executed multiple times. Preferably, each execution of the second operation includes a randomly modified operation, which may be the same as or different from a previously executed modified operation. Optionally, the modified operation includes multiple additional operations.

[0017] Typically, when an operation (such as a second, third, or fourth operation) includes a first operation and more than one modification operation, it is preferable to randomly select multiple modification operations, and the selection can be different each time the operation is executed.

[0018] The first operation may include multiple operations performed sequentially. Similarly, the second operation may include multiple operations performed sequentially. A quantum device is typically used to perform the first operation. Preferably, the second operation is performed at different time points using the same quantum device with a modified error rate. Likewise, it is preferred that the third and fourth operations are performed at different time points using the same quantum device with another modified error rate.

[0019] Preferably, performing the first operation multiple times includes performing multiple first operations, and performing the second operation multiple times includes performing multiple first operations and multiple modification operations. Preferably, performing the third and fourth operations multiple times includes performing multiple first operations and multiple one or more modification operations. Typically, the first measurement of the average state of the qubit is obtained by averaging the measurements recorded after each of the multiple first operations. Similarly, the second, third, and fourth measurements of the average state of the qubit can be obtained by averaging the measurements recorded after the execution of the second, third, and fourth operations, respectively. Typically, the first operation is performed multiple times to obtain the first measurement of the average state of the qubit before performing the second operation multiple times to obtain the second measurement of the average state of the qubit. Similarly, the second measurement is typically obtained before performing the third operation, and the third measurement is typically obtained before performing the fourth operation. This advantageously reduces the measurement uncertainty in the first, second, third, and fourth measurements of the average state of the qubit.

[0020] The second operation can be performed multiple times by executing each of the multiple first operations and each of the multiple modification operations in any order. Preferably, performing the second operation multiple times involves executing one of the multiple modification operations after each of the multiple first operations. The advantage of performing the second operation multiple times by executing each of the multiple first and modification operations in an alternating manner is that it advantageously results in a fixed second error rate. It should be noted that the error rate is an expected value, and the actual number of errors that occur will change each time an experiment with a fixed error rate is performed.

[0021] Typically, for a second, third, or fourth operation that includes more than one modification operation in addition to the first operation, multiple executions of the second, third, or fourth operation include executing two or more modification operations in any order after each of the multiple first operations.

[0022] The first, second, third, and / or fourth operations can transform the state of a qubit from an input state to a measurable output state. The state of the qubit can be initialized. Specifically, the state of the qubit can be initialized before multiple executions of the first, second, third, and / or fourth operations, thereby providing a uniform initial state. Qubit initialization provides a uniform initial qubit state. The initial state can affect the measurement of the qubit's output state; therefore, the qubit preferably enters the quantum circuit in an initialized state or a zero state. The state of the qubit can be initialized before the first execution of the first operation, before the first execution of the second operation, before the first execution of the third operation, and before the first execution of the fourth operation.

[0023] Further operations can be performed on the state of the qubit. Optionally, each additional operation is performed multiple times on the state of the qubit to obtain a measurement of the average state of the qubit. Preferably, the state of the qubit is initialized before each of the further operations is performed for the first time. Typically, the method may include modifying the error rate of the quantum computer to an i-th error rate; performing an i-th operation on the state of the qubit; wherein the i-th operation has an i-th error rate; wherein the i-th error rate is greater than a first error rate; obtaining an i-th measurement of the average state of the qubit; and fitting the first, second, third, and fourth measurements and the i-th measurement to a multi-exponential decay curve. Preferably, the method further includes using the fitted curve to extrapolate the average state of the qubit at a fifth error rate, wherein the fifth error rate is lower than the first, second, third, fourth, and i-th error rates.

[0024] Further measurements at additional error rates in this manner advantageously improve the estimation of the fitting parameters in the multi-exponential decay curve, and thus improve the estimation of the values ​​of the error-free observables. A double-exponential decay curve has four free fitting parameters: A1, γ1, A2, and γ2. Therefore, for a double-exponential decay curve, at least four different operations are performed, each at a different error rate. Similarly, a triple-exponential decay curve has six free fitting parameters. Preferably, for a triple-exponential decay curve, at least six different operations are performed, each at a different error rate. Typically, the number of different error rates is greater than or equal to 2K, where K is the number of exponents in the multi-exponential decay curve. The error rate of the quantum computer can be modified to the i-th error rate using one or more random Pauli gates as described with respect to the second operation. The addition of random Pauli gates can increase the error due to Pauli noise.

[0025] Another aspect of the present invention provides an apparatus for performing quantum computing. The apparatus includes a quantum processor configured to: perform a first operation multiple times on the state of a qubit, wherein the first operation has a first error rate; perform a second operation multiple times on the state of the qubit, wherein the second operation has a second error rate; perform a third operation multiple times on the state of the qubit, wherein the third operation has a third error rate; and perform a fourth operation multiple times on the state of the qubit, wherein the fourth operation has a fourth error rate. The apparatus also includes a quantum measurement gate configured to: obtain a first measurement of the average state of the qubit after the first operation; obtain a second measurement of the average state of the qubit after the second operation; obtain a third measurement of the average state of the qubit after the third operation; and obtain a fourth measurement of the average state of the qubit after the fourth operation. The apparatus also includes a classical processor configured to fit the first, second, third, and fourth measurements to a multi-exponential decay curve, and use the fitted curve to extrapolate the average state of the qubit at a fifth error rate, wherein the fifth error rate is lower than the first, second, third, and fourth error rates.

[0026] Advantageously, the first, second, third, and fourth measurements obtained using this device at different error rates can be used to predict the average state of qubits at lower error rates using extrapolation. The fifth error rate can be selected as zero to estimate the average state of noise-free qubits.

[0027] Another aspect of the present invention provides a computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to perform the following steps on a quantum computer: performing a first operation multiple times on the state of a qubit; wherein the first operation has a first error rate; obtaining a first measurement of the average state of the qubit; modifying the error rate of the quantum computer from the first error rate to a second error rate; performing a second operation multiple times on the state of the qubit; wherein the second operation has a second error rate; obtaining a second measurement of the average state of the qubit; modifying the error rate of the quantum computer from the second error rate to a third error rate; performing a third operation multiple times on the state of the qubit; wherein the third operation has a third error rate; obtaining a third measurement of the average state of the qubit; modifying the error rate of the quantum computer from the third error rate to a fourth error rate; performing a fourth operation multiple times on the state of the qubit; wherein the fourth operation has a fourth error rate; obtaining a fourth measurement of the average state of the qubit; fitting the first, second, third, and fourth measurements to a multi-exponential decay curve; and using the fitted curve to extrapolate the average state of the qubit at a fifth error rate, wherein the fifth error rate is lower than the first, second, third, and fourth error rates.

[0028] It is difficult to eliminate the errors that occur in quantum computers using physical improvements. Advantageously, the execution of these steps leads to a mathematical estimate of the average state of the qubits with a reduced error rate.

[0029] Another aspect of the present invention provides a method for reducing errors when using a quantum computer. The method includes: performing a first operation on the state of a qubit; wherein the first operation has a first error rate; obtaining a first measurement of the average state of the qubit; modifying the error rate of the quantum computer from the first error rate to a second error rate; performing a second operation on the state of the qubit; wherein the second operation has a second error rate; obtaining a second measurement of the average state of the qubit; fitting the first and second measurements to a multi-exponential decay curve; and using the fitted curve to extrapolate the average state of the qubit at a third error rate, wherein the third error rate is lower than the first and second error rates.

[0030] Another aspect of the present invention provides an apparatus for performing quantum computing computation, wherein the apparatus includes: a quantum processor, a quantum measurement gate, and a classical processor. The quantum processor is configured to perform a first operation on the state of a qubit, wherein the first operation has a first error rate, and to perform a second operation on the state of the qubit, wherein the second operation has a second error rate; the quantum measurement gate is configured to obtain a first measurement of the average state of the qubit after the first operation, and to obtain a second measurement of the average state of the qubit after the second operation; the classical processor is configured to fit the first and second measurements to a multi-exponential decay curve, and to use the fitted curve to extrapolate the average state of the qubit at a third error rate, wherein the third error rate is lower than the first and second error rates.

[0031] Another aspect of the present invention provides a computer-readable storage medium comprising instructions which, when executed by a computer, cause the computer to perform the following steps on a quantum computer: performing a first operation on the state of a qubit; wherein the first operation has a first error rate; obtaining a first measurement of the average state of the qubit; modifying the error rate of the quantum computer from the first error rate to a second error rate; performing a second operation on the state of the qubit; wherein the second operation has a second error rate; obtaining a second measurement of the average state of the qubit; fitting the first and second measurements to a multi-exponential decay curve; and using the fitted curve to extrapolate the average state of the qubit at a third error rate, wherein the third error rate is lower than the first and second error rates. Attached Figure Description

[0032] Embodiments of the present invention will now be described with reference to the accompanying drawings, in which:

[0033] Figure 1 This is a flowchart of the error reduction method according to the first embodiment;

[0034] Figure 2A This is a schematic diagram of the first operation according to the second embodiment;

[0035] Figure 2B It is a schematic diagram of the second operation according to the second embodiment; and

[0036] Figure 3 This is a graph showing the values ​​of observable quantities as a function of the error rate according to the third embodiment. Detailed Implementation

[0037] Figure 1 This is a flowchart depicting an embodiment of a method for reducing errors. Quantum computing typically involves: initializing a set of qubits; performing a sequence of quantum operations on the set of qubits; and measuring the output state of each qubit.

[0038] In step S101, a quantum processor is used to perform a first operation on the state of the qubit. In this embodiment, the qubit is initialized to a zero state before performing the first operation. The first operation transforms the state of the qubit. A qubit is one in a set of qubits. Each qubit in the set of qubits can be operated on simultaneously, but can undergo different state transitions. The first operation includes multiple operations of quantum logic gates, including Pauli gates, Hadamard gates, and controlled-NOT (CNOT) gates. Multiple operations are performed sequentially. The first operation has a first error rate n1, where n1 is the number of errors expected to occur when performing the first operation. It is assumed that the number M of possible error locations is large, and the first error rate is of order 1, i.e., M >> 1; n ~ 1. Examples of errors that may occur in a quantum device include Pauli errors, such as dephase errors and depolarization errors. Pauli errors are a form of Markovian errors. Pauli rotation techniques can be used to convert Markov errors that are not Pauli errors into Pauli errors.

[0039] The first operation is executed multiple times. After each execution of the first operation, the state of the qubit is measured. In step S102, a first measurement of the average state of the qubit is obtained using a quantum measurement device. The measurement of the state of the qubit is obtained after the first operation has been executed. The state of the qubit is typically a superposition of the first state |0> and the second state |1>. However, the measured state of the qubit after each execution of the operation will be either the first state or the second state. If the qubit is in the first state, the recorded measurement is -1. Or, if the qubit is in the second state, the measurement is recorded as +1. By executing the same operation multiple times, the average state of the qubit can be determined. In S102, the first measurement is obtained by averaging the measurements recorded after each of the multiple executions of the first operation.

[0040] Depending on the type of qubit, the first and second states are different. Therefore, the measurement properties of a qubit depend on its type. A quantum measurement device is selected to correspond 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 as -1. Therefore, to obtain a measurement of an electron spin qubit by measuring its electron spin, a quantum measurement device is configured to measure electron spin.

[0041] If the qubit is an electron charge qubit, then the electron charge is measured, where the first and second states are no electron and one electron. If the qubit is a superconducting phase qubit, then the excited state is measured, where the first and second states are the ground state and the first excited state. Any quantum system having first and second measurement states can be used as a qubit. A suitable quantum measurement device capable of distinguishing between the first and second states is used to obtain the measurement.

[0042] The first operation is repeated multiple times, and the average state of the qubit is calculated by averaging the individual measurements recorded after each execution of the first operation. Therefore, the first measurement is the expected value of the qubit's state at the first error rate. This corresponds to a noisy measurement of the observable, i.e., a property of the system.

[0043] Once the first measurement has been obtained, the error rate of the quantum computer is modified at S103. The error rate is changed from a first error rate to a second error rate. The first error rate is typically the minimum achievable error rate; modifying it to the second involves deliberately increasing the physical error rate. Importantly, to accurately mimic the effect of noise on the measurement, any additional error should originate from the same noise model. Experimenters should understand that there are many ways to achieve this in practice. Typically, the experimenter's goal is to reduce the noise level to its minimum possible severity. Many experimental techniques can be used to achieve this. To increase the noise level, the quantum computer's operations can be modified to remove or reduce the effects of noise-reducing elements applied by the experimenter. For example, the time between operations can be increased to increase the effective level of decoherence, the magnetic shielding level can be reduced to increase random magnetic noise, or additional operations simulating noise can be performed.

[0044] After the error rate of the quantum computer has been modified in step S103, a second operation is performed on the state of the qubit using a quantum processor in step S104. The second operation is performed on the same quantum device used to perform the first operation, but with a modified error rate, namely a second error rate n2. In this embodiment, the second error rate is greater than the first error rate. Before performing the second operation, the state of the qubit is initialized. Initialization of the qubit places it in a "zero" state, which can be used as a baseline or reference point from which the output state of the qubit can be measured.

[0045] In this embodiment, the modification takes the form of a modification operation performed in addition to the first operation. The second operation includes both the first and modification operations. In this embodiment, the assumed noise model is a Pauli noise model, therefore the modification operation is a randomly selected Pauli gate. The execution of the additional operation after the first operation increases the noise level. Examples of Pauli gates include: a Pauli-X gate that performs a rotation around the x-axis. Perform a Pauli-Y gate to rotate around the y-axis. And the Pauli-Z gate that performs rotations around the z-axis, The axis referred to here is the axis of the Bloch sphere. The Bloch sphere can be used to geometrically represent qubit states. It has been found that the change in the expected value of a Pauli observable with increased Pauli noise using random Pauli gates can be approximated using multi-exponential decay.

[0046] The second operation is executed multiple times. The second operation comprises the first operation and a modification operation. Each execution of the first operation can be modified by performing a modification operation such as a random Pauli gate. The second operation can be performed by alternating between the first and modification operations, such that each operation of the first operation is followed by one operation of the modification operation. These operations can be executed in any order. Importantly, the different error rates associated with the first and second operations allow the state of the qubit to be measured as a function of the error rate.

[0047] In step S105, a second measurement of the average state of the qubit is obtained using a quantum measurement device. The second measurement is obtained in a similar manner to the first measurement but using a modified quantum computer. The state of the qubit is measured after each execution of the second operation, and the expected value of the qubit's state is calculated by averaging the measurements recorded for each repeated second operation. The second measurement provides a considerable second noise measurement with a different error rate.

[0048] In order to estimate error-free observables with a suitable high level of accuracy, a factor C, which depends on the first and second error rates, must be used. EThis increases the number of circuit operations, i.e., the execution of the first and second operations and their measurements. For example, for the first error rate n1 and the second error rate n2 = λn1, the cost factor C E Approximately equal to:

[0049]

[0050] Where γ is the observable decay rate of the expected value of an observable with noise.

[0051] Based on cost factor C E After a suitable number of circuit runs, the expected value of the observable can be plotted as a function of the error rate. A first measurement of the expected value of the qubit's state is taken at the first error rate, and a second measurement is taken at the second error rate. As the error rate increases, the expected value decreases. The effect of the increasing error rate on the expected value of the observable can be used to estimate the error-free value.

[0052] The expected value, as a function of the error rate, will follow a trend. To estimate the error-free value, the measurements (i.e., the first, second, third, and fourth measurements) can be fitted to a suitable trend, and extrapolation can be used to estimate the error-free value. Currently, experimental quantum systems cannot perform operations with an error rate below a certain threshold. Therefore, mathematical estimation of the error-free value is important in quantum computing in the near future, and fault-tolerant or error-free quantum computing is under development.

[0053] Once the second measurement has been obtained, the error rate of the quantum computer is modified at S106. The error rate of the quantum computer is modified from the second error rate to a third error rate. The error rate modification from the second error rate to the third error rate is similar to the error rate modification from the first error rate to the second error rate described in step S103. In this example, the third error rate is greater than the second error rate. In an alternative example, the third error rate may be lower than the second error rate but greater than the first error rate.

[0054] In step S107, a third operation is performed on the state of the qubit. The third operation is performed multiple times, and the state of the qubit is measured after each execution. The third operation has a third error rate. In this example, the third operation includes a first operation and first and second modification operations. The first and second modification operations are randomly selected from the Pauli set of operations described with respect to step S104 above. The first and second modification operations may be the same or different, and may vary for each execution of the third operation.

[0055] In step S108, a third measurement of the average state of the qubit is obtained using a quantum measurement device. This third measurement is obtained in a similar manner to the first and second measurements, but using a modified quantum computer. The expected value of the qubit's state is calculated by averaging the measurements recorded in each repeated third operation. The third measurement provides a third noise measurement of the observable with a different error rate.

[0056] Once the third measurement has been obtained, then as described above (S103, S106), in S109, the error rate of the quantum computer is modified from the third error rate to the fourth error rate. In this example, the fourth error rate is greater than the third error rate.

[0057] In step S110, a fourth operation is performed on the state of the qubit. The fourth operation is performed multiple times, and the state of the qubit is measured after each execution. The fourth operation has a fourth error rate. In this example, the fourth operation includes the first operation and first, second, and third modification operations, which are randomly selected from the Pauli set of operations described above (S104). In this example, the fourth error rate is greater than the third error rate. The first, second, and third modification operations can be performed in any order and are independent of any specific operation. Before each execution of the fourth operation, the first, second, and third modification operations are randomly selected.

[0058] In step S111, a fourth measurement of the average state of the qubit is obtained using a quantum measurement device. This fourth measurement is obtained in a similar manner to the first, second, and third measurements, but using a modified quantum computer. The expected value of the qubit's state is calculated by averaging the measurements recorded for each repeated fourth operation. The fourth measurement provides a fourth noise measurement of the observable with a different error rate.

[0059] In step S112, the first, second, third, and fourth measurements are fitted to a multi-exponential decay curve, which is in the form of... E is the average state of the qubit, n is the error rate, and A is the error rate. k and γ k These are the fitting parameters. Fitting parameter γ k Greater than or equal to 0 and less than or equal to 1, i.e., 0 ≤ γ k ≤1. The form of this decay curve is based on the assumption that quantum circuits are affected by general Pauli noise (i.e., random Pauli error). For example, Pauli noise can occur due to environmental interactions and incomplete qubit control. The fitting parameters are determined using a classical processor in a conventional computer.

[0060] It has been found that including multiple exponential components in the decay curve can improve the estimation of error-free observables. However, the use of additional exponents in the summation has a cost, C. EYes, because additional measurements are needed to reduce the uncertainty in the calculation of the expected value. It has been found that the use of multiple exponential components provides a balance between improved estimation of observability and overfitting, which can occur when the decay curve is fitted to noise beyond the underlying trend. Without sufficient measurements to adequately determine the fitting parameters A... k and γ k Overfitting may occur if the minimum number of measurements required for a good fit is equal to the number of free parameters.

[0061] In this embodiment, K=2, and the multi-exponential decay curve is the sum of the two exponents, i.e. There are four free parameters: A1, A2, γ1, and γ2, therefore, the minimum number of measurements required for a good fit is four. Although the cost (i.e., the number of runs required when fitting the data using a double exponential decay curve) is slightly higher than K=1, it significantly improves the absolute uncertainty in the estimation of observables.

[0062] Finally, in step S113, the average state of the qubits is extrapolated using the fitted curve with a fifth error rate. The extrapolation is performed using a classical processor. The fifth error rate is lower than each of the first, second, third, and fourth error rates. In this embodiment, the first error rate is the lowest experimentally achievable error rate using the quantum device. The first, second, third, and fourth error rates should be different to perform the fitting, but none of them need to be the minimum achievable error rate. In this embodiment, the fifth error rate is chosen to be zero, where n = 0. At the zero error rate, the estimated expectation is equal to the sum of two defined amplitudes, i.e., E. n＝0 =A1+A2. This extrapolation technique returns an estimate of the observable quantity that is error-free or noise-free.

[0063] In an alternative embodiment, the estimation of error-free observables can be further improved by increasing the number of measurements taken, where each measurement is at a different error rate. The quantum device can be modified before performing additional operations, each modification resulting in a new error rate. The state of the qubits is initialized before performing the i-th operation, where i > 4. There is no theoretical upper limit on i; the larger the value of i, the better the error estimation. However, each additional measurement requires additional experimental costs, so the total number of measurements obtained is typically finite for practical reasons. The same quantum device is used to perform each i-th operation, with different modifications to the hardware to modify the error rate. Each i-th operation has an i-th error rate. In one example, each i-th operation includes a first operation and a modification operation, where the modification operation can include one or more operations such as Pauli gates.

[0064] Each i-th operation is performed in a manner similar to the second, third, and fourth operations described above with respect to steps S104, S107, and S110. After the i-th operation has been performed, the i-th measurement of the average state of the qubit is obtained, similar to the acquisition of the second, third, and fourth measurements described with respect to steps S105, S108, and S111. In the described alternative embodiment, each obtained measurement is fitted to the multi-exponential decay curve described with respect to step S112.

[0065] Figure 2A This is a schematic diagram of the first operation 21 according to the second embodiment. The first operation 21 is performed on the state of the qubit by a quantum processor. Subsequently, a quantum measurement device 22 obtains a measurement of the state of the qubit. The quantum processor is configured to perform the first operation 21 multiple times, each time using the quantum measurement device 22 to measure the state of the qubit. The expected value of the measurement is determined by averaging these measurements.

[0066] Figure 2B This is a schematic diagram of the second operation according to the second embodiment. In this embodiment, the second operation 23 includes a first operation 21 and a modification operation 25. After the quantum processor performs the second operation 23 on the state of the qubit, the quantum measurement device 26 obtains a measurement of the state of the qubit. The second operation 23 is executed multiple times, each time using the quantum measurement device 26 to measure the state of the qubit. The modification operation 25 is used to modify the quantum operation by adding a certain level of noise. The addition of the modification operation 25 increases the probability of error occurring. The modification operation 25 is selected from a set of operations that contribute to the noise model of the first operation 21. The expected value of the measurement taken by the measurement device 26 in this way under the condition of increased noise is determined by averaging the measurements taken by the measurement device 26 after the execution of the second operation 23.

[0067] In an alternative embodiment of performing the i-th operation, Figure 2B The diagram of the second operation can generally be interpreted as a schematic depiction of the i-th operation. Each i-th operation may include more than one modification operation in addition to the first operation.

[0068] Figure 3 This is a schematic diagram of the fitting and extrapolation process according to the third embodiment. A classical computer processor is used to perform the fitting and extrapolation. The method described above is used to obtain a first measurement 31 at a first error rate of 32 and a second measurement 33 at a second error rate of 34. Third and fourth measurements (not shown) are also obtained. In this embodiment, when the first error rate 32 is lower than the second error rate 34, the first measurement 31 is greater than the second measurement 33.

[0069] The multi-exponential decay curve 35 is fitted to the first measurement 31, the second measurement 33, and the third and fourth measurements using a classic processor. The multi-exponential decay curve 35 takes the form of: After determining the fitting parameters A1, A2, γ1, and γ2, the curve is extrapolated to a zero error rate of 37, n = 0, using a classical processor. The error-free value of the observable 36 is estimated by extrapolating to a zero error rate.

[0070] In an alternative embodiment, further measurements can be performed at an additional error rate greater than the minimum circuit error rate to improve the estimation of the fitted parameters in the double exponential decay curve. At least four distinct error rates should be probed to provide a good estimate of the fitted parameters, thereby providing a good estimate of the error-free observable. While more measurements are required when fitting measurements to a double exponential decay curve instead of a single exponential decay curve, the improved fit results in an estimation of the error-free observable with significantly lower estimation error.

[0071] As will be understood, an improved error reduction method is provided, which significantly improves the estimation of error-free observables. By increasing the error rate of the quantum computer and sampling at different error rates by fitting the resulting measurements to a multi-exponential decay curve, extrapolation can be used to estimate the error-free value of the observable.

Claims

1. A method for reducing errors when using a quantum computer, comprising: Perform the first operation on the state of the qubit multiple times; The first operation has a first error rate; A first measurement to obtain the average state of the qubit; The error rate of the quantum computer is modified from the first error rate to the second error rate; The second operation is performed multiple times on the state of the qubit; wherein the second operation has the second error rate; A second measurement to obtain the average state of the qubit; The error rate of the quantum computer is modified from the second error rate to the third error rate; The third operation is performed multiple times on the state of the qubit; wherein the third operation has the third error rate; A third measurement to obtain the average state of the qubit; The error rate of the quantum computer is modified from the third error rate to the fourth error rate; The fourth operation is performed multiple times on the state of the qubit; wherein the fourth operation has the fourth error rate; A fourth measurement to obtain the average state of the qubit; The first measurement, the second measurement, the third measurement, and the fourth measurement are fitted to a multi-exponential decay curve; as well as The fitted curve is used to extrapolate the average state of the qubit at the fifth error rate, wherein the fifth error rate is lower than the first error rate, the second error rate, the third error rate, and the fourth error rate; Each of the second operation, the third operation, and the fourth operation includes the first operation and at least one modification operation, wherein the at least one modification operation is selected based on the predicted noise model.

2. The method for reducing errors according to claim 1, wherein, The multi-exponential decay curve is the sum of two or more exponential curves, and the multi-exponential decay curve has the following form: ,in E The average state of the qubit. n For the error rate, and and These are the fitting parameters.

3. The method for reducing errors according to claim 2, wherein, K =2。 4. The method for reducing errors according to claim 2 or 3, wherein, 。 5. The method for reducing errors according to claim 1, wherein, The second error rate is higher than the first error rate.

6. The method for reducing errors according to claim 1; wherein, The at least one modification operation includes the Pauli operation.

7. The method for reducing errors according to claim 1, wherein, Executing the first operation multiple times includes executing multiple first operations, wherein executing the second operation multiple times includes executing multiple first operations and multiple modification operations.

8. The method for reducing errors according to claim 7, wherein, Performing the second operation multiple times includes performing one of the plurality of modification operations after each of the plurality of first operations.

9. The method for reducing errors according to claim 1, further comprising: Initialize the state of the qubit.

10. The method for reducing errors according to claim 9, wherein, The state of the qubit is initialized before the first operation, the second operation, the third operation, and the fourth operation are performed for the first time.

11. The method for reducing errors according to claim 1, further comprising: The error rate of the quantum computer is modified to the i-th error rate; Perform the i-th operation on the state of the qubit; The i-th operation has the i-th error rate; and Wherein the i-th error rate is greater than the first error rate; Obtain the i-th measurement of the average state of the qubit; as well as The first measurement, the second measurement, the third measurement, the fourth measurement, and the i-th measurement are fitted to the multi-exponential decay curve.

12. A device for performing quantum computing calculations, comprising: A quantum processor, wherein the quantum processor is configured to: A first operation is performed on the state of a qubit multiple times, wherein the first operation has a first error rate; A second operation is performed on the state of the qubit multiple times, wherein the second operation has a second error rate; A third operation is performed multiple times on the state of the qubit, wherein the third operation has a third error rate; and A fourth operation is performed on the state of the qubit multiple times, wherein the fourth operation has a fourth error rate; Each of the second operation, the third operation, and the fourth operation includes the first operation and at least one modification operation, wherein the at least one modification operation is selected based on the predicted noise model; A quantum measurement gate, wherein the quantum measurement gate is configured as follows: A first measurement is taken to obtain the average state of the qubits after the first operation; A second measurement is obtained to obtain the average state of the qubit after the second operation; A third measurement of the average state of the qubit after the third operation; and A fourth measurement is taken to obtain the average state of the qubit after the fourth operation; and A classical processor configured to fit the first, second, third, and fourth measurements to a multi-exponential decay curve, and to use the fitted curve to extrapolate the average state of the qubit at a fifth error rate, wherein the fifth error rate is lower than the first, second, third, and fourth error rates.

13. A computer-readable storage medium comprising instructions that, when executed by a computer, cause the computer to perform the following steps on a quantum computer: A first operation is performed on the state of the qubit multiple times; wherein the first operation has a first error rate; A first measurement to obtain the average state of the qubit; The error rate of the quantum computer is modified from the first error rate to the second error rate; The second operation is performed multiple times on the state of the qubit; wherein the second operation has the second error rate; A second measurement to obtain the average state of the qubit; The error rate of the quantum computer is modified from the second error rate to the third error rate; The third operation is performed multiple times on the state of the qubit; wherein the third operation has the third error rate; A third measurement to obtain the average state of the qubit; The error rate of the quantum computer is modified from the third error rate to the fourth error rate; The fourth operation is performed multiple times on the state of the qubit; wherein the fourth operation has the fourth error rate; A fourth measurement to obtain the average state of the qubit; The first measurement, the second measurement, the third measurement, and the fourth measurement are fitted to a multi-exponential decay curve; as well as The fitted curve is used to extrapolate the average state of the qubit at the fifth error rate, wherein the fifth error rate is lower than the first error rate, the second error rate, the third error rate, and the fourth error rate; Each of the second operation, the third operation, and the fourth operation includes the first operation and at least one modification operation, wherein the at least one modification operation is selected based on the predicted noise model.

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