Controlling noise in quantum systems is crucial for the development of practical technologies. Such noise can occur due to unwanted interactions of a passive qubit with the environment, or due to imperfections in the use of circuit elements that compose the algorithm (qubit initialization, gates, and measurement). In all cases the result is errors occurring at the level of physical qubits. The theory of quantum fault tolerance (QFT) reveals that the introduction of logical qubits, composed of numerous physical qubits, can allow one to detect and correct errors at the physical level; however, this capacity comes at an enormous multiplicative cost in resources. A recent estimate suggests that a Shor algorithm operating on a few thousand logical qubits would require several million physical qubits [1]. While it is encouraging to know that such techniques exist, hardware on this scale is probably at least a decade away. The timely (indeed, urgent) question is, to what extent can we control the impact of errors in computing devices that are too small to support full QFT?

It may prove to be the case that deep quantum algorithms, such as Shor’s factoring algorithm and Grover’s search algorithm, cannot be successfully executed on classically intractable problems without the support of QFT. However, fortunately there are other algorithms of potential practical significance that focus on shallow circuits, with the output typically being fed into a classical supervising algorithm so as to form a hybrid system. Such approaches have been proposed for the simulation to aid discovery in chemistry and materials science; see Refs. [2–11] for examples. Hybrid systems may be capable of yielding significant results, surpassing conventional computers, even when finite error rates are present because of their error resilience [12,13]. In order to achieve this it is desirable to suppress or mitigate errors to the greatest extent possible while keeping the qubit count ideally unchanged, or increasing only modestly compared to the high cost of full QFT.

Recently two techniques were introduced for quantum error mitigation (QEM) in generic hybrid quantum algorithms where the expected value of an observable—say, a -basis measurement of a given qubit—is the quantity of interest. The goal is to estimate the value that this observable would take given an error-free circuit, despite the reality that the real experimental system cannot perform operations with less than a certain error rate. Reference [2] introduced a hybrid algorithm simulating quantum dynamics, which featured an active error minimization technique involving extrapolation. The experimentalist would execute the circuit with all errors at their minimum achievable severity, obtain the expected value of the observable, and then repeat the exercise having deliberately increased the physical error rate (or having applied additional quantum gates to achieve the same effect). By noting the effect of the increased errors on the observable, the experimentalist would be able to make an extrapolated estimate of the zero-error value, presuming that the error sources had scaled proportionately. The technique was found to be very advantageous in the numerical simulations of few-qubit experiments presented in that paper (see, e.g., Fig. 5 in Ref. [2]).

A paper that appeared online at almost the same time was Ref. [14] by the IBM-based team of Temme, Bravyi, and Gambetta. That Letter presented a comprehensive analysis of the extrapolation technique, which the authors had independently conceived, and moreover, it introduced a second technique with using (what we will call) a “quasiprobability” formalism. The authors explained that by replacing operations in the quantum circuit and assigning parity to each operation following a certain probability distribution dependent on the noise, an experimentalist can obtain the unbiased estimator, at the cost of an increase in the variance. Their method was shown to be applicable to specific noise types, including homogeneous depolarizing errors and damping errors. The authors found both methods to be promising in few-qubit numerical simulations (see, e.g., Fig. 2 in Ref. [14]).

As exciting as these studies were, open questions remained to be answered before these two techniques could be considered to be fully practical. First, both techniques rely on the full knowledge of the error model, whereas an experimentalist will have imperfect knowledge and the real noise will generally differ from the canonical types considered in these first papers. Second, we need an explicit method to derive the QEM circuits, i.e., a specification of how to algorithmically increase the error rate in the error extrapolation or how to sample circuits in the quasiprobability decomposition. In this paper, we solve these two problems. We find that gate set tomography (GST) [15,16] provides sufficient information to enable full elimination of the impact of localized Markovian errors. As with other process tomography protocols, GST cannot determine the exact physical error model due to noise associated with state preparation and measurement. However, we determine that preparation and measurement noise in GST is not harmful to the overall QEM approach. We also find that single-qubit Clifford gates and measurements are universal in computing expected values. Each quantum operation is a linear map, and single-qubit Clifford gates and measurements yield a complete set of linearly independent maps (quantum operations). Therefore, any error can be simulated or subtracted by decomposing the error using the complete operation set, which is the standard linear decomposition. We prove that, by combining GST and the complete set decomposition, any localized and Markovian errors in the quantum computer can be systemically mitigated, so that the error in the final computational output is only due to unbiased statistical fluctuation.

For the quasiprobability method, we provide an upper bound of the cost in QEM, and we describe the utility of “twirling” operations [17–19] in minimizing this cost. For the extrapolation method, which is a relatively straightforward technique, our optimization is to observe that typically for the classes of noise most common in experiments it is appropriate to assume that the expected value of the observable will decay exponentially with the severity of the circuit noise. Adopting this underlying assumption, rather than a polynomial (e.g., linear) fit, proves to be quite advantageous.

Having thus optimized both the quasiprobability and the extrapolation techniques, we make a series of numerical simulations to study their efficacy. We opt for a specific circuit, a realization of the “swap test” that is often employed in quantum algorithms as a means for estimating the similarity of quantum states [20,21]. Our swap test operates on qubits, and we simulate a total of 15 qubits over a comprehensive set of cases as well as 19 qubits for specific cases. We numerically simulate the actions of the experimentalist, who must perform many circuit trials in order to make a single estimate of the observable (we choose trials). But in order to evaluate our QEM techniques we must then repeat this entire process to determine the distribution of values that the experimentalist might obtain. We perform at least repetitions so that the distribution becomes clear; thus, at least individual numerical experiments are performed for each of the curves that we presently report.

In this paper, we focus on computing the expected value of an observable in a state (the final state of a quantum circuit) using a quantum computer. It is typical of a number of quantum algorithms and subroutines that the desired output is the expected value of a qubit or qubits—the swap test [20,21] itself, which is a component of algorithms including the recently introduced autoencoder [22], and several proposed hybrid algorithms for simulating chemical or materials systems [2–8].

Without using QEM as shown in Fig. 1(a), the quantum circuit is repeated for many times, and the measurement outcome of each time is collected. Then, we can calculate the average as our best estimate of the expected value. Given that the number of repetitions is finite, the value of is a random variable with an associated distribution. Because the implementation of the quantum circuit is imperfect, it is likely that the distribution of is not even centered at the ideal value, i.e., the exact expected value when the quantum circuit is perfectly implemented without error.

FIG. 1.

When we use QEM as shown in Fig. 1(b), instead of the original quantum circuit, we implement a set of modified circuits. The scheme depicted in the figure is relevant to the quasiprobability method for QEM, but can also apply to the extrapolation method as a means to deliberately boost errors. Each modified circuit is determined by a set of random numbers . The distribution of random numbers, i.e., modified circuits, depends on the error model, which is measured using GST before the quantum computing. In each run of the quantum experiment, firstly the random number set is generated, then depending on a specific circuit is implemented, and finally the measurement outcome is collected. Rather than calculating the average , we use both and to calculate the average of an effective outcome , which is given explicitly later. If QEM is successful, the distribution of is centered at the ideal value, but the distribution is wider than because of the factor , which is greater than 1 and can be efficiently computed. Thus, only error due to the statistical fluctuation remains, although it is amplified. By repeating the quantum experiment enough times, we can obtain an accurate computing result of the expected value.

In Sec. IV, we explicitly give the effective outcome and the factor . Modified circuits and their distribution are given in Sec. VIII.

We use the notation commonly used in quantum tomography (e.g., in Refs. [15,16]).

In quantum theory, a quantum state is usually represented by a density matrix , and an observable is represented by a Hermitian operator . The expected value of the observable quantity in the state is . An operation is a map on the space of states, , expressed in the Kraus form.

Because an operation is a linear map, we can always express the operation as a square matrix, e.g., using the Pauli transfer matrix representation, acting on the state expressed as a column vector . Similarly, an observable can be expressed as a row vector , and the expected value is . Throughout this paper, we use the Pauli transfer matrix representation; see Appendix A for details. In quantum tomography, usually we focus on observables that are positive-operator valued measure (POVM) operators, which is not necessary here.

In the Pauli transfer matrix representation, vectors representing states or observables and matrices representing operations are all real. For qubits, vectors and matrices are -dimensional. The expected value of the observable in the state going through a sequence of operations reads as follows:

We suppose that the initial state is , which goes through a sequence of operations , and in the final state the observable is measured. Each time the experimentalist implements this circuit, the measurement returns an eigenvalue of , and the probability distribution of eigenstates is determined by the final state. By repeating such a circuit for many times, we can estimate the expected value , where , and is the measurement outcome. Generally in this paper we use the superscript 0 to denote the ideal noise-free realization of a state, operation, or observable quantity.

In the case that the quantum computation has errors, the actual initial state is , actual operations are , and the actually measured observable is . As a result, the estimation of the expected value converges to rather than . Here, , and we have assumed that errors are Markovian.

The central idea introduced by the IBM team in Ref. [14] is that one can exactly compensate for the effect of errors by sampling from a set of (real, error-burdened) circuits, each labeled for , provided that their outputs satisfy

Reference [14] describes how the real numbers , which represent quasiprobabilities, can be efficiently derived given specific error models, assuming that the experimentalist has full knowledge of the model. Note that each denotes the total operation composed by a sequence of operations in the th circuit.

We can use the Monte Carlo method to compute . We note that , where is the measurement outcome in the th circuit. Then . To compute , we randomly choose a circuit to implement, and the th circuit is chosen with the probability , where . Then, the computing result is given by the expected value of effective measurement outcomes, i.e., , where the effective outcome is if the th circuit is chosen to be implemented, and is the outcome directly obtained in the th circuit.

We can correct errors in each operation using the quasiprobability method, which is the primary focus for the following several sections. We suppose that we have a set of initial states satisfying , and a set of operations satisfying for each error-free operation , and a set of observables satisfying . Then, computing with error mitigation can be expressed as

When we sample circuits to compute , the initial state is with probability , the th operation is with probability , and the observable is with probability . Here, , and accordingly. To calculate , we use .

The presence of quasiprobabilities taking negative values amplifies the variance of the expected value of the observable. We consider the case that is a Pauli operator (maybe with error) and the measurement reports two kinds of outcomes denoted by , respectively. In this case, the distribution is binomial. The standard deviation of the average of outcomes in the Monte Carlo calculation is . Here, is the total number of samples; i.e., the total number of circuits of all kinds which the experimentalist performs is . We compare this to the error-free computing; i.e., the ideal original circuit is repeated for times to estimate . For the error-free computing, the standard deviation is given by . Therefore, to achieve the same accuracy, i.e., , the error-mitigated computation needs times more samples than the error-free computation. Here, we have used the fact that the error-mitigated computation and the error-free computation should converge to the same value of ; i.e., .

In order to limit the standard deviation to be , we can choose . Therefore, if the factor is larger, the computing takes longer.

Because if errors are corrected for each operation, we call the cost for mitigating error in the corresponding operation. The overall cost therefore increases with the number of operations; thus it is important to reduce the operation number. For example, in a quantum computer with qubits fully connected [23], operations for communication are not required, which may significantly reduce the cost.

The set of operations including measurement and single-qubit Clifford gates is universal in computing expected values of observables. The relevant measurement operation reads , which projects a qubit to the state . Here, denotes a superoperator. Such a nondestructive measurement can be realized using a destructive measurement followed by initializing the qubit in the state . Single-qubit Clifford gates include the Hadamard gate , the phase gate , and all other single-qubit Clifford gates can be derived from these two.

The measurement superoperator [ ] also means postselection; i.e., if the outcome of the measurement corresponding to [ ] (which is not the final measurement on the observable ) is in a trial, the value of the observable is noted as , but the trial is counted in the total number of samples in the Monte Carlo calculation. If has two values , we can estimate the value of by calculating . Here, we have supposed that the circuit is implemented for total times; for times, the circuit does not pass postselections (i.e., ), and for times, the circuit passes all postselections and reports (i.e., ). It is the same when we compute using the Monte Carlo method. If the effect outcome is with the probability , then the standard deviation of the Monte Carlo calculation becomes .

In Table I, we list 16 linearly independent operations that can be derived from the minimum universal operation set . In the following, we use to denote these 16 operations. Because they are linearly independent, any single-qubit operation , which is a real matrix, can be expressed as a linear combination of 16 basis operations; i.e., . Similarly, multiqubit operations can be expressed as a linear combination of tensor products of basis operations. Using the quasiprobability method, any computation of expected values of observables can be realized using this operation set.

Note that these basis operations are universal, as one can verify by constructing a non-Clifford gate or an entangling gate: We can decompose the gate using our basis operations as (see Appendix B for controlled-not as a second example). However, this construction would not be used in practice—it is not an efficient means to actually implement a desired in the basic circuit, since the corresponding cost would imply an unacceptably steep exponential in the time overhead, as one would expect from, e.g., Refs. [24–26]. Instead we rely on the assumption that the experimental system can directly implement a universal set of gates (including entangling and non-Clifford gates) with a reasonably high fidelity. Then, rather than fully synthesizing any of the basic gates using our basis, we need only compensate for slight imperfections. The cost for doing so, for each imperfect gate, is then , as we presently discuss.

Having obtained the complete operation set in Table I we can use it in deriving the protocol that will compensate for errors. In this paper, we focus on the case that errors are localized: An (error-free) operation that is applied on a set of qubits is a -dimensional real matrix, then the corresponding operation in real (i.e., error-burdened) is also a -dimensional real matrix acting on the same set of qubits. The overall operation on the entire system can be expressed as , where is the identity acting on all other qubits. It is similar for the initialization and measurement. If each qubit is initialized individually, the overall initial state is , where is a two-dimensional real vector representing the th qubit’s initial state. Similarly, individual measurement of qubits implies that the overall measured observable is , where is a two-dimensional real vector representing the measured observable for the th qubit. In this case, a single-qubit operation with error can still be expressed using a real matrix. We suppose that for a qubit, 16 basis operations with errors are , which are all real matrices. When errors are not significant, these 16 bases should still be linearly independent; i.e., the set of basis operations with errors is still universal.

To make this statement more precise, we consider the real matrix

Here, denotes the th column of the matrix of the basis operation . Sixteen basis operations are linearly independent if the matrix is invertible. We use as the measure of the error severity in basis operations. When , is always invertible (see Appendix C). We remark that even if exceeds the threshold, basis operations are still likely to be linearly independent.

Given an operation with error , we can use 16 basis operations to correct the error, i.e., realize the operation without error . There are two ways for correcting the error.

Compensation method.—The operation is close to . Therefore, we can keep the correct component of and only decompose the error component using basis operations. We decompose the operation without error as , where is an arbitrary real number. If basis operations are linearly independent, the decomposition always exists, and there is only one solution of coefficients when is determined.

Inverse method.—If the matrix is invertible, we can express as followed by a noise operation, i.e., , where the noise operation . In order to correct the error, we can decompose the inverse of the noise as . By applying the inverse of the noise after the operation , we can realize the operation without error, i.e., . Similar to the compensation method, if basis operations are linearly independent, the decomposition always exists, and there is only one solution of coefficients . However, the inverse method can be applied only if the matrix is invertible.

For multiqubit operations, the decomposition is performed using tensor products of basis operations, as described explicitly in Appendix D. Although basis operations are not entangling, we can use basis operations to efficiently mitigate multiqubit errors and errors that can entangle qubits. As an example, we show how to decompose the controlled-not gate only using basis operations in Appendix B, which suffices to imply that any error in the form of the controlled-not gate can be mitigated using basis operations.

Initialization and measurement errors can also be corrected using basis operations. Taking first the case of initialization errors: If is the error-burdened initial state, and it is a nonzero vector, we can always find a transformation that satisfies , where is the error-free initial state. Thus, by decomposing using basis operations and applying it after the initialization, we can prepare the initial state without error. Actually, given an initial state that is close to , we can generate a complete set of linearly independent vectors using basis operations. With these vectors, we can decompose the initial state without error as .

A similar approach yields the corresponding result for measurement: For an observable there will be some where is the error-free quantity. If an observable is close to , then a linearly independent set can be generated; then the error-free observable .

Circuits for QEM are shown in Fig. 2(a). Given quasiprobabilities, we can compute the corresponding probability in sampling circuits as shown in Sec. V. More details of QEM using basis operations are given in Appendix D.

FIG. 2.

Using the same technique, we can also increase the error in an operation, as required by the alternative error extrapolation method for QEM. Instead of decomposing the error-free operation using and basis operations, we can also decompose the error-boosted operation ( ) using and basis operations. It is similar for initial states and observables. We have noted that in the decomposition of an error-free operation, there are always some negative quasiprobabilities, i.e., the factor is greater than 1, which leads to greater time costs. But fortunately, when we merely wish to decompose an error-boosted operation we can do so without introducing negative quasiprobability, e.g., by boosting Pauli errors using Pauli gates [2].

We can measure a set of initial states , observables , and operations (including basis operations) using GST [15,16]. These vectors and matrices with the bar notation describe the actual physical system. Because there are errors in both initial states and observables, and initialization and measurement errors cannot be distinguished, we may not obtain exactly these vectors and matrices describing the actual physical system. Instead, the vectors and matrices obtained using GST are , , and , which are estimations of , , and , respectively.

If we know , , and because the physical system is well understood, we can directly use them in QEM. If our knowledge about the physical system is not enough, we can use GST to obtain , , and . We show that, although the estimations may not be exact, we can exactly correct errors by using these estimations in QEM.

Using the protocol in Refs. [15,16] (also see Appendix E), the estimation of an operation and the actual physical operation are similar matrices, i.e., , where is a matrix determined by initial states (i.e., ), and is an arbitrary invertible matrix. We note that and are independent of the operation , and cannot be determined by GST. By choosing , we can obtain different estimations of the operation set. Similarly, and .

All operations are transformed by the same similarity transformation, and initial states and observables are also transformed accordingly. As a result, these estimations obtained by GST can exactly predict the expected value of an observable; i.e., . Therefore, we can directly use these estimations in QEM, and the similarity transformation does not lead to any computing error.

Using GST estimations in QEM, the actual operations realized in this way differ from operations without error, but the computing result is correct. To correctly obtain , we decompose the initial state, the observable, and the operation using , , and , respectively. Here, , , , and GST estimations are all known to us. The decompositions are , , and . Accordingly, we actually realize , , and in the physical system. We have , , and . Therefore, the physical system gives the computing result , i.e., the desired error-free output. The cost of this adaption lies in the potential increase to the number of samples required, as shown in Fig. 3 and discussed in the caption.

FIG. 3.

We remark that, when errors in actual operations are small, errors in estimations of operations are also small. If we take a proper strategy for choosing , and errors in initial states and observables are small, the estimation of an operation is close to the operation without error when the actual operation is close to . It is similar for estimations of initial states and observables. See Appendix F for details.

In general, when the error in an operation is more significant, there is a higher cost for mitigating the error (to a given level of suppression). We take as the measure of the error severity in the operation, where ( ) is the -qubit operation with (without) error. An upper bound of the cost for correcting error in is

Here, is the maximum error in all basis operations for all qubits, and . Similar upper bounds can be obtained for correcting errors in initial states and observables. See Appendix G for details.

There are several ways for reducing the cost. The upper bound of the cost is obtained using the compensation method and taking . In general, we can optimize the value of or use the inverse method to minimize the cost. For example, for the depolarizing error model (see Appendix H), the cost of using the inverse method is lower than using the compensation method [see Fig. 3(a)]. We remark that, to obtain data for the compensation method in Fig. 3, we have optimized the value of . If we use estimations obtained from GST to correct errors, we can optimize the matrix to minimize the cost. In Fig. 3(a), we can find that, without optimizing matrices, the cost using estimations obtained from GST is higher than using actual operations. If we choose the matrix in the form , where is a four-dimensional real matrix corresponding to the th qubit, there are total parameters to be optimized for a -qubit quantum computer, which is a nontrivial task. Under some reasonable conditions, we can also use the Pauli twirling [17–19] to reduce the cost.

In our numerical simulation, we apply QEM to the swap-test circuit [20] shown in Fig. 4, in which we realize each controlled-swap gate using Toffoli gates and realize each Toffoli gate using gates, gates, Hadamard gates, and controlled-not gates [31]. We note with interest that very recently the implementation of a swap test using shallow circuit has been proposed [21]. However, for present purposes, it is not essential to use an optimized realization of the swap circuit; its role is simply to act as a real test case for our technique, and indeed the considerable depth of our nonoptimal circuit is helpful here. The number of gates scales as , where is the number of qubits (e.g., in Fig. 4). Without error, the expected value of the observable ( of the probe qubit) in the swap-test circuit in Fig. 4 is 0.5.

FIG. 4.

We consider error models according to which the same noise is applied after the initialization to the state , before the measurement, and before and after each gate. For the controlled-not gate, the noise applied is on two qubits. We remark that basis operations are also affected by noise likewise. We consider two types of noise: inhomogeneous Pauli error and leakage error, which can be, respectively, described as

and

where is the probability of the error , and is the probability of the leakage error from the state . It is worth mentioning that the leakage error is a non-trace-preserving error. In our simulations, we set , , and . Thus, in both models the total error rate is 0.08% for initialization and measurement, 0.16% for single-qubit gates, and 0.32% for two-qubit gates, which is achievable with two-qubit gates in ion traps [29] and can be far surpassed for one-qubit gates [28]. Moreover, with these numbers the expected total number of error events in circuits of the depth and breadth that we consider here is approximately unity; this is a challenging domain for error mitigation.

In addition to quasiprobability decomposition (see Appendix I for an instruction of the implementation), we also study the extrapolation technique introduced in Ref. [2]. The expected value of obtained by running the swap-test circuit in a quantum computer with noise depends on the error rate; i.e., it is a function that can be denoted as , where is the overall error rate. For our first set of numerical experiments we consider linear extrapolation to the error-free value as follows: We obtain the expected value with the lowest attainable error rate , and by increasing error rate to with , we obtain another expected value . Using these two values, we can infer , as shown in Fig. 2(b), which is the final estimation of . Here, we set .

The first set of numerical results is shown in Fig. 5. We assume that the experimentalist makes their overall estimate of the after they perform individual experiments. We take this number of runs as a fixed constraint (effectively, we are constraining their overall time resource), and they may choose to employ those runs using one of three alternative approaches: no error correction, linear extrapolation, and quasiprobability decomposition (using basis operations and incorporating GST). In each experiment the swap-test circuit or its variant for the purpose of QEM is implemented. Because of the finite number of samples, the estimation is stochastic. Therefore, in our numerical simulation we perform the appropriate series of experiments, mirroring the actions of the experimentalist, and then we repeat times in order to determine the distribution of final estimations that may be obtained. The distribution for each case is plotted in Fig. 5.

FIG. 5.

We can observe that both QEM approaches can improve the result; i.e., the corresponding distributions are shifted closer to the ideal value 0.5 compared to the approach without QEM. For the inhomogeneous Pauli error model, the means of distributions are at 0.1961, 0.3415, and 0.5011 for the three approaches, respectively. The distribution of the quasiprobability approach is centered at the ideal value, which clearly shows its desirable property of completely removing any systematic bias. However, the distribution is wider (as we expected) compared to the other two approaches. A fairer metric would be the expected absolute error versus ideal value (i.e., ). Given an ideal error-free computer and trials, this metric would evaluate to 0.006910. Using the error-prone computer with our three protocols the three corresponding values are 0.3039, 0.1853, and 0.0491. Similarly, for the leakage error model, the means for three approaches now lie at 0.3819, 0.4710, and 0.5007, while the expected absolute error evaluates to 0.1181, 0.0294, and 0.0434.

From these results it may appear that (given a large but reasonable number of samples) the quasiprobability technique outperforms the extrapolation method, with the latter unable to approach the mean of the error-free circuit. However, here the extrapolation method was limited to linear interpolation whereas the physical error rates are high enough that the linear assumption is poor. One could fit a higher-order polynomial using more data points (here, we have only used two: one derived from the actual lowest possible error rate and one boosted to twice the error rate); however, since we are limiting the total number of experimental runs to this would lead to greater noise in each data point. Moreover, as we now argue, the underlying trend is likely to be well approximated by an exponential decay rather than a polynomial one (i.e., the expected value of the observable falls exponentially with the physical error rate) and two data points will suffice to estimate the zero-error observable under that assumption.

In Fig. 6, we show the results when the experimentalist indeed assumes that the expected value changes exponentially with respect to the error rate and converges to 0 in the limit of . Then they will infer the error-free value as

Here we take .

FIG. 6.

As shown in Fig. 6, the distribution of the final result using the exponential extrapolation approaches the ideal value of (which is 0.5 for the swap-test circuit) much better than the linear extrapolation. Given the same experimental runs, the mean of the experimentalist’s estimate is now 0.5111 for the inhomogeneous Pauli error model and 0.4986 for the leakage error model. These numbers almost rival those of the quasiprobability technique but do so with a smaller variance. The expected absolute error for inhomogeneous Pauli error and leakage error are 0.065 01 and 0.018 82, respectively. For the latter, the expected absolute error comes within a factor of 3 of the shot-noise limit that would be achieved by error-free ideal hardware (0.00691). This is despite the fact that our error-burdened circuits have error rates corresponding to at least one error event per circuit. We emphasize that this suppression results purely from the QEM protocol; i.e., it is achieved at no cost in terms of the qubit count or the total number of runs (constrained to ).

Because of the limited power of the classical computer we utilized, our exact numerical simulations did not involve more than 19 qubits. However, it is of course very interesting to assess the relevance of our techniques to quantum computing using over 50 qubits, which is in the so-called “quantum supremacy” regime. Therefore, we estimate the cost of quantum error mitigation in the swap-test circuit, using the same error models in our numerical simulation and error rates achievable in ion trap experiments [28,29]; i.e., the error rate of the two-qubit gate is 0.1% and error rates of single-qubit operations are 0.01%. Take for example the swap test with qubits (the number of gates is 1152). For the inhomogeneous Pauli error model, the overall cost is , which implies that we can attain the same computing precision as the ideal case if we have times more repetitions of the experiment, which is experimentally feasible. For the leakage error model, the cost for the 51-qubit swap test is , which means times more repetitions. A plot showing how the cost scales versus qubit count is shown in Fig. 7.

We also evaluate for a fully paralleled circuit, whose circuit depth is , and each layer has single-qubit gates and controlled-not gates, which means the quantum circuit has single-qubit gates and controlled-not gates. As single-qubit gates, we use the gate, gate, and Hadamard gate, because these gates plus a controlled-not gate constitute a universal gate set, and we equally assign the number of qubits to these three types of single-qubit gates. We plot versus the number of qubits for the swap-test circuit and the fully paralleled circuit in Fig. 7. We observe that for the swap-test circuit, it is feasible to venture into the supremacy regime with today’s best fidelities; for the more demanding case of full parallelism (so that the gate count scales as ), we see that today’s error rates would not suffice much beyond 50 qubits, but that error rates 10 times lower would easily suffice for 80 qubits and beyond.

FIG. 7.

Intuitively, the explanation for the success of the exponential extrapolation is as follows. We express the th noise event occurring in the quantum circuit as

where is the error component. The only assumption is that the error component only weakly depends on the error rate (see Appendix J). Now, for simplification we ignore the computing operations, which do not affect our general argument. The total noise that the entire quantum circuit experiences is

Here, is the total number of the noise-burdened operations. Expanding the overall noise, we get

where

Note that the coefficient of in the overall noise corresponds to a binomial distribution, which can be approximated by the Poisson distribution. We have

We can find that the impact of the overall noise on the expected value of some observable is proportional to , which implies that exponential extrapolation works better than linear extrapolation.

We demonstrate that, following our protocol step by step, an experimentalist can derive an algorithm to run on a noisy quantum computer so as to estimate an output observable with zero bias versus the ideal observable. The experimentalist does not require any prior knowledge of the physical property of the noise, and the only condition is that the noise is localized and Markovian. For this purpose, we show that quantum gate set tomography is a perfect tool for measuring the noise in a quantum computer, if the aim is only to compensate the effect of the noise in quantum computing, and we also show that single-qubit Clifford gates and measurement can derive a complete set of operations that can compensate any noise in quantum computing.

The price of using such a systematic method to negate computing errors is that the quantum computation needs to run for a longer time than an error-free system. We verify the protocol with numerical simulations of up to 19 qubits, in which an alternative method, i.e., exponential error extrapolation, is introduced and studied. We find that the estimation using exponential error extrapolation is also very accurate, while the computing time could be shorter. An approach combining two methods may optimize both accuracy and efficiency.

In Appendix I, we describe in detail the steps that an experimentalist would take in order to realize the quasiprobability method. We hope that this compact summary, presented in a single section, will indeed be useful to researchers who are interested in demonstrating the QEM technique with their hardware.

Our general conclusion is that these quantum error mitigation techniques can dramatically enhance the performance of quantum computers, especially at the small-to-medium scale where full code-based quantum error correction is impossible. Our simulations consider circuits up to 19 qubits, but with error rates considerably worse than the state of the art. Extrapolating from the trends that we observe in these smaller systems, we anticipate that hybrid algorithms involving qubits, i.e., beyond the reach of classical emulation, will benefit from QEM techniques if the hardware fidelity matches today’s state-of-the-art error or modestly improves upon it.

This work was supported by the EPSRC National Quantum Technology Hub in Networked Quantum Information Technologies. S. E. is supported by Japan Student Services Organization (JASSO) Student Exchange Support Program (Graduate Scholarship for Degree Seeking Students). Y. L. is also supported by NSAF (Grant No. U1730449). Numerical simulations were performed using QuEST, the Quantum Exact Simulation Toolkit [32]. The authors would like to acknowledge the use of the University of Oxford Advanced Research Computing (ARC) facility in carrying out this work [33].