Purity-Assisted Zero-Noise Extrapolation for Quantum Error Mitigation

Quantum computation is expected to surpass classical computation for specific problems; nevertheless, the vulnerability of qubits to environmental noise poses a significant challenge to realizing a practical quantum computer. Consequently, the development of fault-tolerant quantum computation methodologies is crucial for enabling the widespread application of quantum computation. Quantum error correction code (QECC) provides a systematic solution to fault-tolerant quantum computation [1], while its application requires many physical qubits and the error rate should be lower than a threshold [2, 3, 4]. With remarkable progress [5, 6], however, it is challenging for the state-of-the-art technique to meet the requirements for the full implementation of QECCs. Therefore, fault-tolerant quantum computation based on high-precision logic qubits is still a far-reaching task.

In the noisy intermediate-scale quantum era [7], quantum error mitigation (QEM) provides an alternative approach to dealing with noise. QECCs detect and correct errors that occur in the quantum process to ensure that there are no errors in the coded logic qubits. On the contrary, QEMs allow errors to occur, but use some post-processing techniques to reduce the bias between the output information of noisy quantum circuit and the ideal quantum circuit [8, 9].

In recent years, several QEM schemes have been proposed, including zero-noise extrapolation (ZNE) [10, 11, 12, 13, 14], probabilistic error cancellation (PEC) [13, 14, 15, 16], symmetry verification methods [17, 18, 19], purification methods [20, 21, 22, 23], subspace expansion [24, 25, 26, 27], and learning-based error mitigation [28, 29]. These methods have been applied on different quantum computation platforms and interesting physics [15, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 31, 44]. In addition, there have been some efforts to integrate various QEM schemes in a generalized framework [45, 19, 46, 47, 48, 49]. Among these various error mitigation schemes, the ZNE and PEC are used to deal with quantum circuits, which based on specific error models of noise. The PEC method mitigates the error by post-cancelling the noise, which is previously estimated from experiments in the selected error basis. Although the Pauli basis is sufficient with Pauli twirling technique [50, 51, 52, 34], in practice, the selected basis should be truncated for large systems, such as the $2$-weighted Pauli basis [34]. Thus, the effectiveness of PEC depends on the error model described by the error basis. Recently, an attempt was made to integrate ZNE into the PEC framework [47].

However, the ZNE method is a subtle approach based on the analyticity of the observable expectation related to the error rate, which is a fundamental physical condition. The relation to the error model emerges when the error rate is considered. To perform the ZNE, the error must be amplified, and the ideal value is obtained by extrapolating the expectation backward to the point of zero noise. In the early application of ZNE, the error is amplified by stretching the duration of pulse for each gate [15]. However, this amplification is not suitable for digital quantum gates. Another implementation is to deliberately add redundant gates in the circuit [30], which is improved with a treatment called unitary folding [53]. In the unitary folding method, the redundant gates are implemented by the target circuit of interest. Specifically, the circuit evolves forward and backward successively, and the error rate is characterized by the number of folds. Although the amplifications of the error are developed more feasible, the estimation of the error rate is a priori, which has limitations on the noise type. Moreover, for the complex circuit in the experiment, the effect of errors cannot be sufficiently described by a single parameter of the error rate alone.

In the ZNE method, dependence of the error model in noisy amplification is contrary to the universal idea of the extrapolation, which limits the effectiveness of this method. In this paper, we try to release the limitation of noise in the ZNE method with the assistance of the purity of the noisy output state. Here, we introduce the purity-assisted zero-noise extrapolation (pZNE) method. Specifically, we propose a modified purification method based on the Pauli diagonal error model. By estimating the bias of the modified purification methods, it is shown that the bias is controlled by error rate, which can be represented by purity of noisy state. The pZNE method mitigates the noisy expectation by extrapolating the mitigated expectations of the modified purification method with different error rates to a noise-free pure state. Equivalently, it can be applied by extrapolating the noisy expectations with purity.

This paper is organized as follows. In Sec. 2, we review the routine ZNE method and analyze how this method is based on the error model of noise. Then, we show that the purity of the output state can assist the extrapolation in zero-noise error mitigation in Sec. 3. We describe the detailed procedures of pZNE in Sec. 4, including the fitting model, purity estimation methods, and the measurement overhead. We also verify this method by numerical simulations and experiments on Quafu cloud-based quantum computation platform in Sec. 5. The conclusion and discussion are given in Sec. 6.

The ZNE method consists of two steps:

1. 1.

   Noise amplification: Collecting the raw data of different error rates via measuring on modulated circuits.

2. 2.

   Extrapolation: Post-processing the experiment data to obtain the mitigated expectation value based on some specific fitting models.

In this section, we will review the ZNE method, the fitting models, and the noise amplification method of unitary folding. In addition, we discuss how the assumption that the error rate is proportional to the number of folds is based on the error model of noise.

Consider the ideal unitary circuit $\mathcal{U}$ of interest. However, in practice, we can only perform a noisy circuit $\mathcal{U}_{\lambda}$ instead. Then the experimental expectation value $\braket{\hat{O}}_{\lambda}=\mathrm{Tr}[\hat{O}\rho_{\lambda}]$ of observable $\hat{O}$ of interest is different from the ideal one $\braket{\hat{O}}_{0}=\mathrm{Tr}[\hat{O}\rho_{0}]$, where $\rho_{\lambda}=\mathcal{U}_{\lambda}(\rho)$ is the noisy output state, and $\rho_{0}=\mathcal{U}(\rho)$ is the ideal output state. Now, we are going to infer the ideal value $\braket{\hat{O}}_{0}$ from some noisy expectation values $\braket{\hat{O}}_{\lambda}$.

This method is inspired by a simple idea that the expectation value $\braket{\hat{O}}_{\lambda}$ is an analytic function of a parameter $\lambda\geq 0$, which characterizes the noise level of noisy circuit $\mathcal{U}_{\lambda}$, and assume $\braket{\hat{O}}_{\lambda}=\braket{\hat{O}}_{0}$ when $\lambda=0$. According to analyticity, the expectation value can be expanded in a power series of $\lambda$ as

and by definition of $\lambda$, we have $O_{0}=O(0)=\braket{\hat{O}}_{0}$. Thus, by interpolation the function $O(\lambda)$ with expectations $\braket{\hat{O}}_{\lambda_{i}}$ of different error rate $\lambda_{i}$, one can infer the noise-free expectation value $\braket{\hat{O}}_{0}$.

For the number of experimental data is finite, the expansion of $O(\lambda)$ should be truncated at some order $M$ of $\lambda$, which requires that the error rate $\lambda$ should small enough, and this is the polynomial model of function $O(\lambda)$ [10, 13]. In addition, there are also (multi-)exponential model [11]

and poly-exponential model [53]

where $f(\lambda)$ is a polynomial of $\lambda$. These fitting models are put forward under the consideration that the physical observable should be bounded.

Then, the noisy data of different error rates $\lambda_{i}$ is obtained by error amplification. The unitary folding performs the circuit sequence in the experiment

where $\circ$ denotes the composition of quantum operations, the corresponding error rate is expected as $\lambda_{n}=(2n+1)\lambda_{0}$, and the resolution is $2\lambda_{0}$. The circuit is shown in Figure 1. To have a more fine-grained resolution, it can also perform the layer (gate) folding [53], which assumes the gate can be decomposed into layers $\mathcal{U}_{\lambda}=\prod_{i=1}^{d}\circ\mathcal{L}_{i}$, and performs the folding on these layers

in the circuit. By doing so, the resolution can be promoted to $2\lambda/d$.

To consider the procedure of unitary folding in more detail, we assume that the circuit is a gate only, and set the noise in forward and backward evolutions as

which can be interpreted as the definition of forward error $\mathcal{E}_{f}$ and backward error $\mathcal{E}_{b}$. The error rate for both errors is denoted as $\lambda_{f}$ and $\lambda_{b}$ respectively. Then, we formally have

thus the noisy gate is

where $\mathcal{E}_{n}=\mathcal{E}^{n-1}\circ\mathcal{E}_{f}$ is the error channel with the error rate $\lambda_{n}=(n-1)\tilde{\lambda}+\lambda_{f}$, where $\tilde{\lambda}$ is the error rate of error channel $\mathcal{E}$.

To obtain $\lambda_{n}=(2n-1)\lambda_{f}$, we can assume $\tilde{\lambda}=2\lambda_{f}$, which is true when the error channel satisfies $\mathcal{E}_{f}=\mathcal{E}_{b}$. However, this assumption is not always realized [54] (or see Appendix A). If we assume $\tilde{\lambda}\approx 2\lambda_{f}$, it will introduce an additional error in the inference of the ideal value $O(0)$ before fitting the experimental data. As a result, the ZNE method is constrained by the error model of noise. The direct way to solve this problem is to determine the error rate $\lambda$ from experiments.

Fidelity is always used to witness the difference between the state with an ideal state, however, the ideal state $\rho_{0}$ cannot be prepared precisely due to the inevitable noise in the experiment. Therefore, measuring the fidelity of the noisy state and the ideal state requires the classical simulation of the quantum circuit, which is generally impractical for large systems. In this section, we will describe the idea of using the purity of the output state of the noisy circuit

to reflect the influence of noise in the circuit. The ideal unitary circuit will not influence the purity of the state. We assume that the error is not unitary, otherwise, it can be canceled by the folding of the well-calibrated circuit. Moreover, we assume the error is in the form of Pauli diagonal

where $\mathcal{P}_{i}(\cdot)=\hat{P}_{i}\cdot\hat{P}_{i}$ is Pauli operations, and $r=f,b$ distinguish the forward and backward errors. This assumption can be realized by the Pauli twirling technique.

Let the ideal state be decomposed into Pauli basis $\rho_{0}=\mathcal{U}(\rho)=\frac{1}{D}\sum_{i}\rho_{i}\hat{P}_{i}$. The Pauli diagonal forward error channel acts as

where $\chi_{ri}=\sum_{j}\epsilon_{ij}q_{rj}$ is the eigenvalue of the error, with $\mathcal{P}_{j}(\hat{P}_{i})=\epsilon_{ij}\hat{P}_{i}$. We assume that the operators whose expectation values of interest are Pauli operators. Then, the expectation of $\hat{P}_{i}$ of the folded noisy state $\rho_{n}=\mathcal{U}_{n}(\rho)=\mathcal{E}_{n}(\rho_{0})$ are

By fitting these expectations with the exponential model, only the parameters $\chi_{fi}\chi_{bi}$ and $\chi_{fi}\rho_{i}$ are obtained. To find the ideal expectation $\braket{\hat{P}_{i}}_{0}=\rho_{0}$, we should estimate $\chi_{fi}$. In general, $\chi_{fi}\neq\chi_{bi}$, so the ideal expectation cannot be extracted.

We use purity to do this estimation. The purity of the noisy state $\rho_{n}$ is

where $\overline{\chi_{n}^{2}}=\frac{\sum_{\chi_{ni}<1}\rho_{i}^{2}\chi_{ni}^{2}}{\sum_{\chi_{ni}<1}\rho_{i}^{2}}$ is the average of nontrivial eigenvalues $\chi_{ni}=\chi_{fi}^{n}\chi_{bi}^{n-1}$ of folding circuits, $p_{0}=\frac{1}{D}\sum_{i}\rho_{i}^{2}$ is the purity of ideal state, and $p_{\infty}$ is the purity of the stable state of the forward error channel. For convenience, we can assume the ideal state is pure, $p_{0}=1$, and the stable state is a maximally mixed state, $p_{\infty}=1/D$. If the distribution of $\chi_{ni}$ concentrates, so that the variance $\Delta\overline{\chi}_{n}^{2}=\overline{\chi_{n}^{2}}-\overline{\chi}_{n}^{2}$ is small, then we can approximate

and the ideal expectation is estimated as

The effectiveness of this approximation that $\chi_{ni}\approx\overline{\chi_{n}^{2}}^{1/2}$ requires $\Delta\overline{\chi}_{n}^{2}$ to be small, which is based mainly on the distribution of eigenvalue of error channel and slightly on the state $\rho_{0}$ in large systems. The relative bias of this method is

Given a tolerant error $\epsilon$, the probability of failing is

where $\sigma^{2}=\frac{\Delta\bar{\chi}_{n}^{2}}{\bar{\chi}_{n}^{2}}$, by Chebyshev’s inequality (for detail, see Appendix D).

Denote the failing probability as $\delta$, and select the tolerant error $\epsilon$ so small, we expand the left-hand of inequality to the leading term of $\epsilon$

which means that the pZNE method works if the eigenvalues of forward error satisfy $\sigma\leq 2$, and the tolerant error $\epsilon$ is controlled by $\sigma$. The parameter $\sigma$ relies on the form of the forward error, for example, for the global depolarizing error, all the nontrivial eigenvalues $\chi_{ni},(i\neq 0)$ are the same, so $\sigma=0$ no matter how large the error. Moreover, without specifying the form of errors, this parameter is controlled by the error probability $q_{\lambda}=\sum_{i\neq 0}q_{ni}$, where $q_{ni}$ is the coefficients of Pauli diagonal error $\mathcal{E}_{n}$. Obviously $1-2q_{\lambda}\leq\bar{\chi}_{n}\leq 1$, we have

thus when the error is so small that $q_{\lambda}<0.4$, this method always has the possibility of success, and the relative bias is roughly controlled as

With more detail estimation (see Appendix D), $\epsilon\sim\sqrt{2}\lambda+O(\lambda^{2})$.

In all, the estimator in Equation (16) has bias controlled by the error probability $q_{\lambda}$ in general, so the number of folding. Therefore, we can extrapolate the estimators of different folding circuits to reduce bias. In this step there are two strategies. One is to extrapolate the estimators $\braket{\hat{P}_{i}}_{n\mathrm{est}}$ to $n=\frac{1}{2}$, which is similar to the extrapolation of routine ZNE. Another is to extrapolate the estimators $\braket{\hat{P}_{i}}_{n\mathrm{est}}$ along the effective error rate $s_{n}=-\ln\frac{p_{n}-p_{\infty}}{p_{0}-p_{\infty}}$ to $s_{n_{0}}=0$, and in this effective error rate,

which means that we can also extrapolate the noisy expectation $\braket{\hat{P}_{i}}_{n}$ with the effective error rate, or purity.

In Sec. 3, we discuss the procedure of the pZNE method. Assume our quantum computation task consists of $(\rho,\mathcal{U},\hat{O})$, where $\rho$ is the initial state, $\mathcal{U}$ is the ideal unitary evolution, and $\hat{O}$ is the operator of interest. The outcome we want from this computation task is the expectation $\braket{\hat{O}}_{0}=\mathrm{Tr}[\hat{O}\mathcal{U}(\rho)]$. The procedure of the pZNE method to mitigate the error in this task is following.

1. 1.

   Task decomposing: Decompose the operator $\hat{O}$ of interest into Pauli basis $\hat{O}=\sum_{i\in I}O_{i}\hat{P}_{i}$, where $I$ is the set of indices with nonzero coefficients $O_{i}$.

2. 2.

   Pauli twirling: Construct the Pauli-twirling the ideal circuit $\mathcal{U}_{\mathrm{twirl}}$, which corresponds to the noisy circuit $\mathcal{U}_{\lambda}$ in experiment.

3. 3.

   Task performing: Perform the circuit $\mathcal{U}_{n}$ to estimate the expectation $\braket{\hat{P}_{i}}_{n}=\mathrm{Tr}[\hat{P}_{i}\mathcal{U}_{n}(\rho)]$ and the purity $p_{n}=\mathrm{Tr}[\mathcal{U}_{n}(\rho)^{2}]$.

4. 4.

   Data processing: Estimate $\braket{\hat{P}_{i}}_{n\mathrm{est}}$ with $\braket{\hat{P}_{i}}_{n}$ and $p_{n}$. Extrapolate $\braket{\hat{P}_{i}}_{n\mathrm{est}}$ to noise-free point, $n=\frac{1}{2}$ or $s_{n_{0}}=0$. Calculate the estimation of the ideal expectation $\braket{\hat{O}}_{0}$ with coefficients $O_{i}$.

In this section, we will discuss the steps of task performing and data processing (for the technique of Pauli twirling, see Appendix C).

In this section, we simulate the pZNE method numerically and experimentally.

In this paper, we introduce a method that utilizes purity to assess noise error rates, operating under the assumption that the error channel is Pauli diagonal. By using the Pauli twirling technique, we ensure that this assumption holds universally. Our study involves estimating the bias of the pZNE method based on a specified failing probability and conducting a comparative analysis with other relevant methods. We demonstrate that the pZNE method may not lead to a substantial reduction in bias compared to the routine ZNE method when error rates are low. However, it showcases its utility within a certain threshold where the routine ZNE method might encounter limitations or failures.

Moreover, we analyze the overhead of the pZNE method using different purity estimation techniques, such as the replica measurement method and the QST-like method. Our results reveal distinct behaviors in the upper bounds of pZNE for these estimation methods. Specifically, the overhead of the replica measurement method is constrained by the overhead of ZNE with a finite constant added, while the overhead of a QST-like method escalates exponentially with the growth of system size. In cases with exponentially increasing overhead, the pZNE method may not be efficient for large systems when using a QST-like method. However, if the task requires dense information about the density matrix, the purity estimation will not significantly inflate the overhead.

In addition, we verify the pZNE method with the numerical simulations under the depolarizing error and the Pauli diagonal errors, as well as the experiments on the quantum computation platform, Quafu. We compare it with the routine ZNE and the VD/ESD method. The results show that the bias of both routine ZNE and pZNE increases with the error rate, and divergence when the error rate is large enough. The bias of pZNE is slightly lower than routine ZNE when the error rate is moderate, and the critical point of error rate for pZNE is larger than routine ZNE method. The standard deviation of pZNE is lower than that of ZNE, indicating that the pZNE method is more efficient when dense state information is of interest.

Our results show that the purity of noisy quantum states can be used to assist the ZNE method, thus extending its effectiveness. It can be expected that pZNE will reinforce the wide usage of the ZNE method in the NISQ era.