Relationship between costs for quantum error mitigation and non-Markovian measures

It is now widely accepted that quantum computing will enable us to perform classically intractable tasks such as Shor’s algorithm for prime factorization Nielsen and Chuang (2002), quantum simulation for quantum many-body systems Georgescu et al. (2014), and Harrow-Hassidim-Lloyd (HHL) algorithm for solving linear equations Harrow et al. (2009). However, the effect of decoherence does impose an inevitable impact on the reliability and efficiency of quantum computation, and hence suppressing physical errors is crucial to obtain reliable results Georgescu et al. (2014); Hauke et al. (2012); Bylander et al. (2011); Reiner et al. (2019); Ma et al. (2019). Although fault-tolerant quantum computing based on quantum error correction can resolve that difficulty, it is not likely to happen for a while because of the large number of physical qubits required per single logical qubit.

Quantum error mitigation (QEM) methods have been proposed to mitigate errors in digital quantum computing, which is compatible with near-term quantum computers with the restricted number of qubits and gate operations as it does not rely on the encoding required in fault-tolerant quantum computing Preskill (2018); Temme et al. (2017); Endo et al. (2018); Li and Benjamin (2017); McArdle et al. (2020). For example, probabilistic error cancellation can perfectly cancel the effect of noise if the complete description of the noise model is given Temme et al. (2017); Endo et al. (2018). We apply recovery quantum operations $\mathcal{E}_{R}$ to invert noise processes of gates $\mathcal{N}_{G}$ such that $\mathcal{E}_{R}=\mathcal{N}_{G}^{-1}$. Since an inverse channel of noisy processes is generally an unphysical channel, we need to realize this by applying single-qubit operations with a classical postprocessing of measurement outcomes. Also, the repetition of quantum circuits needs to be $C^{2}$ times greater to achieve the same accuracy as before QEM, where $C$ is an overhead factor determined by the noise model and operations used in the QEM procedure. We refer to this overhead factor as “QEM costs” throughout this paper. We can write QEM costs of the error mitigation of the quantum circuit as $C=\prod_{k=1}^{N_{g}}c_{k}$, where $c_{k}$ is the QEM cost of the $k$th noisy gate and $N_{g}$ is the number of gates. Therefore, in order to suppress costs of QEM, we need to investigate the property of $c_{k}$ and optimize it.

Recently, Sun et al. (2020); Endo et al. (2021) proposed a general quantum error mitigation scheme that can also be applied to continuous quantum systems such as analog quantum simulators. The continuous time evolution of a quantum system is described by

where $H(t)$ is the Hamiltonian, $\rho_{N}(t)$ denotes a density matrix under noisy dynamics, and $\mathcal{L}\big{[}\rho_{N}(t)\big{]}$ is the superoperator describing the effect of the environment, which should be mitigated. Even when $\mathcal{L}$ consists of local Lindblad operators, the effects of them easily propagate to the entire system, resulting in highly correlated noise. Note that Eq. (1) can be rewritten as $\rho_{N}(t+\delta t)=\mathcal{E}_{N}(\rho_{N}(t))$, where $\mathcal{E}_{N}$ denotes a superoperator of noisy dynamics for a small time interval $\delta t$. Thus, similarly to probabilistic cancellation, denoting $\mathcal{E}_{I}$ as the ideal process, we can apply the recovery channel $\mathcal{E}_{R}$ such that $\mathcal{E}_{R}\mathcal{E}_{N}=\mathcal{E}_{I}$ using additional single-qubit operations and classical postprocessing of measurement results. In Ref.  Sun et al. (2020), the stochastic QEM method was introduced to implement recovery operations in the limit of $\delta t\rightarrow 0$ via Monte Carlo sampling. Note that QEM costs corresponding to $\mathcal{E}_{R}(t)$ can be described as $c(t)\approx 1+c^{\prime}(t)\delta t$. Therefore, a QEM cost from $t=0$ to $t=T$ can be described as $C(T)=\exp[\int_{0}^{T}dtc^{\prime}(t)]$.

So far, QEM for Markovian noise has been mainly considered and non-Markovian noise has not been investigated well. The development of the QEM for non-Markovian noise is practically important, because non-Markovian noise is relevant in most of the solid-state systems such as superconducting qubits, nitrogen vacancy centers in diamond, and spin qubits in quantum dots Yoshihara et al. (2006); Kakuyanagi et al. (2007); Bar-Gill et al. (2013); De Lange et al. (2010); Kawakami et al. (2014); Watson et al. (2018).

In addition to the practical motivation for non-Markovian noise, the concept of quantum non-Markovianity has been extensively studied from a fundamental interest in the characterization and the quantification of the backflow of the information from an environment Rivas et al. (2014); Breuer et al. (2016). There are some definitions of non-Markovianity such as the semigroup definition Breuer et al. (2002), the divisibility definition Wolf and Cirac (2008); Rivas et al. (2010), and that proposed by Breuer, Laine and Piilo Breuer et al. (2009) (Ref. Rivas et al. (2014) discusses a hierarchical relation between these definitions). Throughout this paper, we adopt divisible maps as Markovian processes and we will show that precise definition in a later section.

Moreover, there are many applications to utilize non-Markovianity in a positive way for quantum information processing, including quantum Zeno effects Misra and Sudarshan (1977); Itano et al. (1990), dynamical decoupling Viola and Lloyd (1998), Loschmidt echo and criticality Haikka et al. (2012), continuous-variable quantum key distribution Vasile et al. (2011), time-invariant discord Haikka et al. (2013), quantum chaos Žnidarič et al. (2011), quantum resource theory Wakakuwa (2017), and quantum metrology Matsuzaki et al. (2011); Chin et al. (2012). These motivate researchers to investigate the properties of non-Markovianity.

In this paper, we investigate QEM costs for the case of non-Markovian noise. The stochastic QEM can be naturally applied to time-dependent non-Markovian noise to fully compensate for physical errors. We show that QEM costs reduce in the non-Markovian region. We also find a clear relationship between costs of QEM and previously reported non-Markovian measures, decay rate measure Hall et al. (2014), and Rivas-Huelga-Plenio (RHP) measure Rivas et al. (2010); Hall et al. (2014). We calculate QEM costs for two experimental setups showing non-Markovianity as examples: One is a controllable open quantum system, which consists of a long-lived qubit coupled with a short-lived qubit. This system has been realized in NMR experiments Binho et al. (2019); Kukita et al. (2020). The other example is a qubit dispersively coupled with a dissipative resonator Govenius et al. (2015); Blais et al. (2004); Bertet et al. (2005). This discovery may illuminate how to construct efficient QEM procedures.

The rest of this paper is organized as follows. In Sec. II, we review the stochastic QEM proposed in Ref. Sun et al. (2020). In Sec. III, we review the definition and the measure of non-Markovianity. In Sec. IV, we discuss the relation between QEM costs and the measure of non-Markovianity, and study QEM costs for specific models. Finally, we summarize and discuss our results in Sec. V.

In this section, we review the stochastic QEM Sun et al. (2020). Suppose that the dynamics of the system of interest can be described by Eq. (1). Here, we assume that the local noise and the coupling to the environment is sufficiently weak and the continuous dynamics of the system can be described by the time-dependent Lindblad master equation. Now we express the evolution of the state from $t$ to $t+\delta t$ as $\rho(t+\delta t)=\mathcal{E}_{N}(t)(\rho(t))$ and $\rho(t+\delta t)=\mathcal{E}_{I}(t)(\rho(t))$, corresponding to the noisy process and the ideal process, respectively. The ideal process represents the unitary dynamics without any noisy operators in Eq. (1). We hope to emulate the ideal evolution $\mathcal{E}_{I}$ by mitigating errors of the process, $\mathcal{E}_{N}$. When the evolution is affected by local noise operators, i.e., $\mathcal{L}$ can be decomposed as a linear combination of local noise operators, by using a recovery operation $\mathcal{E}_{Q}(t)$, we can efficiently find a decomposition:

Here, $c(t)=\sum_{i}|\mu_{i}|$, $p_{i}=|\mu_{i}|/c(t)$, and $\{\mathcal{R}_{i}\}$ is a set of a polynomial number of physical operations applied for QEM. Each $\mathcal{R}_{i}$ is a tensor product of single-qubit operations. For a given decomposition, the ideal process $\mathcal{U}_{T}$ from $t=0$ to $t=T$ can be decomposed as

where $N_{d}=T/\delta t$, $\vec{i}=(i_{1},i_{2},...,i_{N_{d}})$, $p_{\vec{i}}=\prod_{n=0}^{N_{d}-1}p_{i_{n}}$, $s_{\vec{i}}=\prod_{n=0}^{N_{d}-1}\mathrm{sgn}(\mu_{i_{n}})$, and the QEM cost can be described as $C(T)=\prod_{n=0}^{N_{d}-1}c(n\delta t)$. Supposing that the initial state for the quantum circuit is $\rho_{in}$, we have

where $\rho_{I}(T)$ is the density operator after the ideal process $\rho_{I}(T)=\mathcal{U}_{T}(\rho_{in})$ and $\rho_{\vec{i}}=(\prod_{n=0}^{N_{d}-1}\mathcal{R}_{i_{n}}\mathcal{E}_{N}(n\delta t))(\rho_{in})$. When measuring an observable $M$, since the expectation value for the state $\rho$ is equal to $\braket{M}_{\rho}=\mathrm{Tr}[\rho M]$, we obtain

We can obtain $\braket{M}_{\rho_{I}}$ of Eq. (6) from the actual experiment as follows: First, we generate the recovery operation $\mathcal{R}_{i}$ with a probability $p_{i}$ with a time interval $\delta t$ until time $T$, and measure the observable $M$, and we record the measurement outcome after multiplying the factor of $s_{\vec{i}}$. Second, we repeat the same procedure to reduce the statistical uncertainty. Finally, we estimate the value of $C(T)\sum_{\vec{i}}p_{\vec{i}}s_{\vec{i}}\braket{M}_{\rho_{\vec{i}}}$ from the measurement results, and this approximates the error-free expectation value.

Since $\mathcal{E}_{N}\approx\mathcal{E}_{I}$ for a small $\delta t$ and the recovery operation becomes an identity operation in almost all the cases, we can use the Monte Carlo method to stochastically realize continuous recovery operations $\mathcal{R}_{i}$ corresponding to $\delta t\rightarrow+0$ to eliminate a discretization error $O(T\delta t)$. This procedure is similar to the one employed in the simulation of the stochastic Schrödinger equation (refer to Ref. Sun et al. (2020) for details). Letting $c(t)=1+c^{\prime}(t)\delta t$, the QEM cost becomes

In this section, we review typical properties of non-Markovianity Rivas et al. (2014); Breuer et al. (2016).

We derive QEM costs for non-Markovian dynamics and discuss the direct relation between costs and the measure of non-Markovianity. Note that, since the time-local master equation [Eq. (9)] is derived from a given Hamiltonian, modifications of the Hamiltonian for applying recovery operations could affect the form of time-local master equation. However, to derive a relation between QEM costs and non-Markovian measures, we assume that recovery operations do not change Eq. (9).

In this work, we discuss a relationship between the QEM and non-Markovian measures. Non-Markovianity is characterized by a negative decay rate of the dissipator. Interestingly, the QEM cost does not increase at all when all decay rates are negative. This demonstrates that non-Markovianity can contribute to reduce the cost of the QEM. We show specific physical systems as examples that support our theoretical analysis.

We provide an intuitive explanation for this below: QEM mitigates physical errors which can be attributed to quantum information escaping to the environment, and in the case of Markovian dynamics, such information never comes back to the system. Therefore, the QEM cost monotonously increases. However, in the case of non-Markovian dynamics, the quantum information can return to the system from the environment, which contributes to the reduction of the sampling cost of QEM.

Furthermore, we mention the difference between QEM for non-Markovian cases and dynamical decoupling. Dynamical decoupling is a method to cancel the interaction between the system and the environment for non-Markovian cases, and we can in principle obtain a noiseless state. On the other hand, QEM is the method to obtain just an expectation value of a noiseless state. Importantly, QEM does not recover the quantum state itself, which is a significant difference from the dynamical decoupling.

We focus on the case where the decoherence operators can be described by a set of orthogonal operators such as Pauli operators, and leave more general cases for a future work. Moreover, the intentional introduction of the non-Markovianity might contribute to the QEM. For this purpose, we need to control not the system but the environment. Actually, there are some theoretical proposals and experimental demonstrations about the control of the non-Markovianity Kukita et al. (2020); Urrego et al. (2018); Valente et al. (2016), which might lead to the potential use of the non-Markovianity to improve the efficiency of the QEM. It is worth noting that the intentional introduction of the non-Markovianity is different from dynamical decoupling, because dynamical decoupling is not a control of the environment but a control of the system. Our work helps in understanding the properties of QEM and may lead to sophisticated construction of QEM for realistic quantum systems with non-Markovian noise.