Unified multivariate trace estimation and quantum error mitigation

Fault-tolerant quantum (FTQ) computers may provide novel computational advantages over classical computers for many tasks Shor1994 ; Harrow2009Quantum ; Biamonte2017Quantum ; Liang2019Quantum . As the perfect FTQ computers are not available yet, a preferable substitute is the the noisy intermediate-scale quantum (NISQ) devices with limited quantum resources Cerezo2021Variational ; Bharti2022Noisy . Nevertheless, the implementation of both FTQ and NISQ devices hinges on the effective control of noise. Thus, error correction and mitigation are of fundamental importance in quantum computation. Aiming at efficiently approximating the desired output states, the quantum error correction (QEC) provides a theoretical blueprint enabling quantum computation in an arbitrary small error level Devitt2013Quantum ; Lidar2013Quantum . However, duo to the larger qubit count, extra circuit complexity and other additional operations Preskill2018Quantum , the overhead of QEC is too large to be available for practical applications. Therefore, a variety of quantum error mitigation (QEM) approaches Endo2021Hybrid ; Zhang2021Variational ; Bultrini2021Unifying ; Yang2021Accelerated ; Cai2021A ; Takagi2022Universal ; Huo2022Dual ; Czarnik2022Improving for NISQ algorithms have been presented instead for QEC. Different from QEC, QEM focuses on recovering the ideal measurement statistics (usually the expectation values) Cao2022NISQ and can be directly employed in the ground state preparation Liang2022Improved ; McArdle2019Variational . For instance, the error extrapolation technique utilizes different error rates to the zero noise limit Temme2017Error ; Li2017Efficient ; Endo2018Practical . In particular, probabilistically implementing the inverse process can mitigate the noise effect on computation for some noise channels Temme2017Error . However, such error mitigation techniques rely on the prior knowledge on the noise model whose characterization is expensive.

The generalized quantum subspace expansion (GSE) Yoshioka2022Generalized and the virtual distillation (VD) methods Koczor2021Exponential ; Huggins2021Virtual do not require any information of the noise. The GSE method optimizes states from a subspace expanded by a small subset of Pauli operators. It is effective against coherent errors Endo2021Hybrid . The VD method prepares $m$ copies $n$-qubit noisy state $\rho=E_{0}|u_{0}\rangle\langle u_{0}|+\sum_{k=1}^{2^{n}-1}E_{k}|u_{k}\rangle\langle u_{k}|$ in spectral decomposition, $E_{0}>E_{1}\geq\cdots\geq E_{2^{n}-1}$, and calculates the expectation value of an observable $O$ with respect to the state $\rho^{m}/\textrm{Tr}(\rho^{m})$,

which exponentially gets closer to the ideal value $\langle O\rangle_{\mbox{exact}}=\langle u_{0}|O|u_{0}\rangle$. The core idea behind this claim is that the state $\rho^{m}/\mbox{Tr}(\rho^{m})$ approaches to the desired pure state $|u_{0}\rangle$ exponentially in $m$. In order to obtain an exponential error suppression, an efficient quantum algorithm for measuring Eq. (1) is required.

Generally, instead of the $m$ copies of the $n$-qubit noisy state $\rho$, one may consider $m$ different $n$-qubit states $\rho_{1},\cdots,\rho_{m}$. The $\textrm{Tr}(\rho^{m})$ in Eq. (1) is then replaced by a more general quantity $\mbox{Tr}(\rho_{1}\cdots\rho_{m})$ which is called multivariate trace (MT) first introduced in the work Quek2022Multivariate . Measuring MT is of fundamental and practical interest in quantum information processing such as the calculation of the Rényi entropy, entanglement entropy Yao2010Entanglement and the entanglement spectroscopy Johri2017Entanglement . Broadly speaking, the estimation of MT on a quantum computer is currently tackled by using either the qubit-optimal approach Ekert2002Direct or depth-optimal method Quek2022Multivariate . Here, the qubit-optimal means that the number of ancilla qubits is optimal, and the depth-optimal stands for that the circuit depth is minimal. The qubit-optimal approach requires only single ancilla qubit and a $\Theta(m)$-depth circuit, whereas the depth-optimal method requires $\lfloor m/2\rfloor$ ancilla qubits and a constant depth circuit, where $\lfloor\cdot\rfloor$ denotes the floor function. The former method is prohibited due to linear depth in $m$. Meanwhile, the latter method has an attractive depth for NISQ devices, but the linear number of needed qubits in $m$ would restrict its application to small $m$. Although a recent work Czarnik2021Qubit drastically reduces the qubit resource by utilizing qubit resets technique, the circuit depth is still $\Theta(m)$. Thus, limited qubit number and circuit depth may hinder the advantage of the VD method in current NISQ era.

Accounting to the fact that the error accumulates with the increasing of the number of qubits and the depth of quantum circuit, in this work we provide a quantum algorithm to calculate the corrected expectation value Eq. (1) by constructing a family of circuits which have different circuit depth and number of qubits. As the authors Quek2022Multivariate pointed out that their circuit is flexible and can be adjusted as different depth and number of qubits. Thus, in this work we first mathematically establish a specific trade-off relation between the number of qubits and the circuit depth. The circuit depth and the number of qubits can be denoted as a function of a free parameter $s$. From the variation $s$, there are $\lfloor m/2\rfloor$ different circuit structures with the same number of quantum gates. Based on the constructed circuits, we propose an unified multivariate estimation (UMT), which is capable of calculating $\mbox{Tr}(\rho_{1}\cdots\rho_{m})$ with a tunable circuit depth and number of qubits. The existing qubit-optimal and depth-optimal algorithms are two extremal cases of our algorithm for $s=1$ and $\lfloor m/2\rfloor$. Furthermore, we apply UMT to achieve the exponential error suppression and give a family of concrete circuits for $8$ and $9$ $n$-qubit density matrices. Finally, we simulate the effects of the global depolarizing channel in the process of estimating $\langle O\rangle_{\textrm{vd}}^{(5)}$ for a two qubits state $\rho$.

The MT is defined as

where $S^{(m)}$ is a unitary representation of the cyclic shift permutation $\pi=(1,2,\cdots,m)$: $S^{(m)}|\psi_{1}\rangle\otimes|\psi_{2}\rangle\otimes\cdots\otimes|\psi_{m}\rangle=|\psi_{m}\rangle\otimes|\psi_{1}\rangle\otimes\cdots\otimes|\psi_{m-1}\rangle$ for pure states $|\psi_{1}\rangle,\cdots,|\psi_{m}\rangle$. Note that for $m=2$, $\mbox{Tr}[\rho_{1}\rho_{2}]=\mbox{Tr}[S^{(2)}(\rho_{1}\otimes\rho_{2})]$ with $S^{(2)}$ the SWAP operator. Eq. (2) means that the MT can be estimated by calculating the real and imaginary parts of $\mbox{Tr}[S^{(m)}(\rho_{1}\otimes\cdots\otimes\rho_{m})]$. Following the framework of Ref. Ekert2002Direct , a crucial step is to perform the controlled unitary $\mathcal{C}(S^{(m)})$ with respect to $S^{(m)}$, $\mathcal{C}(S^{(m)})=|0\rangle\langle 0|\otimes I+|1\rangle\langle 1|\otimes S^{(m)}$, where $I$ denotes the identity. In this section, we construct a sequence of alternative circuits to achieve the operation $\mathcal{C}(S^{(m)})$.

A direct application of UMT is to the quantum error mitigation Huggins2021Virtual ; Koczor2021Exponential . Suppose that the near-term quantum devices aim to prepare an $n$-qubit pure state $|\phi\rangle$. However, owing to the effect of environment noise, one prepares instead a mixed state $\rho=\mathcal{C}(|\phi\rangle\langle\phi|)$, where the operation $\mathcal{C}$ is a map containing a unitary evolution and a noise channel such as depolarizing channel. The error-free expected value of an Hermitian operator $O$ is $\langle O\rangle=\mbox{Tr}(O|\phi\rangle\langle\phi|)$. However, the noisy expected value is $\langle O\rangle_{\textrm{noise}}=\mbox{Tr}(O\rho)\neq\langle O\rangle$. Virtual distillation provides a method to approximate $\langle O\rangle$ as a corrected expectation value

by $m$ copies of the mixed state $\rho$.

We have proposed an unified quantum algorithm for estimating the multivariate trace. Our results depend on a qubit-depth trade-off relation which helps us to construct a family of circuits. The designed circuits have flexible depth and number of qubits, but the total number of quantum gates is always $\mathcal{O}(mn)$. These proposals can be used as an important subroutine for estimating entanglement spectroscopy Yirka2021Qubit , quantum metrology Yamamoto2022Error and calculating the nonlinear function of density matrix Ekert2002Direct . Moreover, we have applied the UMT to achieve the exponential error suppression for quantum error mitigation and numerically find that our circuits still work in the noise situation of the global depolarizing channel. Notice that the recent work Zhou2022A estimated the MT with randomized measurement and further reduced the overhead of qubits Elben2022The . However, the number of depth remains the same as the qubit-optimal method Ekert2002Direct . Our algorithm gives also an alternative of the quantum parts in Zhou2022A .

There are two proposals involving dual-state purification Huo2022Dual ; Cai2021Resource , in which only the single ancillary qubit and the implementation of the dual channel are required. The related framework utilizes qubit reset technique to reduce the number of qubits Cai2021Resource compared with our circuit for $s=1$. However, the circuit depth is still $\mathcal{O}(m)$. Our family of circuits provides alternatives to reduce the circuit depth under the consume of the number of qubits. In general, depth is a more important index than qubit overhead for a quantum circuit. It is also worth to remark that our approaches utilize ancillary system to estimate $\textrm{Tr}(O\rho^{m})$. For $m=2$, a well-known destructive SWAP test Escartin2013SWAP ; Cincio2018Learning achieves the same goal without ancillary qubit and, at the same time, also reduces the circuit depth to a constant. However, one requires to measure multiple qubits which may increase the measurement overhead.

Notably, we have dealt with the problem for arbitrary positive integer $m$. Estimating a nonlinear function of quantum states is also of fundamental and practical interests. For instance, the fidelity involves a square root of a quantum state Nielsen2000Quantum , while the Tsallis entropies are defined by $S^{T}_{\alpha}(\rho)=\frac{1}{1-\alpha}[\textrm{Tr}(\rho^{\alpha})-1]$ for $\alpha\in(0,1)\cup(1,\infty)$ Tsallis1988Possible . Although direct estimation is hard on a quantum computer, a hybrid quantum-classical framework Tan2021Variational makes the computation plausible by combining quantum state learning Lee2018Learning and approximating fractional powers. The core idea is to minimize the quantum purity which involves an estimation of MT. Thus, our proposals can be generalized to calculate a nonlinear function of quantum states in a roundabout way.

Several interesting issues should further be investigated in the near future. The first one is to simulate the effects of different types of noise in the circuit implementation of estimating $\textrm{Tr}(O\rho^{m})$ such as the amplitude damping and phase damping channels. It would be also interesting to explore circuit structures that reduce the width and depth simultaneously. However, for the calculation of MT estimation, the qubit-depth trade-off shows that the circuit depth and width are complementary computational resources. In particular, a reduction in circuit depth is often accompanied by an increase in width, and vice versa. Current available quantum computation devices often have small size in number of qubits and circuit depth. Thus, from the view of computational resources the circuit depth and width should be reduced as much as possible in quantum algorithm design. Due to the friendly decomposition of the cyclic permutation, the circuit depth in our algorithms is reduced coincidentally, although the number of ancilla qubits is increased in a control range. Our results may highlight further investigations on the depth reduction for general quantum circuits.

Acknowledgements: This work is supported by the National Natural Science Foundation of China (NSFC) under Grants 12075159 and 12171044; Beijing Natural Science Foundation (Grant No. Z190005); the Academician Innovation Platform of Hainan Province.