Unified multivariate trace estimation and quantum error mitigation

TL;DR

This paper introduces a unified approach for multivariate trace estimation in quantum computing, offering flexible circuit designs that adapt to hardware constraints and improve error mitigation techniques.

Contribution

It proposes a unified multivariate trace estimation method that unifies previous approaches with tunable circuit depth and qubit count, adaptable to different hardware.

Findings

Enables flexible circuit design for various hardware constraints.

Achieves exponential error suppression via virtual distillation.

Demonstrates noise mitigation in numerical simulations.

Abstract

Calculating the trace of the product of $m$ $n$-qubit density matrices (multivariate trace) is a crucial subroutine in quantum error mitigation and information measures estimation. We propose an unified multivariate trace estimation (UMT) which conceptually unifies the previous qubit-optimal and depth-optimal approaches with tunable quantum circuit depth and the number of qubits. The constructed circuits have $\lceil(m-1)/s\rceil$ or $n\lceil(m-1)/s\rceil$ depth corresponding to $(s+m)n$ or $s+mn$ qubits for $s\in\{1,\cdots,\lfloor m/2\rfloor\}$, respectively. Such flexible circuit structures enable people to choose suitable circuits according different hardware devices. We apply UMT to virtual distillation for achieving exponential error suppression and design a family of concrete circuits to calculate the trace of the product of $8$ and $9$ $n$-qubit density matrices. Numerical example shows that the additional circuits still mitigate the noise expectation value under the global depolarizing channel.