Unified multivariate trace estimation and quantum error mitigation

Abstract

Calculating the trace of the product of $m\phantom{\rule{4pt}{0ex}}n$-qubit density matrices (multivariate trace) is a crucial subroutine in quantum error mitigation and information measure estimation. We propose a unified multivariate trace estimation (UMT) which conceptually unifies the previous qubit-optimal and depth-optimal approaches with tunable quantum circuit depth and number of qubits. The constructed circuits have $\ensuremath{\lceil}(m\ensuremath{-}1)/s\ensuremath{\rceil}$ or $n\ensuremath{\lceil}(m\ensuremath{-}1)/s\ensuremath{\rceil}$ depth corresponding to $(s+m)n$ or $s+mn$ qubits for $s\ensuremath{\in}{1,\ensuremath{\cdots},\ensuremath{\lfloor}m/2\ensuremath{\rfloor}}$, respectively. Such flexible circuit structures enable people to choose suitable circuits for different hardware devices. We apply UMT to virtual distillation to achieve exponential error suppression and design a family of concrete circuits to calculate the trace of the product of eight and nine $n$-qubit density matrices. A numerical example shows that the additional circuits still mitigate the noise expectation value under the global depolarizing channel.