Multivariate trace estimation in constant quantum depth

TL;DR

This paper demonstrates that multivariate trace estimation, previously believed to require deep quantum circuits, can be achieved with constant-depth circuits using local gates, making it more feasible for near-term quantum devices.

Contribution

The authors construct a constant-depth quantum circuit for multivariate trace estimation, challenging the conventional belief that deep circuits are necessary, and show its implementation on near-term quantum architectures.

Findings

Constant-depth quantum circuit for multivariate trace estimation.

Implementation feasible on architectures similar to Google's Sycamore.

Application to estimating nonlinear functions of quantum states.

Abstract

There is a folkloric belief that a depth-$\Theta(m)$ quantum circuit is needed to estimate the trace of the product of $m$ density matrices (i.e., a multivariate trace), a subroutine crucial to applications in condensed matter and quantum information science. We prove that this belief is overly conservative by constructing a constant quantum-depth circuit for the task, inspired by the method of Shor error correction. Furthermore, our circuit demands only local gates in a two dimensional circuit -- we show how to implement it in a highly parallelized way on an architecture similar to that of Google's Sycamore processor. With these features, our algorithm brings the central task of multivariate trace estimation closer to the capabilities of near-term quantum processors. We instantiate the latter application with a theorem on estimating nonlinear functions of quantum states with "well-behaved" polynomial approximations.