# Multivariate trace estimation in constant quantum depth

arXiv: https://arxiv.org/abs/2206.15405 · HTML: https://mlsanity.com/paper/2206.15405/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jun 2022.

## Built on by (most cited first, 2 of 2)

- [Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2307.01171/) (2023)
- [Multivariate trace estimation using quantum state space linear algebra](https://mlsanity.com/paper/2405.01098/) (2024)

## Abstract (arXiv)

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.