Estimation of multivariate traces of states given partial classical information

TL;DR

This paper introduces generalized quantum circuits for estimating multivariate traces of quantum states using partial classical information, reducing resource requirements and expanding applicability in quantum metrology and nonclassicality certification.

Contribution

It proposes a novel generalization of the cycle test for multivariate trace estimation with partial classical data, improving efficiency and scope.

Findings

Reduced qubit and gate requirements in quantum circuits.

Extended estimation capabilities to higher-order invariants.

Applicable to quantum metrology and nonclassicality certification.

Abstract

Bargmann invariants of order $n$, defined as multivariate traces of quantum states $\text{Tr}[\rho_1\rho_2 \ldots \rho_n]$, are useful in applications ranging from quantum metrology to certification of nonclassicality. A standard quantum circuit used to estimate Bargmann invariants is the cycle test. In this work, we propose generalizations of the cycle test applicable to a situation where $n$ systems are given and unknown, and classical information on $m$ systems ($m\leq n)$ is available, allowing estimation of invariants of order $n+m$. Our main result is a generalization of results on 4th order invariants appearing in double weak values from Chiribella et al. [Phys. Rev. Research 6, 043043 (2024)]. The use of classical information on some of the states enables circuits on fewer qubits and with fewer gates, decreasing the experimental requirements for their estimation, and enabling multiple applications we briefly review.