Polynomial-time certification of fidelity for many-body mixed states and mixed-state universality classes

TL;DR

This paper introduces a polynomial-time algorithm to efficiently estimate and certify bounds on the fidelity between many-body mixed quantum states, enabling scalable verification and analysis of quantum systems.

Contribution

The authors develop a novel variational optimization framework that provides certified bounds on fidelity for matrix product density operators, improving precision and scalability.

Findings

Accurately track universal fidelity scaling behavior

Extract critical exponents from fidelity data

Achieve exponential improvement over previous bounds

Abstract

Computation of Uhlmann fidelity between many-body mixed states generally involves full diagonalization of exponentially large matrices. In this work, we introduce a polynomial-time algorithm to compute certified lower and upper bounds for the fidelity between matrix product density operators (MPDOs). Our method maps the fidelity estimation problem to a variational optimization of sequential quantum circuits, allowing for systematic improvement of the lower bounds by increasing the circuit depth. Complementarily, we obtain certified upper bounds on fidelity by variational lower bounds on the trace distance through the same framework. We demonstrate the power of this approach with two examples: fidelity correlators in critical mixed states, and codeword distinguishability in an approximate quantum error-correcting code. Remarkably, the variational lower bound accurately track the universal scaling behavior of the fidelity with a size-consistent relative error, allowing for the extraction of previously unknown critical exponents. Our results offer an exponential improvement in precision over known moment-based bounds and establish a scalable framework for the verification of many-body quantum systems.