Quantum Algorithm for Fidelity Estimation

TL;DR

This paper introduces a quantum algorithm that efficiently estimates the fidelity between two unknown mixed quantum states, achieving exponential speedup over previous methods by leveraging quantum oracles and low-rank assumptions.

Contribution

The paper presents a novel quantum algorithm for fidelity estimation that operates in polylogarithmic time relative to the dimension, surpassing classical tomography-based approaches.

Findings

Achieves exponential speedup over classical algorithms

Operates efficiently for low-rank states

Requires quantum oracles for state preparation

Abstract

For two unknown mixed quantum states $\rho$ and $\sigma$ in an $N$-dimensional Hilbert space, computing their fidelity $F(\rho,\sigma)$ is a basic problem with many important applications in quantum computing and quantum information, for example verification and characterization of the outputs of a quantum computer, and design and analysis of quantum algorithms. In this paper, we propose a quantum algorithm that solves this problem in $\operatorname{poly}(\log (N), r, 1/\varepsilon)$ time, where $r$ is the lower rank of $\rho$ and $\sigma$, and $\varepsilon$ is the desired precision, provided that the purifications of $\rho$ and $\sigma$ are prepared by quantum oracles. This algorithm exhibits an exponential speedup over the best known algorithm (based on quantum state tomography) which has time complexity polynomial in $N$.