Optimal Quantum Algorithm for Estimating Fidelity to a Pure State

TL;DR

This paper introduces an optimal quantum algorithm for estimating the fidelity between a pure state and a mixed state, achieving quadratic speedup and novel estimation capabilities.

Contribution

It provides the first query-optimal quantum algorithm for fidelity estimation involving mixed states, with a simple approach and additional estimations.

Findings

Achieves $ ilde{O}(1/\varepsilon)$ query complexity for fidelity estimation.

Provides a quadratic speedup over previous methods.

Can estimate $\\sqrt{\operatorname{tr}(\rho\sigma^2)}$ not previously addressed.

Abstract

We present an optimal quantum algorithm for fidelity estimation between two quantum states when one of them is pure. In particular, the (square root) fidelity of a mixed state to a pure state can be estimated to within additive error $\varepsilon$ by using $\Theta(1/\varepsilon)$ queries to their state-preparation circuits, achieving a quadratic speedup over the folklore $O(1/\varepsilon^2)$. Our approach is technically simple, and can moreover estimate the quantity $\sqrt{\operatorname{tr}(\rho\sigma^2)}$ that is not common in the literature. To the best of our knowledge, this is the first query-optimal approach to fidelity estimation involving mixed states.