Optimal Trace Distance and Fidelity Estimations for Pure Quantum States

TL;DR

This paper introduces optimal quantum algorithms for estimating the trace distance and fidelity between pure states with significantly fewer queries than previous methods, advancing quantum state distinguishability measurement.

Contribution

It presents the first algorithms achieving $ heta(1/\varepsilon)$ query complexity for estimating trace distance and fidelity between pure states, improving over the traditional $O(1/\varepsilon^2)$ bound.

Findings

Quadratic improvement in query complexity for state distinguishability

Development of quantum square root amplitude estimation technique

Enhanced understanding of quantum state measurement precision

Abstract

Measuring the distinguishability between quantum states is a basic problem in quantum information theory. In this paper, we develop optimal quantum algorithms that estimate both the trace distance and the (square root) fidelity between pure states to within additive error $\varepsilon$ using $\Theta(1/\varepsilon)$ queries to their state-preparation circuits, quadratically improving the long-standing folklore $O(1/\varepsilon^2)$. At the heart of our construction, is an algorithmic tool for quantum square root amplitude estimation, which generalizes the well-known quantum amplitude estimation.