Optimal Trace Distance and Fidelity Estimations for Pure Quantum States

TL;DR

This paper develops optimal quantum algorithms that estimate both the trace distance and the (square root) fidelity between pure states to within additive error.

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 $Θ(1/\varepsilon)$ 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.

Figures (2)

• Figure 1: Quantum circuit for estimating the squared fidelity.
• Figure 2: Quantum circuit for estimating the squared fidelity with state-preparation oracles.

Theorems & Definitions (38)

• Theorem 1.1: Optimal pure-state trace distance and square root fidelity estimations
• Proposition 1.2: Testing quantum state tomography
• Proposition 1.3: Computing the minimum error of quantum hypothesis testing
• Proposition 1.4: Bounding the sample complexity for quantum state discrimination
• Theorem 1.5: Square root amplitude estimation, \ref{['thm:sqrt-ampl-est']} restated
• Theorem 1.6: $\mathsf{BQP}$-completeness, adapted from RASW23 and WZ23
• Theorem 3.1: Quantum amplitude estimation, BHMT02
• Proposition 3.2
• proof
• Theorem 3.3: Quantum phase estimation, NC10