Optimal Quantum Algorithm for Estimating Fidelity to a Pure State

2025-06-30

AI Summary

This work addresses the problem of efficiently estimating quantum state fidelity, specifically when one state is pure (σ = |ψ⟩⟨ψ|). We propose the first query-optimal quantum algorithm for estimating the fidelity F(ρ, σ) = ⟨ψ|ρ|ψ⟩—equivalently, √tr(ρσ²)—to additive precision ε. Our algorithm achieves query complexity Θ(1/ε) to the state preparation oracles for ρ and |ψ⟩, breaking the prior O(1/ε²) barrier inherent in both classical and earlier quantum approaches. This quadratic speedup matches the fundamental lower bound, establishing asymptotic optimality. The method operates within the standard quantum query model and employs a simple, shallow-depth circuit that avoids controlled evolutions or intricate subroutines. As such, it combines theoretical optimality with practical implementability.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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 $Θ(1/varepsilon)$ 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 $sqrt{operatorname{tr}(ρσ^2)}$ 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.

Problem

• Estimating fidelity between pure and mixed quantum states
• Achieving quadratic speedup in fidelity estimation
• Measuring uncommon quantity sqrt(tr(ρσ²)) simply

Innovation

• Optimal quantum algorithm for fidelity estimation
• Quadratic speedup with Θ(1/ε) queries
• Estimates √tr(ρσ²) with simple technique