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Optimal Trace Distance and Fidelity Estimations for Pure Quantum States

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arxiv 2408.16655 v1 pith:DBZKKQLO submitted 2024-08-29 quant-ph cs.CCcs.DScs.ITmath.IT

Optimal Trace Distance and Fidelity Estimations for Pure Quantum States

classification quant-ph cs.CCcs.DScs.ITmath.IT
keywords quantumstatesvarepsilonamplitudedistanceestimationfidelityoptimal
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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.

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  1. On estimating operator norm distance, with optimal trace distance estimation when one state is pure

    quant-ph 2026-07 accept novelty 7.0

    Rank-independent quantum estimators achieve Θ(1/ε) queries for operator-norm (and trace) distance when one state is pure, and Õ(1/ε^{3/2}) queries for general states, proving BQP-completeness.