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arXiv · 2410.21201

Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer

Qisheng Wang·Zhicheng Zhang
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Abstract

We settle the problem of estimating the trace distance and (square root) fidelity between $n$-qubit pure quantum states to within additive error $\varepsilon$, given their independent samples, which was raised as an open question by Wang (IEEE Trans. Inf. Theory 2024). This is achieved by a quantum algorithm with optimal sample complexity $Θ(1/\varepsilon^2)$, improving the long-standing folklore with sample complexity $O(1/\varepsilon^4)$. At the heart of our algorithm is a samplized phase estimation of the product of two Householder reflections. This is realized by an improved (multi-)samplizer for pure states, through which any quantum query algorithm using $Q$ queries to the reflection operator $I - 2|ψ\rangle\!\langleψ|$ can be converted to a $δ$-close (in the diamond norm distance) quantum sample algorithm using $Θ(Q^2/δ)$ samples of the state $|ψ\rangle$. This samplizer for pure states is also shown to be optimal.

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Qisheng Wang, Zhicheng Zhang. 2026-07-06. Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer. https://doi.org/10.4230/lipics.icalp.2026.154

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