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

Understand

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 $\Theta(1/\varepsilon^2)$, improving the long-standing folklore with sample complexity $O(1/\varepsilon^4)$.