Fidelity Estimation to a Known Quantum State Is Nearly Quadratic in the Smaller Rank
Abstract
We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most s, we give an estimator using O(s2/ε2) copies, uniformly in the ambient dimension and reference rank. The estimator applies a random-purification channel and a covariant pure-state measurement, then rescales the observed amplitude before evaluating a weighted nuclear norm. Combined with the lower bound established independently in our first version and in concurrent work of Wang, this determines the worst-case complexity under rank bounds r,s as min{r,s}2/ε2 up to logarithmic factors in the smaller rank. The same estimator gives the upper bound O(r(tr√σ)2/ε2) for a rank-r reference σ. We combine spectral truncation with lower bounds obtained by embedding hard instances for the maximally mixed reference into spectral subspaces. For spectra λi∝ i−α with fixed 1<α≤2, as ε↓0 with r≥ Cαε−2/(α−1), the bounds determine the complexity as Θ̃α(ε−4/(α−1)). In particular, inverse-square spectra have accuracy exponent four. The lower-bound construction uses exact moment matching and an explicitly computable Schur measure.
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BibTeX
@misc{lee2026fidelity,
title = {{Fidelity Estimation to a Known Quantum State Is Nearly Quadratic in the Smaller Rank}},
author = {Gye Jin Lee and Sunghyeon Jo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.01770},
eprint = {2608.01770},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01770},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fidelity Estimation to a Known Quantum State Is Nearly Quadratic in the Smaller Rank
AU - Gye Jin Lee
AU - Sunghyeon Jo
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2608.01770
UR - https://arxiv.org/abs/2608.01770
AB - We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most s, we give an estimator using O(s^2/ε^2) copies, uniformly in the ambient dimension and reference rank. The estimator applies a random-purification channel and a covariant pure-state measurement, then rescales the observed amplitude before evaluating a weighted nuclear norm. Combined with the lower bound established independently in our first version and in concurrent work of Wang, this determines the worst-case complexity under rank bounds r,s as min{r,s}^2/ε^2 up to logarithmic factors in the smaller rank. The same estimator gives the upper bound O(r(tr√σ)^2/ε^2) for a rank-r reference σ. We combine spectral truncation with lower bounds obtained by embedding hard instances for the maximally mixed reference into spectral subspaces. For spectra λ_i∝ i^−α with fixed 1<α≤2, as ε↓0 with r≥ C_αε^−2/(α−1), the bounds determine the complexity as Θ̃_α(ε^−4/(α−1)). In particular, inverse-square spectra have accuracy exponent four. The lower-bound construction uses exact moment matching and an explicitly computable Schur measure.
ER -