Exact Critical Curve for Uniform Stabilizer-State Identification
Abstract
The sample complexity of identifying an unknown pure stabilizer state has been a longstanding problem in quantum information. In this paper, we resolve the entire exact critical crossover for the uniform ensemble of pure n-qubit stabilizer states under arbitrary collective measurements and without side information. For k=n+c, with fixed integer c, the optimal success probability converges to an explicit positive series P∞(c). In particular, P∞(0)=0.1760837896…, while its two tails obey a quadratic-exponential law below threshold and 1−P∞(c)∼2−c above threshold. The pretty-good measurement is exactly optimal for every finite n and k. The proof diagonalizes the stabilizer Gram operator through Clifford tensor-power duality and a defect filtration, including the exceptional binary sectors. The resulting integer-scale law quantifies the n+O(1) copy threshold.
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BibTeX
@misc{hayashi2026exact,
title = {{Exact Critical Curve for Uniform Stabilizer-State Identification}},
author = {Masahito Hayashi and Yimin Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00266},
eprint = {2610.00266},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00266},
}
RIS
TY - UNPB
TI - Exact Critical Curve for Uniform Stabilizer-State Identification
AU - Masahito Hayashi
AU - Yimin Lu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00266
UR - https://arxiv.org/abs/2610.00266
AB - The sample complexity of identifying an unknown pure stabilizer state has been a longstanding problem in quantum information. In this paper, we resolve the entire exact critical crossover for the uniform ensemble of pure n-qubit stabilizer states under arbitrary collective measurements and without side information. For k=n+c, with fixed integer c, the optimal success probability converges to an explicit positive series P_∞(c). In particular, P_∞(0)=0.1760837896…, while its two tails obey a quadratic-exponential law below threshold and 1−P_∞(c)∼2^−c above threshold. The pretty-good measurement is exactly optimal for every finite n and k. The proof diagonalizes the stabilizer Gram operator through Clifford tensor-power duality and a defect filtration, including the exceptional binary sectors. The resulting integer-scale law quantifies the n+O(1) copy threshold.
ER -