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 $P_{\infty }(c)$. In particular, $P_{\infty }(0)=0.1760837896\dots$, while its two tails obey a quadratic-exponential law below threshold and $1-P_{\infty }(c)\sim 2^{-c}$ 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.

arXiv commentRelated arXiv paper: arXiv:2609.13923,"Partial Stabilizer Learning under Fixed Commuting Constraints and the Absence of a Copy-Rate Discount,"establishes the first-order copy threshold for unrestricted stabilizer-state identification, but leaves the fixed-offset regime k=n+c untreated. This submission resolves that regime by determining the limiting optimal success probability