Exact Critical Curve for Uniform Stabilizer-State Identification

arXiv:2610.00266v1 Announce Type: new 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, $P_{\infty}(0)=0.1760837896\ldots$, while its two tails obey a quadratic-exponential law below threshold and $1-P_{\infty}(c)\sim2^{-c}$ 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.

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, $P_{\infty}(0)=0.1760837896\ldots$, while its two tails obey a quadratic-exponential law below threshold and $1-P_{\infty}(c)\sim2^{-c}$ 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.

Related 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

arXiv:2610.00266 [quant-ph]

(or arXiv:2610.00266v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.00266 arXiv-issued DOI via DataCite (pending registration)

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Exact Critical Curve for Uniform Stabilizer-State Identification

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