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arXiv:2609.13923

Partial Stabilizer Learning under Fixed Commuting Constraints and the Absence of a Copy-Rate Discount

Masahito Hayashi, Yimin Lu

quant-phSubmitted Sep 12, 2026• Announced Sep 15, 2026• 0 citations• Revised Oct 16, 2026 (v2)
v1 · Sep 12, 2026v2 · Oct 16, 2026v3 · Oct 28, 2026
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Abstract

We consider an unknown pure stabilizer state of \(n\) qudits with a fixed prime local dimension \(p\), together with prescribed commuting Pauli observables. Rather than learning the whole state, the learner must recover only complementary stabilizers that commute with the prescribed observables, together with their eigenvalues. Once the prescribed measurement is performed, this information and the observed outcome determine the corresponding conditional stabilizer state. We compare the number \(k\) of copies available to the learner with the number \(n\) of qudits and refer to their ratio \(k/n\) as the copy rate. Suppose that the dimension \(m\) of the requested complementary stabilizer subspace satisfies \(m=βn+o(n)\) for a fixed \(0<β\leq1\). If the copy rate converges to a value below one, the optimal exact recovery probabilities and verification scores converge to zero, both in the worst case and under the uniform prior. Conversely, if the number \(k\) of copies minus the number \(n\) of qudits tends to positive infinity, the entire stabilizer state can be identified with probability tending to one, and the requested complementary information can then be extracted. Thus, restricting the learning target to the stabilizer information needed after the prescribed measurement yields no copy-rate discount.

Comments
A Lean 4 formalization of our main theorem, Theorem 2.2, is available. The formalization, verification records, and reproducibility instructions are publicly available in release v1.0.0 of the accompanying repository: https://github.com/Imine20/Lean-formalization-for-stabilizer-state-identification/releases/tag/v1.0.0

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We consider an unknown pure stabilizer state of (n) qudits with a fixed prime local dimension (p), together with prescribed commuting Pauli observables. Rather than learning the whole state, the learner must recover only complementary stabilizers that commute with the prescribed observables, together with their eigenvalues. Once the prescribed measurement is performed, this information and the observed outcome determine the corresponding conditional stabilizer state. We compare the number (k) of copies available to the learner with the number (n) of qudits and refer to their ratio (k/n) as the copy rate.
Suppose that the dimension (m) of the requested complementary stabilizer subspace satisfies (m=βn+o(n)) for a fixed (0<β\leq1). If the copy rate converges to a value below one, the optimal exact recovery probabilities and verification scores converge to zero, both in the worst case and under the uniform prior. Conversely, if the number (k) of copies minus the number (n) of qudits tends to positive infinity, the entire stabilizer state can be identified with probability tending to one, and the requested complementary information can then be extracted.
Thus, restricting the learning target to the stabilizer information needed after the prescribed measurement yields no copy-rate discount.
We consider the problem of testing whether an unknown n-qubit quantum state is a stabilizer state, with only single-copy access. We give an algorithm solving this problem using O(n) copies, and conversely prove that Ω(√n) copies are required for any algorithm. The main observation behind our algorithm is that when repeatedly measuring in a randomly chosen stabilizer basis, stabilizer states are the most likely among the set of all pure states to exhibit linear dependencies in measurement outcomes. Our algorithm is designed to probe deviations from this extremal behavior. For the lower bound, we first reduce stabilizer testing to the task of distinguishing random stabilizer states from the maximally mixed state. We then argue that, without loss of generality, it is sufficient to consider measurement strategies that a) lie in the commutant of the tensor action of the Clifford group and b) satisfy a Positive Partial Transpose (PPT) condition. By leveraging these constraints, together with novel results on the partial transposes of the generators of the Clifford commutant, we derive the lower bound on the sample complexity.
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