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Sharp noise transitions for an adaptive single-copy stabilizer-learning process

Scott Schubr ORCID

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Abstract

We determine the noise-dependent behavior of the adaptive single-copy stabilizer-learning update of Bittel, Eisert, Gong, Mele and Schatzki on globally depolarized stabilizer states. For every fixed prime local dimension p, the number of diagonal stabilizers still missing follows an exact birth-death chain. Starting with all n diagonal stabilizers missing and writing c for the retained pure-state weight, the expected time to record the optimizing stabilizer is linear in n when c^2>p/(p+1), quadratic at equality, and exponential when c^2<p/(p+1). We obtain the exact supercritical speed, subcritical exponential prefactors, and the complete 1/n critical-window mean profile. An exact regeneration identity distinguishes first exposure of the computational-basis target from its actual recording. A finite-time bound shows that below threshold the unchanged algorithm has exponentially small success probability in any polynomial round budget, even on this simple noise family. The analysis separates the first hitting time from the actual recording time, retaining their dependence.

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Edition files

Available editions

  1. Edition 3 · 2026-11-02 · public
  2. Edition 2 · 2026-10-27 · public
  3. Edition 1 · 2026-10-22 · public

Research details

Subjects
Mathematics · Quantum Information Theory
Keywords
stabilizer learning, adaptive measurements, global depolarizing noise, birth–death chains, critical windows, quantum learning
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Exact rank and candidate-recording laws for the specified unchanged adaptive single-copy update under global depolarization, with sharp transition, critical window and finite-time barrier from initial missing rank n.

Disclosures

OpenAI Codex assisted with derivation, programming, literature comparison, proof checking and manuscript preparation. The author takes responsibility for the mathematics and accompanying files.

Reproduction

Eleven-page manuscript and nine-member package containing nested TeX source, portable fixed-date Tectonic build, exact Python controls, expected output, canonical citations and checksums. The packaged source reproduces the supplied PDF exactly. Run verify.py with ordinary assertions enabled; optimized execution deliberately rejects. Finite controls exhaust 632 unsigned Lagrangian spaces, 158490 local gate updates and 1896 mixed transition models, and solve 192 hitting-time and 128 candidate-recording systems. General and asymptotic claims rest on analytic proofs. Archive edition 2.

References and related resources

  1. L. Bittel, J. Eisert, W. Gong, A. A. Mele and L. Schatzki. Adaptivity is all you need: Optimal stabilizer learning using just single-copy measurements. arXiv:2610.02031v1 (2026).
  2. S. Chen, W. Gong, Q. Ye and Z. Zhang. Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation. STOC 2025, 429-438.
  3. W. Qi, B. Xu, S. Feng and L. Li. Agnostic learning of qudit stabilizer states. arXiv:2607.15559v1 (2026).
  4. S. Schubr. Mixed-state escape bounds and proper agnostic learning of stabilizer states in odd-prime dimensions. Luminary Research Archive, edition 3 (2026).
  5. L. Tang and T. Steckmann. Learning Noise-Robust Stabilizer Structure via Bell Sampling. arXiv:2610.03307v2 (2026).
  6. G. Cho and D. Kim. Single-copy stabilizer learning: average case and worst case. arXiv:2604.24099v1 (2026).
  7. A. Chen, A. De and A. Vijayaraghavan. Learning a mixture of two subspaces over finite fields. arXiv:2010.02841v2 (2021).