Sharp noise transitions in adaptive single-copy stabilizer learning
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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. Writing c for the retained pure-state weight, the expected time to record the optimizing stabilizer is linear in the number n of qudits when c² > p/(p+1), quadratic at equality, and exponential when c² < 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 obstruction concerns this update and candidate generation rule.
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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
The theorems analyze the specified Bittel et al. adaptive update and its candidate-recording rule on globally depolarized stabilizer inputs, with ideal operations and fixed prime local dimension. The lower obstruction starts at missing-diagonal rank n; some targets begin at lower rank and are easier. The threshold is in retained signal weight c²; finite-n fidelity is c+(1−c)p^(−n). Candidate generation and fresh-copy fidelity validation have separate costs.
Disclosures
AI tools assisted derivation, programming, literature discovery and proof checking, as disclosed in the manuscript. This is an author-published preprint. The paper credits the adaptive algorithm of Bittel et al. and distinguishes its analysis from prior multi-copy learning results.
Reproduction
The six-member package contains the 11-page paper, self-contained LaTeX source with bibliography, README.txt, verify.py, expected_checks.json and SHA256SUMS.txt. Run python verify.py using Python 3.10 or later with ordinary assertions enabled. The exact controls cover 632 small prime-field stabilizer spaces, 158490 gate updates, 1896 mixed-transition models, 192 hitting-time systems and 128 candidate-time systems. These finite controls supplement the analytic all-dimension proofs. Source compilation instructions are in the README.