Research Article · Preprint Mathematics
Sharp noise transitions in adaptive single-copy stabilizer learning
- Primary subject
- Mathematics
- Subject subfields
- Quantum Information Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
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; it is not a lower bound for arbitrary adaptive single-copy measurements and does not resolve fully agnostic stabilizer learning.
Keywords stabilizer learning · adaptive measurements · global depolarizing noise · birth–death chains · critical windows · quantum learning
Scope of the work
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. The results do not establish a lower bound for arbitrary adaptive single-copy learners, a hardware error threshold, or a solution of fully agnostic stabilizer learning.
Reproduction notes
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.
The manuscript
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Author disclosures
AI tools assisted derivation, programming, literature discovery and proof checking, as disclosed in the manuscript. This is an author-published preprint. AI-assisted internal checking is not external human refereeing or formal proof-assistant verification. The paper credits the adaptive algorithm of Bittel et al. and distinguishes its analysis from prior multi-copy learning results.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Sharp noise transitions in adaptive single-copy stabilizer learning (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/sharp-noise-transitions-adaptive-single-copy-stabilizer-learning/v/1
BibTeX
@techreport{luminary_sharp_noise_transitions_adaptive_single_copy_stabilizer_learning_v1,
title = {Sharp noise transitions in adaptive single-copy stabilizer learning},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/sharp-noise-transitions-adaptive-single-copy-stabilizer-learning/v/1}
}Loading discussion…