Luminary Research Archive
Luminary Research ArchiveIndependent scholarship
Luminary Portal
PublicEdition 2

Single-copy learning of stabilizers with correlated diagonal noise

Scott Schubr

Original PDF · First-page preview

Loading original first page…

Abstract

We study exact identification of an unknown qubit stabilizer from a state diagonal in its stabilizer eigenbasis. A positive lower bound on every target-signed stabilizer expectation unifies global white noise, strict-majority diagonal mixtures, and additional nonmajority correlated mixtures. For a supplied gap μ_0≥2^(-(log_2 n)^γ), fixed γ<1, recovery uses O(n^(1+ε)) original copies at fixed confidence, polynomial ideal quantum work, and 2^(O(n/log log n)) classical time. The new step is a uniformly spread successful component in a learned, verified Pauli direction, giving contraction of surviving error odds without independent syndrome errors. We also give polynomial-time recovery when the background is an unknown target-containing stabilizer code projector and a relative diagonal-visibility promise holds. Dense CSS targets with sufficiently small background-code rank are included. Known noisy-parity solvers, polynomial subspace lifts and high-fidelity learning are explicit ingredients. A bounded-weight local-Pauli corollary supplies another polynomial regime.

Read the full paper in the original PDF.

Edition files

Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-24 · public

Research details

Subjects
Mathematics · Quantum Information Theory
Keywords
stabilizer learning, correlated diagonal noise, quantum state learning, noisy parity, positive spectrum
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

Single-copy stabilizer identification under a target-signed expectation gap for correlated diagonal noise, with near-linear copy/subexponential classical bounds and separate polynomial-time relative-visibility regimes.

Disclosures

AI systems assisted research exploration, literature discovery, drafting and internal checking. The author retains responsibility for the manuscript. Internal AI-assisted and computational checks are distinct from external human peer review.

Reproduction

10-page manuscript and 7 source-package members. Compile main.tex. Python 3.10+ and its standard library run python verify.py, also under -O; compare its parsed JSON with expected_checks.json for the finite stabilizer/filtering controls. These fixtures accompany the analytic learning bounds.

References and related resources

  1. S. Arunachalam, S. Bravyi, A. Dutt and T. J. Yoder, Optimal algorithms for learning quantum phase states , TQC 2023, LIPIcs 266, 3:1–3:24. doi:10.4230/LIPIcs.TQC.2023.3. Full version: arXiv:2208.07851v2.
  2. V. Lyubashevsky, The parity problem in the presence of noise, decoding random linear codes, and the subset sum problem , APPROX/RANDOM 2005, LNCS 3624, 378–389. doi:10.1007/11538462_32. Author manuscript.
  3. 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).
  4. A. Chen, A. De and A. Vijayaraghavan, Learning a mixture of two subspaces over finite fields , ALT 2021, PMLR 132, 481–504. Proceedings paper . Full version: arXiv:2010.02841v2.
  5. I. Ben-Eliezer, R. Hod and S. Lovett, Random low-degree polynomials are hard to approximate , Computational Complexity 21 (2012), 63–81.
  6. P. Keevash and B. Sudakov, Set systems with restricted cross-intersections and the minimum rank of inclusion matrices , SIAM Journal on Discrete Mathematics 18 (2005), 713–727.
  7. R. Koenig and J. A. Smolin, How to efficiently select an arbitrary Clifford group element , Journal of Mathematical Physics 55 (2014), 122202. doi:10.1063/1.4903507. arXiv:1406.2170.
  8. S. Arunachalam and L. Schatzki, Optimal stabilizer testing and learning with limited quantum memory, arXiv:2607.02444v2 (2026).
  9. L. Tang and T. Steckmann, Learning noise-robust stabilizer structure via Bell sampling , arXiv:2610.03307v2 (2026).
  10. S. Schubr, Single-copy learning of noisy stabilizers by certified filtering , Luminary Research Archive, edition 1 (2026). Public predecessor .
  11. S. Chen, W. Gong, Q. Ye and Z. Zhang, Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation , arXiv:2408.06967 (2024), STOC 2025.
  12. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Mixed-state entanglement and quantum error correction , Phys. Rev. A 54 (1996), 3824–3851. quant-ph/9604024.
  13. H.-Y. Huang, R. Kueng and J. Preskill, Predicting many properties of a quantum system from very few measurements , Nature Physics 16 (2020), 1050–1057. arXiv:2002.08953.
  14. Y. Ouyang and M. Tomamichel, Learning quantum graph states with product measurements , arXiv:2205.06432 (2022).
  15. H. Leitch, S. Tirukkovalluri and Y. Ouyang, Learning unknown stabilizer codes using product measurements, arXiv:2609.04997v2 (2026).