Mixed-state escape bounds and proper agnostic learning of stabilizer states in odd-prime dimensions
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Abstract
For every odd prime p, we give a proper agnostic learner for arbitrary mixed states on n qudits. Given a stabilizer-overlap promise F ≥ τ, the learner returns a pure stabilizer of overlap at least F − ε, with confidence 1 − δ, using exp(O(log²(2/τ))) poly(n,p,ε⁻¹,log(2/δ)) resources. The exponential constants and polynomial degree are independent of p. The access model uses independent input copies, exact Clifford operations and coherent Weyl eigenspace measurements. The proof combines a mixed-state escape estimate with a constant-threshold spectral test and selection of an observed eigenvalue. An even quotient-convolution argument gives the escape bound (p−1)F^r/(4p), with r=6 for p≡3 (mod4) and r=4 for p≡1 (mod4), through known weighted stabilizer measurements. Explicit rational counters bound every conditioning attempt and the final validation. These ingredients extend stabilizer bootstrapping with an absolute quasipolynomial inverse-overlap exponent and fixed-polynomial local-dimension dependence.
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Research details
- Subjects
- Mathematics · Quantum Information Theory
- Keywords
- stabilizer states, agnostic learning, quantum learning, mixed states, odd-prime dimensions
- Manuscript date
- 2026-10-19
- Reader reuse
- No additional permission granted
Scope
Proper agnostic learning for arbitrary mixed inputs in every odd-prime local dimension with dimension-uniform exp(O(log²(2/τ))) resource exponents, and stronger quotient-convolution escape, in the stated exact Clifford/coherent-measurement model.
Disclosures
AI assistance supported derivation, programming, literature discovery and proof checking. The author is responsible for the analytic arguments and supporting computations. The manuscript credits the prior measurement constructions, uncertainty ingredients, convolution tools and stabilizer-bootstrapping framework where they are used.
Reproduction
The 11-page manuscript and fourteen-member source package include the complete analytic proof, editable LaTeX source, citation metadata, verification programs and an exact member checksum manifest. Compile manuscript/main.tex. Reproducibility.md describes Python/NumPy integrity checks, finite diagnostic replay and robustness controls using fresh external output directories.