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. Its original-copy bound is independent of p: at fixed τ and δ, it is O_(τ,δ)(n log²(n+1)+ε^(-2)). Gates and classical time are bounded by exp(O(log²(2/τ)))poly(n,p,ε^(-1),log(2/δ)), with absolute exponential constants and fixed polynomial degree. 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. Isotropic rank growth captures the relevant labels, while a distribution-free empirical-CDF bound estimates their spectral masses uniformly in p. An even quotient-convolution argument gives escape at least (p-1)F^r/(4p), with r=6 for p≡3 (mod 4) and r=4 for p≡1 (mod 4), through known weighted stabilizer measurements. Explicit rational counters account for conditioning, rejection and final validation.
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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
- none
Scope
Proper agnostic stabilizer learning for arbitrary mixed states in every odd-prime local dimension. Original-copy complexity is independent of the prime at fixed promise and confidence; gate and classical costs remain dimension-dependent in the specified exact Clifford/coherent-measurement model.
Disclosures
Originality is qualified against the inspected literature; no worldwide priority, optimal-resource or practical-efficiency claim is made. Known weighted/skewed measurements, convolution tools and stabilizer bootstrapping are credited. Exact Clifford and coherent eigenspace operations are assumed; no noisy or compiled finite-gate theorem is proved. AI assistance was used in derivation, programming, literature discovery and proof checking. Internal reviews do not constitute external human refereeing or formal verification. Personal explanations and internal review records are excluded from the public package.
Reproduction
The 12-page manuscript and complete editable source are supplied in a 14-member public package. README.md gives build instructions, dependencies and reproduction scope. Build reproduction is qualified by the compiler, support files and timestamp settings specified in the supplied instructions. The mathematical conclusions have analytic proofs; accompanying finite checks, where present, retain the limited scopes stated in the package.