Geometric Classification of Biased Quantum Capacity via Harmonic Translation

2026-03-21

AI Summary

This work addresses quantum error correction under diagonal local phase noise by moving beyond the conventional stabilizer formalism. Starting directly from the noise model, it leverages harmonic shifts in the Fourier domain to recast the Knill–Laflamme conditions as additive collision-free constraints, thereby establishing a direct connection to classical $q$-ary packing functions and zero-error information theory. Through harmonic analysis and the discrete uncertainty principle, the study demonstrates that nonlinear spectral support can strictly outperform affine constructions, proving that the maximal logical dimension equals the classical code parameter $A_q(n,2t+1)$; when $A_q > B_q$, nonlinear schemes are provably superior. Furthermore, it links the biased quantum capacity to the Lovász theta function and derives a rate upper bound $R \leq 1 - (\gamma_X + \gamma_Z)/2$ for mixed Pauli noise.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We establish an exact noise-model-derived characterization of quantum error correction under diagonal local phase noise. Under uniform locality, the maximal logical dimension under t-local phase errors equals Aq(n,2t+1), the classical q-ary packing function. Because no affine or stabilizer structure is imposed, nonlinear spectral supports achieve this bound and strictly exceed all affine constructions whenever Aq(n,2t+1)>Bq(n,2t+1). This follows from a harmonic translation principle: diagonal phase operators act as rigid translations in the Fourier domain, reducing the Knill-Laflamme conditions exactly to an additive non-collision constraint (S-S) cap Et={0}. For structured phase noise, exact correction is equivalent to independence in an additive Cayley graph, connecting biased quantum capacity to classical zero-error theory and the Lovasz theta function. Under mixed Pauli noise, simultaneous protection in conjugate domains incurs an intrinsic rate penalty R <= 1-(gamma_X+gamma_Z)/2, exposing a discrete harmonic uncertainty principle. In contrast with stabilizer- or graph-based frameworks, this classical correspondence is derived directly from the phase-noise model itself rather than from an auxiliary algebraic construction.

Problem

• quantum error correction
• biased quantum noise
• phase errors
• quantum capacity
• zero-error theory

Innovation

• harmonic translation
• biased quantum capacity
• nonlinear spectral codes
• zero-error classical correspondence
• discrete harmonic uncertainty principle