Geometric Classification of Biased Quantum Capacity via Harmonic Translation

TL;DR

This paper characterizes quantum error correction under diagonal local phase noise using harmonic translation, revealing a deep connection to classical zero-error information theory and establishing bounds on quantum capacity.

Contribution

It introduces a harmonic translation principle for phase noise, providing a novel, model-based characterization of quantum error correction without relying on stabilizer structures.

Findings

Maximal logical dimension equals classical q-ary packing function under uniform locality.

Exact correction relates to independence in additive Cayley graphs for structured phase noise.

Protection in conjugate domains under mixed Pauli noise incurs a rate penalty, revealing a harmonic uncertainty principle.

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.