Geometric Classification of Biased Quantum Capacity via Harmonic Translation

• URL: http://arxiv.org/abs/2603.22336v1
• Date: Sat, 21 Mar 2026 04:17:42 GMT
• Title: Geometric Classification of Biased Quantum Capacity via Harmonic Translation
• Abstract summary: Under uniform locality, the maximal logical dimension under t-local phase errors equals Aq(n,2t+1)>Bq(n,2t+1)<n>For structured phase noise, exact correction is equivalent to independence in an additive Cayley graph.<n>For mixed Pauli noise, simultaneous protection in conjugate domains incurs an intrinsic rate penalty R = 1-(gamma_X+gamma_Z)/2.
• Score: 0.20524609401792393
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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