Geometric Classification of Biased Quantum Capacity via Harmonic Translation
This provides fundamental theoretical limits for quantum error correction under specific noise models, connecting quantum capacity to classical coding theory.
The paper establishes an exact characterization of quantum error correction under diagonal local phase noise, showing that the maximal logical dimension equals the classical q-ary packing function Aq(n,2t+1) and that nonlinear spectral supports can exceed affine constructions when Aq(n,2t+1) > Bq(n,2t+1). It also reveals an intrinsic rate penalty R ≤ 1-(γ_X+γ_Z)/2 under mixed Pauli noise due to a discrete harmonic uncertainty principle.
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.