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Geometric Classification of Biased Quantum Capacity via Harmonic Translation

Eliseo Sarmiento Rosales, Egor Maximenko, Dionisio Manuel Tun Molina, Juan Carlos Jimenez Cervantes, Jose Alberto Guzman Vega, Rodrigo Leon Morales

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.

Geometric Classification of Biased Quantum Capacity via Harmonic Translation

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.
Paper Structure (50 sections, 27 theorems, 111 equations, 1 figure)

This paper contains 50 sections, 27 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation: Phase-Biased Noise and Structural Freedom
  3. Main Contributions
  4. Related Work
  5. Structure of the Paper
  6. Harmonic Translation and Exact Characterization
  7. Finite Abelian Harmonic Framework
  8. Phase Errors as Spectral Translations
  9. Fourier-Support Codes and Exact Detection
  10. Classical Reduction and Transfer Principle
  11. Equivalence with Hamming Distance
  12. Classical Transfer and Decoding Equivalence
  13. Beyond Affine Supports: Dimensional Separation
  14. Additive (Affine) Spectral Supports
  15. Strict Separation Theorem
  16. ...and 35 more sections

Key Result

Lemma 2.2

For all $s,\omega\in V$,

Figures (1)

  • Figure 1: Geometric classification of biased quantum capacity (Theorem \ref{['thm:geometric_classification']}). The achievable logical dimension is governed by the additive geometry of the noise difference set $D_\Omega=(\Omega-\Omega)\setminus\{0\}$. Dispersive noise without additive structure yields classical packing capacity, additive subspaces induce dimensional collapse, and simultaneous bit--phase protection produces a harmonic rate tradeoff.

Theorems & Definitions (57)

  • Definition 2.1: Quantum Fourier Transform over $V$
  • Lemma 2.2: Spectral Translation
  • proof
  • Definition 2.3: Fourier-Support Code
  • Theorem 2.4: Knill--Laflamme KnillLaflamme1997
  • Theorem 2.5: Exact Harmonic Non-Collision
  • proof
  • Corollary 2.6: Exact Phase Correction
  • proof
  • Theorem 3.1: Distance Equivalence
  • ...and 47 more