Entropic Rigidity in Quantum Memories: How Geometry and Algebra Control the Onset of Degeneracy Corrections

• URL: http://arxiv.org/abs/2608.18420v1
• Date: Wed, 19 Aug 2026 01:17:59 GMT
• Title: Entropic Rigidity in Quantum Memories: How Geometry and Algebra Control the Onset of Degeneracy Corrections
• Abstract summary: We determine the first physical error weight $m$ at which their logical winner sets become disjoint.<n>The onset fixes the leading operational failure gap, proportional to the $m$th power of the physical noise strength.
• Score: 4.226475360842309
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Maximum-probability (MP) decoding selects the most probable microscopic error, whereas degenerate maximum-likelihood (MLD) decoding includes the configurational entropy of an entire logical sector. Using code-capacity Pauli noise to isolate rigidity intrinsic to the code, we determine the first physical error weight $m$ at which their logical winner sets become disjoint, even under optimal MP tie resolution. Code distance imposes the universal bound $m\geq h=\lceil d/2\rceil$. We define the entropic rigidity depth $r$ through $m=h+r$ and certify a three-level hierarchy: $r=0$ for planar surface codes and two concatenated families, $r=1$ for odd-distance square toric codes and the Gross $[[144,12,12]]$ quantum low-density-parity-check code, and $r=2$ for a separable family with hypergraph product and bivariate bicycle descriptions. The onset fixes the leading operational failure gap, proportional to the $m$th power of the physical noise strength. Geometry and algebra therefore provide quantifiable controls of configurational entropy and an exact benchmark for low-noise decoder selection.

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