Superselection Rigidity of Topological Calderbank--Shor--Steane Codes: From Koszul-Complex Models to Nonsplit Extensions
Abstract
Can excitation structure classify topological quantum codes up to finite-depth local circuits? For translation-invariant topological Calderbank--Shor--Steane (CSS) codes in arbitrary dimensions, we introduce excitation superselection profiles ℰσ (σ=X,Z). Their cohomology layers Eσℓ:=Hℓ(ℰσ) encode excitation types, fusion rules, and translation action, while ℰσ retains cross-degree gluing. Pointlike, looplike, and higher-dimensional excitations correspond to degrees ℓ=1,2,…, respectively. Our matrix-level Schanuel-lemma method proves single-degree superselection rigidity: if ℰσ is supported in a single degree ℓ, then Eσℓ and ℓ fix the stable translation-invariant Clifford class under a finite-resolution hypothesis automatic for prime qudits. If Eσℓ is also finite, prime-qudit codes decouple into toric-code stacks by Clifford circuits after coarse graining, with the exact minimal threshold set by the translation kernel. We develop a unified topological theory of Koszul-complex models (encompassing toric codes, bivariate-bicycle codes, and Haah's cubic code): for regular models, we compute Eσℓ, prove ℰσ is single-degree, express finite-torus logical modules as Tor groups, and obtain coarse-grained toric-stack normal forms at full length. Two constructions extend this framework to realize any admissible single-degree profile and build multi-degree extensions. The latter yields eight 3D qubit toric-code extensions with identical layers in both Pauli sectors and trivial translation action, yet distinct size-dependent ground-state degeneracies. The seven nonsplit models require blocking to decouple. Thus, cross-degree gluing carries translation-symmetry-enriched topological information invisible to individual layers.
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BibTeX
@misc{song2026superselection,
title = {{Superselection Rigidity of Topological Calderbank--Shor--Steane Codes: From Koszul-Complex Models to Nonsplit Extensions}},
author = {Hao Song},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.09913},
eprint = {2608.09913},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09913},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Superselection Rigidity of Topological Calderbank--Shor--Steane Codes: From Koszul-Complex Models to Nonsplit Extensions
AU - Hao Song
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2608.09913
UR - https://arxiv.org/abs/2608.09913
AB - Can excitation structure classify topological quantum codes up to finite-depth local circuits? For translation-invariant topological Calderbank--Shor--Steane (CSS) codes in arbitrary dimensions, we introduce excitation superselection profiles ℰ_σ (σ=X,Z). Their cohomology layers E_σ^ℓ:=H^ℓ(ℰ_σ) encode excitation types, fusion rules, and translation action, while ℰ_σ retains cross-degree gluing. Pointlike, looplike, and higher-dimensional excitations correspond to degrees ℓ=1,2,…, respectively. Our matrix-level Schanuel-lemma method proves single-degree superselection rigidity: if ℰ_σ is supported in a single degree ℓ, then E_σ^ℓ and ℓ fix the stable translation-invariant Clifford class under a finite-resolution hypothesis automatic for prime qudits. If E_σ^ℓ is also finite, prime-qudit codes decouple into toric-code stacks by Clifford circuits after coarse graining, with the exact minimal threshold set by the translation kernel. We develop a unified topological theory of Koszul-complex models (encompassing toric codes, bivariate-bicycle codes, and Haah's cubic code): for regular models, we compute E_σ^ℓ, prove ℰ_σ is single-degree, express finite-torus logical modules as Tor groups, and obtain coarse-grained toric-stack normal forms at full length. Two constructions extend this framework to realize any admissible single-degree profile and build multi-degree extensions. The latter yields eight 3D qubit toric-code extensions with identical layers in both Pauli sectors and trivial translation action, yet distinct size-dependent ground-state degeneracies. The seven nonsplit models require blocking to decouple. Thus, cross-degree gluing carries translation-symmetry-enriched topological information invisible to individual layers.
ER -