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 $\mathscr E_σ$ ($σ=X,Z$). Their cohomology layers $E_σ^\ell:=H^\ell(\mathscr E_σ)$ encode excitation types, fusion rules, and translation action, while $\mathscr E_σ$ retains cross-degree gluing. Pointlike, looplike, and higher-dimensional excitations correspond to degrees $\ell=1,2,\ldots$, respectively. Our matrix-level Schanuel-lemma method proves single-degree superselection rigidity: if $\mathscr E_σ$ is supported in a single degree $\ell$, then $E_σ^\ell$ and $\ell$ fix the stable translation-invariant Clifford class under a finite-resolution hypothesis automatic for prime qudits. If $E_σ^\ell$ 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 $E_σ^\ell$, prove $\mathscr E_σ$ is single-degree, express finite-torus logical modules as $\mathrm{Tor}$ 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.