From Koszul-Complex Stabilizer Models to Superselection Profiles: Topological Rigidity and Nonsplit Extensions

AI summary

Overview

Research area: Quantum error correction and topological quantum matter — specifically translation-invariant Calderbank–Shor–Steane (CSS) stabilizer codes, studied with homological algebra over Laurent polynomial rings.

Technical level: Advanced. The paper is written for readers comfortable with commutative algebra (Koszul complexes, regular sequences, Ext/Tor, free resolutions, Schanuel's lemma), CSS stabilizer formalism, and the classification of symmetry-enriched topological (SET) phases.

Scope (one sentence): The paper defines a "topological superselection profile" built from Ext modules of a CSS code's stabilizer maps, computes it explicitly for regular Koszul-complex stabilizer models (which include toric codes and bivariate-bicycle codes), and proves a rigidity theorem showing when a single nonzero layer fixes the translation-invariant Clifford class — together with eight explicit counterexample models where it does not.

What This Paper Is About

Translation-invariant stabilizer codes can be described algebraically by polynomial matrices over a Laurent polynomial ring R = ℤ_N[x_1^{±1}, …, x_D^{±1}], where multiplication by a monomial encodes a lattice translation. The paper asks how much of a code's translation-symmetry-enriched topological (SET) order is already determined by its excitation data — the sectors, their fusion, and the translation action on them. To answer this, the author packages all positive-degree Ext layers of the two stabilizer maps into a single derived object (the topological superselection profile), computes it for a large family of models built from Koszul complexes, and then tests the limits of the resulting rigidity statement by explicitly constructing models that defeat it.

Key Contributions

1. A central new definition. Definition 8 introduces the topological superselection profile 𝒮_σ ≔ τ_{≥1} R Hom_R(conj(coker φ_σ), R) for σ = X, Z, whose cohomology layers E_σ^ℓ ≅_R Ext_R^ℓ(conj(coker φ_σ), R) encode sectors, fusion, and the lattice-translation action, while the derived structure retains inter-layer gluing. A table (Table 1) supplies an algebra–physics dictionary.

2. General theorems and a new method. Theorem 1 re-derives Ruba and Yang's Ext-based description of superselection sectors directly from support-sensitive CSS stabilizer maps. A matrix-level Schanuel-lemma method, combined with an algebraic–Clifford correspondence, yields Theorem 2: a topological rigidity result for single-layer profiles whose reduced Pauli-X module admits a finite-length free resolution. For prime qudits, Corollaries 3 and 4 give a stable Clifford-equivalence criterion and the exact minimal coarse graining to a toric-code stack.

3. Two constructions. Construction 1 realizes any prescribed single-layer data satisfying the grade condition grade_R(E) > j, proving the grade criterion is not merely necessary but sufficient. Construction 2 produces multi-degree extensions, including nonsplit ones, taking the analysis beyond the rigidity theorem's single-layer regime.

4. Model-specific results. A sequence of Facts establishes the topological and finite-size properties of regular Koszul-complex stabilizer models, gives their full-length toric-stack normal form, distinguishes eight three-dimensional qubit extensions as distinct translation SET phases, and gives a uniform coarse-splitting construction identifying their common intrinsic toric-code order.

Main Findings

• Koszul-complex models unify two code families. The two-polynomial check maps of bivariate-bicycle (BB) codes have the length-two Koszul form, while the higher-dimensional toric-code complexes arise from the coordinate sequence (x_1 − 1, …, x_D − 1). Color codes (D = 2, sequence (1 + x_1 + x_2, 1 + \bar{x_1} + \bar{x_2})) and Haah's code (D = 3, s < D) also fit the framework; Haah's code is the case s < D, which the author links to fracton rather than conventional topological order.

• Regular models have single-layer profiles with explicitly computed layers. For a degree-j regular Koszul-complex model with S ≔ R/(f), all positive-degree superselection layers vanish except Ext_R^j(conj(coker φ_X), R) ≅_R S and Ext_R^{s−j}(conj(coker φ_Z), R) ≅_R \bar{S}. The abelian-group structure of S gives excitation fusion; the R-module structure records the translation action.

• Finite-size logical spaces come from Tor. On the L-periodic torus, ℋ_j^Z(L) ≅_{R_L} Tor_j^R(S, R_L). This yields an all-degree, all-dimensional logical–superselection correspondence, together with duality, Euler, and arithmetic finite-size relations, extending the earlier anyon–logical result for BB codes.

• Rigidity holds in the single-layer, finite-resolution setting. Under the finite-resolution hypothesis, the nonzero layer together with its degree uniquely determines the code's stable translation-invariant Clifford class. For prime qudits the resolution hypothesis is automatic.

• Exact retained-translation threshold. For a prime-qudit code whose X-sector profile is single-layer with finite nonzero layer E, rigidity gives the exact minimal coarse graining to a toric-code stack with trivial retained-translation action: a translation subgroup is compatible precisely when it acts trivially on E. In the overlapping two-dimensional qubit setting this agrees with Wang et al., who additionally give a polynomial-time construction of the decoupling map and bounds on operator spreading.

• Individual layers are not enough — eight counterexamples. Split and nonsplit extensions of 3D qubit toric codes (Construction 2, coupling degree-one and degree-two Koszul models) produce eight models that share identical layers E_σ^ℓ for σ = X, Z and ℓ = 1, 2, all with trivial translation action, yet have distinct size-dependent ground-state degeneracies — hence distinct translation SET phases, inequivalent under finite-depth local unitaries with the original translation symmetry fixed.

• Inter-layer gluing is physically detectable. Because the eight models collapse to the same decoupled toric-code stack only after uniform 2 × 2 × 2 blocking, the gluing data between layers carries SET information invisible to any individual layer E_σ^ℓ.

Methodology in Plain English

The author encodes a translation-invariant stabilizer code as a pair of polynomial matrices (φ_X, φ_Z) over a ring of Laurent polynomials, where multiplying by x_j means "shift one cell in direction j." The commutation of X- and Z-type stabilizers becomes the algebraic condition that two short sequences of modules form chain complexes; the code being topological becomes the statement that these complexes are exact in the middle.

The main tool is the Koszul complex: given a list of polynomials f = (f_1, …, f_s), its exterior-algebra differentials automatically square to zero, so adjacent differentials can be used directly as the two stabilizer maps. When the list is a regular sequence, the Koszul complex is exact everywhere above degree zero, giving a finite free resolution of the module S = R/(f) and guaranteeing the model is topological on the infinite lattice.

Excitation data is then extracted by applying the Ext functor to the cokernel of each stabilizer map. Rather than looking at each Ext group in isolation, the author bundles the positive-degree ones into a single derived object whose internal gluing maps are retained. Finite systems are handled by changing the base ring to impose periodic boundary conditions, which turns Ext into Tor and yields logical operators as Tor modules.

To prove rigidity, the author uses a matrix-level version of Schanuel's lemma: two presentations of the same module can be compared stably, and combined with an algebraic–Clifford dictionary this converts stable module isomorphism into stable Clifford equivalence. Finally, the limits of the theorem are probed constructively — building explicit multi-degree extensions whose layers agree but whose size-dependent degeneracies do not.

Why This Matters

Impact on research. The paper connects two normally separate literatures: the homological-algebra approach to quantum low-density parity-check (qLDPC) code parameters, and the physics classification of symmetry-enriched topological phases. It sharpens the charge-module programme of Ruba and Yang into a rigidity theorem with explicit hypotheses, gives a constructive proof that the necessary grade condition is also sufficient, and — crucially — exhibits concrete models showing that single-layer invariants are incomplete. The eight-model example is a clean warning that any classification based only on per-degree Ext groups will conflate distinct SET phases.

Potential applications (as motivated by the paper's framing; the paper reports no experimental or hardware demonstrations):

• Code design for qLDPC codes. The Koszul framework covers BB, multivariate-bicycle (MB), trivariate-tricycle (TT), generalized-bicycle (GB), and abelian two-block group-algebra (2BGA) codes, giving designers an algebraic handle on which parameters are fixed by excitation data.
• Coarse graining and decoupling algorithms. The retained-translation threshold identifies exactly which translation subgroups permit reduction to a toric-code stack, which is the kind of structural reduction needed before running decoders or Clifford-circuit simulations.
• Size-dependent degeneracy prediction. The Tor formula predicts how logical space grows with torus dimensions L, relevant to choosing lattice sizes in architecture studies.
• Benchmarking SET invariants. The eight-model family is a ready-made test suite for any proposed classifier of translation SET order — a correct classifier must separate them.

Industry relevance. Companies and academic groups developing qLDPC codes for fault-tolerant quantum computing work directly with BB codes and related two-block constructions; this paper supplies the algebraic invariants and the rigidity/obstruction results that constrain what such codes can and cannot be reduced to. The results are theoretical: no hardware, error-rate, or threshold benchmarks are reported.

Future Directions

• Classify multi-layer profiles. Since individual layers E_σ^ℓ fail to distinguish the eight models, the natural next step is a classification that treats inter-layer gluing (the full derived object 𝒮_σ, not just its cohomology) as the fundamental invariant.
• Remove or weaken the finite-resolution hypothesis. Rigidity is proved under a finite-length free resolution assumption that is automatic only for prime qudits; the composite-ℤ_N case, where finite resolutions are not automatic, remains to be understood.
• Extend beyond the single-layer regime. Construction 2 shows nonsplit extensions exist; a general theory of which extensions are realizable, and what SET data each carries, is not developed.
• Connect to L-theoretic classifications. The author cites Ruba–Yang's bulk–boundary work, Geiko–Shuklin's relative L-theory classification of invertible trivial-charge codes, and Yang–Yu's L-theoretic classification of fully mobile prime-qudit codes; reconciling the presentation-level rigidity here with those interface- and coarse-graining-based classifications is an open comparison.

Target Audience

Advanced graduate students and researchers in quantum information theory, topological quantum matter, and algebraic quantum coding. Readers will benefit most if they are already comfortable with CSS stabilizer codes and Laurent polynomial methods, and have some exposure to commutative algebra (Ext, Tor, regular sequences, free resolutions). The paper is not suitable as an introduction to either quantum error correction or homological algebra; it is aimed at specialists who want a rigorous, presentation-level account of when excitation data determines a translation SET phase and when it provably does not.

Authors’ abstract

Koszul-complex stabilizer models unify the toric-code hierarchy and bivariate-bicycle codes in one homological framework. To study translation-symmetry-enriched topological (SET) phases in these models and, more generally, in translation-invariant Calderbank-Shor-Steane (CSS) codes with stabilizer maps $(\varphi_X,\varphi_Z)$, we introduce the associated topological superselection profile $\mathscr S_σ:=τ_{\geq1}\mathbf R\!\operatorname{Hom}_R(\overline{\operatorname{coker}\varphi_σ},R),\ σ=X,Z$. Its cohomology layers $E_σ^\ell:=H^\ell(\mathscr S_σ)\cong_R\operatorname{Ext}_R^\ell(\overline{\operatorname{coker}\varphi_σ},R)$ encode sectors, fusion, and translation action. When finite, the $\ell=1,2,\ldots$ layers describe pointlike, looplike, and higher-dimensional excitations; $\mathscr S_σ$ retains inter-layer gluing. For regular Koszul models, we compute $E_σ^\ell$ explicitly, find $\mathscr S_σ$ single-layer, and obtain finite-size $Z$-logicals from $\operatorname{Tor}$. Single-layer means that at most one positive-degree layer $E_σ^\ell$ is nonzero. Our matrix-level Schanuel-lemma method proves rigidity: under a finite-resolution hypothesis, the nonzero layer and degree uniquely determine a topological CSS code's translation SET order at the stable translation-invariant Clifford level. For prime qudits the hypothesis is automatic, and a finite layer's translation kernel gives the exact minimal coarse graining to a toric-code stack. We realize admissible single-layer data. Beyond this regime, split and nonsplit extensions of 3D qubit toric codes yield eight models sharing all identical $E_σ^\ell$ layers with trivial translation action but having distinct size-dependent ground-state degeneracies, hence distinct translation SET phases. Thus inter-layer gluing carries SET data invisible to individual layers.