Topological Calderbank-Shor-Steane (CSS) codes in more than two spatial dimensions offer features useful for fault-tolerant quantum computation, such as transversal non-Clifford gates and single-shot decoding, but a systematic understanding of them is still lacking. One approach is to decouple a code: map it to a stack of higher-form toric codes by a finite-depth local CSS-preserving Clifford circuit, possibly after adding ancillas. For translation-invariant $p$-ary CSS codes, where $p$ is prime ($p=2$ for qubits), we require the circuit to commute with the translations of a superlattice. In two dimensions, such a decoupling exists once these translations act trivially on every anyon type. In higher dimensions, excitations are sorted by their dimension into superselection layers, which can remain glued together even when translations act trivially on each layer. We show that this gluing is the only remaining obstruction: a code can be decoupled on a superlattice exactly when, after coarse-graining to it, translations act trivially on every layer and the gluing vanishes. Moreover, for a code with finitely many excitation types in each layer and $m$ nonzero layers, once translations act trivially on every layer, further coarse-graining by a factor of $p^{m-1}$ in every direction removes the aforementioned gluing. Physically, the same criterion determines when the code encodes the maximal number of logical qudits on a finite torus. Finally, we give an explicit algorithm that constructs the decoupling as a circuit of SUM (qudit CNOT) and SWAP gates.
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The paper asks when a translation-invariant topological CSS code in $D\geq2$ dimensions is equivalent to a stack of higher-form toric codes by a finite-depth local CSS-preserving Clifford circuit, allowing ancillas and retaining a prescribed superlattice of translations. It treats prime-dimensional qudits and excitation structures organized into superselection layers, including the derived-category data describing inter-layer “gluing.”
The main criterion is necessary and sufficient: decoupling on a superlattice exists exactly when its translations act trivially on every layer and the restricted gluing vanishes. If every layer has finitely many excitation types, a suitable superlattice always exists. Once translations act trivially, further coarse-graining by $p^{m-1}$ in every direction suffices for $m$ nonzero layers. Minimum-index additional coarse-graining is characterized when at most two layers are nonzero, including all three-dimensional cases. On a torus, the encoded-qudit count satisfies $k\leq\sum_{j=1}^{D-1}\binom{D}{j}\dim_{\mathbb F_p}E_X^j$, with equality exactly when decoupling respects its period lattice.
The proofs eliminate extension classes under restriction to coarser lattices, then construct stable module isomorphisms and explicit SUM/SWAP circuits. Circuit synthesis has polynomial cost for fixed $D$ in the dense size of the already coarse-grained complex; this does not bound the coarse-graining index or its computation. An exact-arithmetic implementation is provided.
The construction does not optimize circuit depth, range, or ancilla count. Minimum coarse-graining with three or more layers remains open. Infinite-layer excitation structures, composite qudit dimensions, and non-CSS codes lie outside the result.
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Topological Calderbank-Shor-Steane (CSS) codes in more than two spatial dimensions offer features useful for fault-tolerant quantum computation, such as transversal non-Clifford gates and single-shot decoding, but a systematic understanding of them is still lacking. One approach is to decouple a code: map it to a stack of higher-form toric codes by a finite-depth local CSS-preserving Clifford circuit, possibly after adding ancillas. For translation-invariant p-ary CSS codes, where p is prime (p=2 for qubits), we require the circuit to commute with the translations of a superlattice. In two dimensions, such a decoupling exists once these translations act trivially on every anyon type. In higher dimensions, excitations are sorted by their dimension into superselection layers, which can remain glued together even when translations act trivially on each layer. We show that this gluing is the only remaining obstruction: a code can be decoupled on a superlattice exactly when, after coarse-graining to it, translations act trivially on every layer and the gluing vanishes. Moreover, for a code with finitely many excitation types in each layer and m nonzero layers, once translations act trivially on every layer, further coarse-graining by a factor of pm−1 in every direction removes the aforementioned gluing. Physically, the same criterion determines when the code encodes the maximal number of logical qudits on a finite torus. Finally, we give an explicit algorithm that constructs the decoupling as a circuit of SUM (qudit CNOT) and SWAP gates.
The paper asks when a translation-invariant topological CSS code in D≥2 dimensions is equivalent to a stack of higher-form toric codes by a finite-depth local CSS-preserving Clifford circuit, allowing ancillas and retaining a prescribed superlattice of translations. It treats prime-dimensional qudits and excitation structures organized into superselection layers, including the derived-category data describing inter-layer “gluing.”
The main criterion is necessary and sufficient: decoupling on a superlattice exists exactly when its translations act trivially on every layer and the restricted gluing vanishes. If every layer has finitely many excitation types, a suitable superlattice always exists. Once translations act trivially, further coarse-graining by pm−1 in every direction suffices for m nonzero layers. Minimum-index additional coarse-graining is characterized when at most two layers are nonzero, including all three-dimensional cases. On a torus, the encoded-qudit count satisfies k≤∑j=1D−1(jD)dimFpEXj, with equality exactly when decoupling respects its period lattice.
The proofs eliminate extension classes under restriction to coarser lattices, then construct stable module isomorphisms and explicit SUM/SWAP circuits. Circuit synthesis has polynomial cost for fixed D in the dense size of the already coarse-grained complex; this does not bound the coarse-graining index or its computation. An exact-arithmetic implementation is provided.
The construction does not optimize circuit depth, range, or ancilla count. Minimum coarse-graining with three or more layers remains open. Infinite-layer excitation structures, composite qudit dimensions, and non-CSS codes lie outside the result.
Two-dimensional translation-invariant topological CSS codes on qubits are known to be locally equivalent, after coarse-graining, to stacks of toric codes. However, existing constructions generally break more translation symmetry than is required to remove anyon-permuting translations, leaving open whether any further obstruction exists. We prove that no such obstruction occurs: after passing to the maximal anyon-preserving superlattice, every such code admits a local unitary decoupling into toric codes and product states. We further provide an efficient algorithm for explicitly constructing the decoupling map, together with bounds on the required supercell size and operator spreading. The decoupling requires no additional ancillas in generic cases and extends to finite systems with suitable boundary conditions.
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.
We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice.
We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group.
For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.
Calderbank--Shor--Steane (CSS) codes support decoding and fault-tolerant logical operations through separate pure X and pure Z stabilizer checks. Generator choices and single-qubit Clifford transformations can obscure this structure. We give a deterministic polynomial-time algorithm that decides whether an arbitrary stabilizer code is locally Clifford-equivalent to a CSS code and constructs the required transformation and CSS checks whenever such a form exists. For m possibly redundant generator rows on n qubits, the algorithm uses O(mn2+n4) binary field operations. We reduce the problem to affine linear systems by characterizing local projectors that preserve the span of the binary stabilizer labels. The conversion preserves code parameters and check sparsity for quantum low-density parity-check (qLDPC) codes. We demonstrate CSS recovery and decoder reuse and resolve previously open LC-to-CSS classification questions for published code constructions. For graph states, LC-to-CSS equivalence corresponds to the existence of a bipartite graph in the local-complementation orbit. We derive a necessary-and-sufficient adjacency-matrix criterion for this property and construct such a graph whenever one exists. Our algorithm makes CSS decoding and fault-tolerant constructions accessible throughout the local Clifford equivalence classes of CSS codes.
The color code model is a crucial instance of a Calderbank-Shor-Steane (CSS)-type topological quantum error-correcting code, which notably supports transversal implementation of the full Clifford group. Its robustness against local noise is rooted in the structure of its topological excitations. From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit. In this work, we analyze the color code model on an infinite lattice within the quasi-local C∗-algebra framework, using a cone-localized Doplicher-Haag-Roberts (DHR) analysis. We classify its irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations. Our results show that the topological order of the color code is described by Rep(D(Z2×Z2))≃Rep(D(Z2))⊠Rep(D(Z2)), which is equivalent to a double layer of the toric code and consistent with established analyses on finite lattices. We also establish the existence and uniqueness of the KMS state at every finite inverse temperature and derive an explicit description of the corresponding thermal distribution of stabilizer excitations.
This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal Z-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal T realizes the logical transversal S†; (b) show that the condition C2∗C1⊆C1⊥ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal T implements the logical identity and the logical transversal T, respectively.