A Classification of Translation-Invariant Quantum Codes in Any Dimension

Abstract: Quantum error-correcting codes with two-dimensional translation invariance are known to be equivalent to copies of the two-dimensional toric code. Such a simple classification is not possible for quantum codes with higher dimensional translation invariance due to the existence of multiple types of toric codes and infinite families of fracton codes. Here, we focus on D-dimensional translation-invariant quantum codes based on length-D chain complexes. This includes multivariate multicycle codes where the number of variables equals the number of cycles. We show that such codes are equivalent to copies of D-dimensional toric codes. This directly generalizes the classification result for two-dimensional translation-invariant codes.

• The paper introduces a Hamiltonian-equivalence classification for $D$-dimensional translation-invariant (TI) CSS quantum codes, showing they can be reduced to copies of $D$-dimensional toric codes.
• The classification is established through a structure theorem that uses exact homological algebra, allowing the conversion of any such code into a simplified, equivalent form.
• Application of coarse-graining, ancilla additions, and symplectic transformations enables this equivalence, demonstrating that all $D$-dimensional TI codes have a gapped quantum liquid phase.

Overview and context

This paper establishes a structure theorem for $D$-dimensional translation-invariant (DDTI) CSS quantum codes whose stabilizer data forms a length-$D$ chain complex of free modules over the Laurent polynomial ring $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$. The main result is that any such code is Hamiltonian-equivalent to finitely many copies of a $D$-dimensional toric code, up to constant coarse-graining, tensoring ancillary product-state qubits, and local symplectic transformations. This directly generalizes the known two-dimensional classification — due to Bombín and formulated algebraically by Haah — in which every 2D translation-invariant code with growing distance is equivalent to copies of the 2D toric code (2608.20981).

The restriction to codes whose chain complex length matches the dimension of the translation group is essential. In higher dimensions this condition is nontrivial because it excludes fracton codes, which arise from complexes of length smaller than the translation-group dimension and exhibit immobile excitations with infinitely many superselection sectors. The theorem therefore carves out precisely the class of higher-dimensional TI codes that retain the "gapped quantum liquid" character of the toric code.

Codes as chain complexes over Laurent polynomial rings

A CSS code is encoded in a length-2 complex of $\mathbb{F}_2$-vector spaces $C_2 \to C_1 \to C_0$, where $C_2$, $C_1$, and $C_0$ are respectively spanned by $Z$ checks, qubits, and $D$0 checks. For a translation-invariant code on $D$1 with $D$2 qubits per site, this becomes a complex of finitely-generated free $D$3-modules $D$4, with $D$5-linearity of the differentials enforcing translational symmetry. Logical operators are homology (resp. cohomology) classes of the complex; exactness at $D$6 and its ZX-dual guarantees no finite-weight logicals, so distance grows under compactification.

The paper's key structural input is that any exact CSS complex embeds into a bounded free resolution of some $D$7-module $D$8. Extending the resolution to one side yields local metachecks — nontrivial products of checks equal to identity — on the corresponding Pauli type. The motivating class is multivariate multicycle (MVMC) codes: taking a length-2 segment of the Koszul complex resolving $D$9 for a regular sequence $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$0 gives an infinite-lattice MVMC code, and when $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$1 these are called $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$2-variate $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$3-cycle codes. Notably, more cycles than variables cannot occur.

Within this framework, the various higher-dimensional toric codes arise uniformly as segments of Koszul resolutions of $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$4. In 2D there is only one choice, but from dimension 3 upward distinct placements along the resolution yield inequivalent codes: the $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$5-toric code occupies degrees $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$6 of the length-$R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$7 Koszul complex. The $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$8- and $R = \mathbb{F}_2[x_1^{\pm 1},\dots,x_D^{\pm 1}]$9-toric codes in 3D are related by ZX-duality, so "the" 3D toric code is unique up to duality; in dimensions $D$0, genuinely distinct types appear — e.g., the $D$1-toric code has only local $D$2 metachecks while the $D$3-toric code has both local $D$4 and $D$5 metachecks. Any classification must therefore specify which toric code type appears, and the number of copies $D$6 is fixed by the charge module.

Hamiltonian equivalence and the classification theorem

Following Haah, two infinite-lattice TI stabilizer Hamiltonians are equivalent if their stabilizer modules become identical after coarse-graining to a sublattice $D$7 (restriction of scalars to $D$8), tensoring ancillas (direct sum with single-layer product-state complexes), and applying symplectic transformations preserving the natural symplectic form on $D$9.

The proof proceeds through three steps:

Mobility of charges. Proposition 1 shows that if $\mathbb{F}_2$0 is a length-$\mathbb{F}_2$1 free resolution and the ZX-dual $\mathbb{F}_2$2 is exact below degree $\mathbb{F}_2$3, then the resolved module $\mathbb{F}_2$4 is zero-dimensional, i.e., $\mathbb{F}_2$5 has codimension $\mathbb{F}_2$6. The proof combines Rees' theorem relating the grade of $\mathbb{F}_2$7 to the grade of its annihilator with the inequality $\mathbb{F}_2$8. Zero-dimensionality implies topological charges fall into finitely many superselection sectors and are freely mobile.

Constant-scale transport. Invoking Haah's Lemma 7.3, there exists a constant $\mathbb{F}_2$9 such that after coarse-graining by $C_2 \to C_1 \to C_0$0, the annihilator becomes $C_2 \to C_1 \to C_0$1, so any translate of any charge is equivalent on the coarse-grained lattice. A lemma then establishes that the restricted module satisfies $C_2 \to C_1 \to C_0$2 with $C_2 \to C_1 \to C_0$3, meaning the coarse-grained charges are generated by $C_2 \to C_1 \to C_0$4 independent generators, each invariant under coarse-lattice translations.

Resolution independence. A homological-algebra lemma shows that any two bounded free resolutions of the same finitely-generated module $C_2 \to C_1 \to C_0$5 become isomorphic after adding contractible summands. The proof uses the comparison theorem (resolutions of the same module are chain-homotopy equivalent), the identification of the stable category of bounded complexes with the bounded homotopy category via Frobenius structure, the decomposition of contractible complexes into disks $C_2 \to C_1 \to C_0$6, and the Quillen–Suslin theorem guaranteeing that finitely-generated projective modules over Laurent polynomial rings are free. The disk summands correspond precisely to tensorable ancillas or trivial stabilizers.

Assembling these ingredients, the main theorem states that the CSS complex extracted at any segment $C_2 \to C_1 \to C_0$7 ($C_2 \to C_1 \to C_0$8) of a qualifying length-$C_2 \to C_1 \to C_0$9 resolution defines a Hamiltonian equivalent to $C_2$0 copies of the $C_2$1-toric code, where $C_2$2. The corollary follows immediately: every $C_2$3-variate $C_2$4-cycle code is Hamiltonian-equivalent to copies of a $C_2$5-dimensional toric code. Importantly, the argument does not require the resolution to be Koszul — it applies to any length-$C_2$6 finite-rank free resolution whose ZX-dual is also a resolution.

An immediate implication is that all such DDTI codes lie in conventional gapped quantum liquid phases of matter, despite being defined by arbitrary translation-invariant stabilizer structures rather than by geometric locality alone.

Limitations and open questions

The theorem's scope is bounded in several respects, which the authors state plainly:

• More variables than cycles excluded: the result does not apply to MVMC-type families where the number of variables exceeds the number of cycles; in that regime charges are typically immobile and the coarse-graining lemma fails.
• Translation invariance assumed: whether the classification extends to general finite-range qLDPC codes in $C_2$7 dimensions described by irreducible length-$C_2$8 chain complexes remains open.
• Complexes of length exceeding $C_2$9: it is unresolved whether any length-$C_1$0 complex with $C_1$1 locally embedded in $C_1$2-dimensional space is equivalent to a length-$C_1$3 complex.
• Fracton phases: extension of the approach to fracton codes, where the length-dimension mismatch is intrinsic, is left open.
• Decoding: whether the structure theorem enables efficient decoders for higher-dimensional TI codes, analogous to matching-based decoders developed for bivariate bicycle codes, is posed as a question rather than answered.
• Completeness: the authors ask whether their result constitutes a full classification of all TI gapped quantum liquid codes.

The authors also note an independent concurrent work with overlapping results posted during preparation, which motivated the addition of the grade-theoretic Proposition 1.

Conclusion

The paper delivers a complete Hamiltonian-equivalence classification for $C_1$4-dimensional translation-invariant CSS codes built from length-$C_1$5 free resolutions with dual resolutions: each such code is a finite direct sum of a single type of $C_1$6-dimensional toric code, determined by the placement within the resolution, with multiplicity given by the $C_1$7-dimension of the charge module. The proof cleanly separates the physical content — charge mobility guaranteed by zero-dimensionality of the resolved module — from purely homological bookkeeping handled by resolution theory and Quillen–Suslin. The result sharpens the boundary between gapped quantum liquid and fractonic behavior in terms of an elementary invariant: the relation between chain-complex length and translation-group dimension.