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arXiv:2606.20548

Topological Codes from Space Groups: A Route beyond Translation Invariance

Chong-Yuan Xu, Ze-Chuan Liu, Yong Xu

quant-phcond-mat.quant-gasSubmitted Jun 18, 2026• Announced Jun 19, 2026• 0 citations• Revised Jul 28, 2026 (v3)
v1 · Jun 18, 2026v2 · Jun 23, 2026v3 · Jul 28, 2026
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Abstract

Translation invariance underlies all algebraic constructions of topological codes with geometrical locality. It has remained an open question whether codes that generically break this invariance can still be topological and simultaneously possess geometrical locality. Resolving this question is important both fundamentally---deepening our understanding of topological phases---and practically, as relaxing translation invariance could vastly expand the design space and potentially reduce resource overhead in fault-tolerant architectures. Here we introduce space-group codes, in which crystallographic point-group operations enter the bulk stabilizer algebra; bivariate bicycle (BB) codes arise as the translation-only limit. The key insight is that the point-group orbit resolves topology and locality together: it yields a computable algebraic criterion for topological order and a folded geometry in which point-group operations become local. We identify space-group codes whose code parameters exceed the reported same-blocklength, same-check-weight BB benchmarks. In five parameter-matched neutral-atom comparisons, reflection codes reduce the optimized movement cost in every case, by up to $60\%$, while folded placements also enable lower-overhead multilayer superconducting layouts. Treating spatial operations as a code-design variable therefore opens a route to topological codes jointly optimized for information protection and hardware geometry.

Comments
24 pages, 6 figures, update the abstract and introduction and correct some typos

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Translation invariance underlies all algebraic constructions of topological codes with geometrical locality. It has remained an open question whether codes that generically break this invariance can still be topological and simultaneously possess geometrical locality. Resolving this question is important both fundamentally---deepening our understanding of topological phases---and practically, as relaxing translation invariance could vastly expand the design space and potentially reduce resource overhead in fault-tolerant architectures. Here we introduce space-group codes, in which crystallographic point-group operations enter the bulk stabilizer algebra; bivariate bicycle (BB) codes arise as the translation-only limit. The key insight is that the point-group orbit resolves topology and locality together: it yields a computable algebraic criterion for topological order and a folded geometry in which point-group operations become local. We identify space-group codes whose code parameters exceed the reported same-blocklength, same-check-weight BB benchmarks. In five parameter-matched neutral-atom comparisons, reflection codes reduce the optimized movement cost in every case, by up to 60%60\%60%, while folded placements also enable lower-overhead multilayer superconducting layouts. Treating spatial operations as a code-design variable therefore opens a route to topological codes jointly optimized for information protection and hardware geometry.
High-rate bivariate bicycle (BB) codes are promising low-overhead quantum memories, but their stabilizer checks typically have weight 666 or higher, making syndrome extraction challenging. We introduce subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes BB-code logical structure using local weight-444 gauge measurements. Their stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes. We further show that nonlocal stabilizers in translation-invariant CSS subsystem codes can be detected using a determinantal-ideal criterion based on the gauge-operator commutation matrix. When this criterion excludes nonlocal stabilizers, a finite-depth Clifford circuit decouples gauge qubits and identifies the protected subsystem with a corresponding BB stabilizer code. An SBB code is topological, meaning that it has no nontrivial local logical operators, if and only if the corresponding BB code is topological. A finite search yields low-overhead examples including [[27,6,3]][[27,6,3]][[27,6,3]], [[75,10,5]][[75,10,5]][[75,10,5]], and [[108,12,6]][[108,12,6]][[108,12,6]]; the latter encodes six times more logical qubits than a subsystem surface code at the same block length and distance. These results show how gauge degrees of freedom can make high-rate BB logical structure compatible with local weight-444 syndrome extraction.
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