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arXiv:2609.28028

Cubical Sheaf Complexes with Constant Expansion with Applications to Asymptotically Good qLTCs

Yeyuan Chen, Miryam Mi-Ying Huang, Yinchen Liu, Er-Cheng Tang

cs.ITcs.CCquant-phSubmitted Sep 23, 2026• Announced Sep 24, 2026• 0 citations
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Abstract

For every fixed integers $r \ge 4$ and $2 \le k \le r-2$, we construct $r$-dimensional cubical sheaf complexes whose degree-$k$ CSS codes have positive constant rate, linear distance, and constant soundness, with bounded row and column weights. Taking $r=4$ and $k=2$ gives a family of asymptotically good binary qLTCs. At the core of our construction is a uniform product-expansion theorem for explicit Reed-Solomon codes on norm-one evaluation sets. The key point is that the expansion constant stays bounded away from zero as the local code lengths grow. We place these codes on arithmetic cubical complexes, obtaining constant local expansion for both the resulting sheaf and its dual. Together with the local-to-global framework of Dinur, Lin, and Vidick (FOCS 2024) and sheaf duality, this gives linear distance and constant soundness, while an asymmetric choice of local code dimensions gives positive rate. The resulting codes are explicit and polynomial-time computable.

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For every fixed integers r≥4r \ge 4r≥4 and 2≤k≤r−22 \le k \le r-22≤k≤r−2, we construct rrr-dimensional cubical sheaf complexes whose degree-kkk CSS codes have positive constant rate, linear distance, and constant soundness, with bounded row and column weights. Taking r=4r=4r=4 and k=2k=2k=2 gives a family of asymptotically good binary qLTCs.
At the core of our construction is a uniform product-expansion theorem for explicit Reed-Solomon codes on norm-one evaluation sets. The key point is that the expansion constant stays bounded away from zero as the local code lengths grow. We place these codes on arithmetic cubical complexes, obtaining constant local expansion for both the resulting sheaf and its dual. Together with the local-to-global framework of Dinur, Lin, and Vidick (FOCS 2024) and sheaf duality, this gives linear distance and constant soundness, while an asymmetric choice of local code dimensions gives positive rate. The resulting codes are explicit and polynomial-time computable.
We introduce a high-dimensional cubical complex, for any dimensiont∈Nt \in \mathbb{N}t∈N, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (caset=2t=2t=2), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codesh1,…,hth_{1}, \ldots,h_{t}h1​,…,ht​as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. Fort=4t=4t=4our construction gives a new family of “almost-good” quantum LTCs - with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.
We construct quantum locally testable codes (LTCs) with constant rate, distance, soundness and locality under a variant of a product expansion conjecture of Bafna and Vyas about Reed-Solomon codes. In particular, we use the high-dimensional expansion framework of Dinur, Lin and Vidick for constructing quantum LTCs, instantiated with the non-Abelian cubical complexes of Rungtanapirom, Stix and Vdovina. Our code is obtained by equipping the complex with carefully chosen Reed-Solomon local codes whose symmetries are compatible with those of the complex.
We study the robust local testability of tensor products of two Algebraic-Geometry (AG) codes. In particular, we prove that constant rate AG codes are robust locally testable. This significantly generalizes the seminal result of Polishchuk-Spielman (1994), which proved robust local testability of Reed-Solomon codes. We establish an algebraic-geometric framework that enables us to geometrically interpret codewords in tensor products of AG codes. Thereby, we use tools from intersection theory of algebraic surfaces to prove a divisibility criterion for AG codes, that generalizes the bivariate divisibility result of Polishchuk-Spielman.Over the years, robust local testability of tensor products has played a key role in the development of classical locally testable codes (LTCs) as well as quantum Low Density Parity Check (qLDPC) codes and quantum Locally Testable Codes (qLTCs). To the best of our knowledge, after Reed-Solomon codes, our result provides the first explicit family of robustly locally testable codes with constant rate and linear dual-distance. Moreover, our result, when combined with Golowich-Guruswami (2024), yields new explicit families of good quantum CSS codes of length N which are locally testable with locality O(N)O(\sqrt N )O(N​) and constant soundness.
Sipser and Spielman constructed LDPC codes from either bipartite \emph{spectral} expanders or one-sided \emph{lossless} expanders. In higher dimensions, \emph{spectral} expansion similarly played a central role in the constructions of asymptotically good classical LTCs and qLDPC codes by Dinur, Evra, Livne, Lubotzky, and Mozes and by Panteleev and Kalachev. Alternatively, Lin and Hsieh constructed classical LTCs and qLDPC codes from two-dimensional \emph{lossless} cubical complexes. In this work we develop the higher-dimensional \emph{lossless} approach. We do not construct the required high-dimensional lossless cubical complexes; rather, we investigate what their existence would imply. We associate with a high-dimensional cubical complex a \emph{level chain complex}, whose chain groups are supported on the level sets of the Boolean cube rather than on its cells. Our main technical contribution is a clean local-to-global theorem for this structure: suitable one-dimensional lossless expansion in the directional graphs implies small-set coboundary expansion of the global level complex. As a consequence, sufficiently imbalanced, two-sided lossless four-dimensional cubical complexes give rise to asymptotically good quantum locally testable codes. We expect the local-to-global principle developed here to have further applications.
Non-Abelian quantum codes connect quantum error correction, phases of matter, and computational resources. In this work, we develop a general framework for constructing non-Abelian quantum low-density parity-check (qLDPC) codes by gauging sheaf codes via cup products and use it to establish code families with constant encoding rate and linear distance, simultaneously achieving optimal asymptotic scaling in both parameters. In particular, we characterize explicit logical representatives of the full gauged code space, and provide a fundamental treatment of non-Abelian code distance in the sense of Knill--Laflamme based on combining expansion with cleaning to establish protection against arbitrary errors. We further construct an almost-good family with entire code spaces exhibiting long-range magic. Moreover, we use linearly addressable cup product to directly generate linearly many encoded magic states from gauging measurements of Clifford operators. These results extend good qLDPC codes beyond the Pauli stabilizer setting, lay the foundation for pursuing the no low-energy trivial magic conjecture and shed light on non-Abelian quantum phases and orders beyond geometric locality.