Cubical Sheaf Complexes with Constant Expansion with Applications to Asymptotically Good qLTCs
Yeyuan Chen, Miryam Mi-Ying Huang, Yinchen Liu, Er-Cheng Tang
cs.IT, cs.CC, quant-ph
2026-09-23
Explicit cubical Reed-Solomon sheaves yield CSS qLTCs with constant rate, linear distance, constant soundness and bounded checks; the 4D binary family has rate at least 1/48.
A quantum code with parameters [[N, K, D]] stores K logical qubits in N physical qubits and can correct errors on fewer than D/2 of them. An asymptotically good family keeps both the rate K/N and the relative distance D/N bounded below by constants independent of N. Quantum LDPC adds sparsity: every check touches a constant number of qubits, and every qubit sits in a constant number of checks.
Sparsity is not enough. A word can sit far from every codeword and still fail only a tiny fraction of those sparse checks. Quantum locally testable codes (qLTCs), defined by Aharonov and Eldar, close that gap. If a word is at relative distance δ from the code, a random check should reject it with probability at least ρδ. The constant ρ is the soundness. The target is all four properties at once: constant rate, linear distance, constant soundness, and bounded check and qubit degree.
Panteleev–Kalachev and Leverrier–Zémor already gave asymptotically good qLDPC codes, without constant soundness. Dinur, Lin, and Vidick (FOCS 2024) reached constant rate and bounded checks, but paid a log³ N factor in both distance and soundness. This paper removes those polylog losses.
The geometry is an arithmetic quotient of a product of r Bruhat-Tits trees: a finite group Γs acts on T1 × ⋯ × Tr and yields a bounded-degree cubical complex. Parallel-face graphs are connected, with spectral gap λ ≤ max 2√qi/(qi+1), independent of the size of the quotient. Raising tree degrees improves mixing while leaving local neighborhoods unchanged. DLV kept degrees bounded with Abelian lifts whose parallel graphs split into many disjoint expanding components. Mixing then only reaches sets of inverse-polylog density, which is where the log³ loss comes from.
Local codes are Reed-Solomon. The q+1 edges around a tree vertex are the projective line P¹(Fq), so homogeneous polynomials evaluate there. After the arithmetic quotient, neighborhoods are identified with permutations and rescalings, which random codes do not survive. A Frobenius twist matches the scaling factors at the two ends of each edge, so the sheaf (and its dual) descend from the trees onto the finite quotient.
Robustness is product expansion: if row and column codewords cancel after they are added on a grid, the same word should still admit a decomposition whose total line length is proportional to the final support. On norm-one evaluation sets, when local rates lie in [β, 1-β], these RS codes and their duals have product expansion ϱ = cr β^r > 0, independent of length. The proof enlarges the support to a low-degree polynomial hull and shows the hull is inner-generated: words supported inside it decompose using only lines inside it. Uniform local expansion lets the construction first raise tree degrees until DLV's spectral condition holds, then grow only the quotient.
Positive rate comes from asymmetric local dimensions: k directions get a small fixed rate, the rest get the complementary rate. DLV's local-to-global argument plus sheaf duality give linear distance and constant soundness on both CSS sides. Restriction of scalars to F2 keeps row and column weights bounded.
For every fixed r ≥ 4 and 2 ≤ k ≤ r−2, the degree-k CSS code satisfies K = Ω(N), D = Ω(N), ρ = Ω(1), with bounded row and column weights, and the check matrices are polynomial-time computable. The case r = 4, k = 2 is an asymptotically good binary qLTC. The rate lower bound is
K/N ≥ 1 / (2 · C(r,k) · 2^{r−k})
which is 1/48 when r = 4 and k = 2. Against DLV, rate stays constant, distance moves from Ω(N/log³ N) to Ω(N), and soundness moves from Ω(1/log³ N) to Ω(1). Block length grows as N = 2^{Θ(s)} in the congruence degree.
Gay and Jeronimo independently construct asymptotically good qLTCs. Bafna, Li, and Nguyen give a closely related cubical RS construction that assumes uniform product expansion as a conjecture; the norm-one theorem covers the parameter family they use. Li, Li, and Liu add transversal multi-controlled-Z gates to the Gay–Jeronimo codes. This paper does not treat logical gates.
Quantum PCP and qLTCs sit on the same line of questions. Constant soundness means sparse checks report global distance faithfully, which is the missing testing property when one moves classical LTCs onto quantum CSS codes. For people who already work on quantum LDPC, the geometric shift is from two-sided group multiplication to arithmetic cubical products of trees, and the local codes shift from random codes to explicit Reed-Solomon sheaves.
This is an existence and polynomial-time constructibility result. It is not a code anyone will decode on hardware soon. The 1/48 rate bound is explicit; the hidden constants in distance and soundness still depend on r, k, and the local parameters, and the paper does not evaluate them.
The authors state that constants may depend on r, k and the fixed local data, not on block length, and that they do not study logical gates. Tree degrees must be large enough to drive the spectral gap down, so the check-weight bound Δ is a constant that may be huge. Block lengths jump exponentially in s; there is no fine-grained family for every N. Product expansion is proved for RS codes on norm-one sets, not for the more general multiplicative-subgroup conjecture of Bafna and Vyas. α, δ, and ρ have no numerical values; 1/48 is only a rate lower bound.
The front page discloses that the main construction and technical ideas were generated with GPT-6 Astra on September 7, after which the authors spent substantial time checking and rewriting the proofs. That is an honest note about readability. It is not a substitute for peer review.