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

1 Introduction

• qLTC soundness requires errors at relative distance δ to violate at least a ρδ fraction of checks, motivating constant rate, linear distance, constant soundness, and bounded degrees.Dinur, Lin, and Vidick [DLV24] brought the goal close with constant rate and bounded check weights but polylogarithmic losses in relative distance and soundness.Source: §1 · p. 3, §1.1 · p. 4

1.1 Main results

• For fixed r >= 4 and 2 <= k <= r - 2, explicit r-dimensional cubical sheaf complexes yield degree-k CSS codes with K = Ω(N), D = Ω(N), ρ = Ω(1), and bounded row and column weights after binary expansion.This removes the polylogarithmic losses in distance and soundness of Dinur, Lin, and Vidick [DLV24] while keeping check weights and qubit degree bounded.Source: §1.1 · pp. 3–4
• For r = 4 and k = 2, the construction gives binary CSS codes with K >= αN and D >= δN for absolute α, δ, ρ > 0 and uniformly bounded matrix row and column weights.The family is polynomial-time computable.Source: §1.1 · p. 4

1.2 Technical overview

• Arithmetic quotients of products of Bruhat–Tits trees avoid inverse-polylog expansion losses: their parallel-face graphs are connected and satisfy the desired spectral bounds independently of quotient-growth parameter s.Fixing tree degrees before growing the finite quotient keeps local neighborhoods unchanged.Source: §1.2 · p. 6
• Compatible Reed–Solomon local codes and their duals have product expansion ϱ = c_rβ^r for rates in [β, 1 − β], uniformly in evaluation-set parameter A and for every nonempty subtuple.A Frobenius twist lets the local codes descend consistently to the arithmetic quotient while preserving them up to coordinate permutations and rescalings.Source: §1.2 · p. 8
• Uniform local expansion for the sheaf and its dual, combined with arithmetic spectral expansion, yields linear distance and constant soundness; asymmetric directional rates ensure positive rate.The local-to-global framework and quantitative sheaf duality provide guarantees on both CSS sides.Source: §1.2 · p. 9

1.3 Related work

• Prior qLDPC constructions achieved constant rate, linear distance, and constant-weight checks, while qLTC constructions explored tradeoffs among rate, distance, soundness, and locality.The closest prior result, Dinur, Lin, and Vidick [DLV24], achieved constant rate and locality with polylogarithmic losses in relative distance and soundness.Source: §1.3 · p. 9
• Kalachev and Panteleev [KP25] proved product expansion for fixed tuples of random codes and introduced inner generation; Bafna and Vyas proved a two-factor result for Reed–Solomon codes on coprime-order multiplicative subgroups.Bafna and Vyas formulated a higher-dimensional conjecture; the norm-one evaluation sets here obey different arithmetic constraints, making the result complementary rather than a resolution of that conjecture.Source: §1.3 · p. 9
• The geometric setting uses arithmetic quotients of products of trees in the function-field framework of [RSV19], with the cubical formulation of [HLM+25].These complexes provide the high-dimensional geometry for placing the Reed–Solomon sheaves.Source: §1.3 · pp. 9–10
• Concurrently, Gay and Jeronimo construct asymptotically good qLTCs, while our proof invokes existing local-to-global and sheaf-duality theorems and centers on Reed–Solomon product expansion.The paper characterizes this as a shorter, more modular route from a local coding statement to qLTC parameters.Source: §1.3 · p. 10
• Against Bafna, Li, and Nguyen, we prove the length-independent norm-one Reed–Solomon expansion their related qLTC construction assumes, partially resolving Conjecture 5.6 and extending to fixed r ≥ 4, 2 ≤ k ≤ r − 2.Their construction uses arithmetic cubical complexes and Frobenius twists for compatibility with the geometry; our proof uses polynomial hulls and inner generation, with the expansion constant fixed as local code lengths grow.Source: §1.3 · p. 10
• Li, Li, and Liu show that Gay and Jeronimo’s good qLTCs support transversal multi-controlled-Z gates; this work instead focuses on code parameters and does not study logical gates.Their construction combines good qLDPC and qLTC properties with compatibility with those gates.Source: §1.3 · p. 10

2.1 Notation

• The notation fixes the coefficient field, array support, and block support and distance conventions used for code spaces.Redundancy and relative redundancy are also defined for finite-length codes.Source: §2.1 · p. 11

2.2 CSS codes and local testability

• A CSS code is quantum locally testable when both check matrices have soundness, and the target families combine constant rate, linear distance, bounded check weights, and constant soundness.Soundness compares the fraction of violated checks with the fraction of coordinates that must change to satisfy all checks.Source: A CSS code is quantum locally testable… · p. 11

2.3 Sheaf codes and local complexes

• A sheaf code assigns a code space to each cell, with restrictions to incident cells; Tanner sheaves are generated by local codes on codimension-one cells.The top-dimensional cell spaces are the coefficient field.Source: §2.3 · p. 12
• The local cochain complex restricts cochains to a cell’s upper star, with differential given by the restriction maps there; local acyclicity requires exactness below the top degree.The upper star consists of cells containing the specified cell.Source: §2.3 · pp. 12–13

2.4 Cubical complexes

• Cubical generating sets commute setwise and have unique factorization across distinct directions, so selecting incident edges in those directions determines a unique cube.The individual generators need not commute.Source: §2.4 · pp. 13–14

3.1 Expansion parameters and standard reductions

• Sheaf duality compares systolic and cosystolic distances and cycle and cocycle expansion constants up to factors depending only on dimension and coface degree.For the decorated cubical complexes considered, the maximum coface degree is bounded by r times the upper directional degree bound.Source: §3.1 · p. 15

3.2 Parallel-face walks and geometric expansion

• Geometric expansion is specified by connected bipartite parallel-face graphs, and the walk moves from a face to a subface, translates it in a free direction, then lifts to a coface.The mixing estimate is formulated for this walk.Source: §3.2 · p. 16

3.3 Local-to-global expansion

• With bounded directional-degree ratio and sufficiently strong spectral expansion, uniform local coboundary expansion gives linear systolic and cosystolic distances and constant cycle and cocycle expansion for 2 ≤ k ≤ r − 2.The criterion assumes fixed field and directional degrees as the complex grows.Source: §3.3 · pp. 17–18

4.1 Tree codes, tensor products, and descent

• Tensor products of tree sheaves are locally acyclic, and actions free on vertices with finitely many vertex orbits descend through cube-injective quotients while preserving cochain complexes and block weight.Source: §4.1 · pp. 19–20

4.2 Products of trees and cubical quotients

• The arithmetic generators act simply transitively on vertices, and their finite quotients are connected and cube-injective, with directional degrees q_i+1 and spectral bound λ ≤ max_i 2√q_i/(q_i+1).Source: §4.2 · p. 22
• For fixed local trees, increasing s grows the quotients while preserving upper stars; |X_s(k)|=Θ(|X_s(0)|) for every k, and the groups, cells, and incidences are polynomial-time computable.Source: §4.2 · pp. 22–23

4.3 Reed-Solomon sheaves

• When (Q−1)|d and d<q+1, the single-tree construction yields a G-equivariant Tanner sheaf whose vertex code is monomially equivalent to projective Reed–Solomon code PRS_q(d).Source: §4.3 · pp. 23–24
• For 0 ≤ d ≤ q_i − 1, PRS_{q_i}(d) is monomially equivalent to RS(Σ_i, d + 1), and its dual to RS(Σ_i, q_i − d).Source: §4.3 · p. 25
• The product Tanner sheaf descends to X_s with direction-i generating codes monomially equivalent to RS(Σ_i, d_i + 1) and type-I cell dimensions ∏_{j∉I}(d_j + 1).The descent and dimension formula hold for degrees satisfying (15).Source: §4.3 · p. 25
• Replacing each d_i by q_i − 1 − d_i gives the Tanner dual F⊥; both F and F⊥ are locally acyclic.Source: §4.3 · p. 25

5 Product expansion

• Product expansion measures whether every word in the span of coordinate-line codes can be decomposed using a small total line length, with lines weighted by their lengths.The inner-generation criterion supplies a sufficient condition for product expansion.Source: §5 · p. 26

5.1 Product expansion for Reed-Solomon codes

• For redundancies h_i at least βn_i, every nonempty subtuple of the Reed-Solomon codes is ϱ-product expanding, with a bound independent of A, h_i, and F.The expansion constant depends only on r and the fixed lower bound β on relative redundancies.Source: §5.1 · pp. 26–27
• Polynomial hulls enlarge a codeword support by at most a constant factor and are inner-generated when the redundancies meet the degree threshold, yielding product expansion.The proof applies the inner-generation result to polynomial zero sets and then uses the product-expansion lemma.Source: §5 · p. 26, §5.1 · pp. 28–29

5.2 From product expansion to local coboundary expansion

• When directional code lengths have ratio at most R, product expansion of every nonempty subtuple implies tensor-complex coboundary expansion with a positive constant depending only on r, R, and ϱ.The result extends the common-length local robustness bound to codes of different directional lengths.Source: §5.2 · p. 29
• For the construction’s parameter choice, both the sheaf and its dual have local coboundary expansion κ(r, β), independent of A and s.The proof obtains this from product expansion of both local-code tuples and the bounded ratio of directional lengths.Source: §5.2 · pp. 29–30

6.1 A lower bound on the code rate

• Asymmetric local-code dimensions at the endpoint colors give dim Ck ⩾ 1/2 binom(r,k)2^(r-k), yielding the stated positive-rate lower bound.The dimension pattern assigns k directions dimensions mi and the others qi + 1 − mi at both endpoint colors.Source: §6.1 · p. 30
• The rate bound follows by controlling low-degree terms in the dimension-counting expressions using a small maximum local dimension fraction.The proof bounds the sums by θ3r/2 and uses ∏i(1 − ui) ⩾ 1 − rθ.Source: §6.1 · p. 30

6.2 Realization in a prescribed degree

• Uniform local expansion lets the geometric degree be chosen after fixing local dimension ratios, enabling realization for each prescribed degree.The geometric and local hypotheses hold for the sheaf and its dual, giving linear distances and cycle and cocycle expansion bounds.Source: §6.2 · p. 31
• The degree-k codes have positive rate, D = Ω(N), and ρ = Ω(1), with both distances and both soundness bounds covered, persisting after binary expansion.The complex has r + 1 nonzero graded terms of dimension Θ(|X(0)|) and bounded-row-and-column-weight differentials.Source: §6.2 · p. 31
• Explicit group operations and polynomial evaluations give polynomial-time construction, with bounded binary row and column weights.The passage attributes bounded weights to bounded incidences and fixed field, local dimensions, and restriction matrices.Source: §6.2 · p. 31
• Restriction of scalars transfers the construction to binary while preserving rate and the four block parameters.The trace pairing identifies the binary dual with the restriction of scalars of the F-linear dual, preserving block support.Source: §6.2 · p. 31
• For r = 4 and k = 2, the proof specializes the theorem to a binary CSS construction with N = Θ(|X(0)|).The passage also gives N = 2^Θ(s).Source: §6.2 · p. 32
• Taking direct sums of finitely many dimension patterns yields good codes in all internal degrees, with only constant losses.Each summand has linear distances and constant soundness in every internal degree, and the summand indexed by k supplies positive rate in degree k.Source: §6.2 · p. 32

A Arithmetic quotients and spectral bounds

• The section verifies the arithmetic and spectral assertions of Proposition 4.3.It keeps the notation of Section 4.2.Source: A Arithmetic quotients and spectral… · p. 32
• Strong approximation is one of the arithmetic inputs used for the verification.The passage names it alongside the Ramanujan theorem.Source: A Arithmetic quotients and spectral… · p. 32
• The Ramanujan theorem is the other named arithmetic input to the verification.The passage identifies both inputs as used for Section 4.2.Source: A Arithmetic quotients and spectral… · p. 32

A.1 Edge transitivity on a Bruhat-Tits tree

• SL2(kν) preserves vertex colors and acts transitively on edges oriented from color 0 to color 1, and therefore on each vertex color class.Every vertex is incident to an edge.Source: §A.1 · p. 32
• An edge with color-0 endpoint [L] is represented using a basis (b1, b2) whose first vector lifts a line in M/πL.The complementary vector is chosen in L so their reductions span L/πL.Source: §A.1 · p. 32
• After rescaling the basis to determinant 1, its matrix in SL2(kν) sends the standard edge to the chosen edge.The rescalings preserve both lattice classes.Source: §A.1 · p. 32

A.2 Useful arithmetic tools

• Strong approximation gives an element of H(k0) meeting prescribed open conditions at finitely many places and integral at the others except νj.Integrality is defined using a fixed integral matrix model, requiring the matrices of the element and its inverse to have entries in Oν.Source: §A.2 · p. 32
• For the specified congruence subgroups acting freely without edge inversions, the tree quotient is finite, connected, and (qj + 1)-regular.The lemma states that its adjacency eigenvalues satisfy a Ramanujan bound.Source: §A.2 · pp. 32–33
• The stated Ramanujan result is the congruence-subgroup form of [LSV05, Theorem 1.2, d = 2].The passage cites [BR17, Theorem 3.2] for the global Jacquet-Langlands correspondence in positive characteristic.Source: §A.2 · p. 33

A.3 The generators and their cubical relations

• Each line in P1(F_qi) determines a unique normalized generator, and the directional sets are inverse-closed.The group acts simply transitively on the vertices of T_Σ.Source: §A.3 · p. 33
• Generators in distinct directions satisfy a unique square relation, and successively recovering factors gives injective ordered products and compatible cube relations.The relation is constructed by choosing a generator with the matching image line modulo the second direction's prime.Source: §A.3 · p. 33
• The same factor-removal operation decomposes any normalized element whose norm is a product of the selected primes into directional generators.At each step the norm degree decreases.Source: §A.3 · p. 33
• An element fixing the standard vertex in every selected tree can be normalized to 1, establishing simple transitivity on T_Σ.The proof uses polynomial coefficients, constant norm, the degree formula, and normalization.Source: §A.3 · p. 33

A.4 Congruence images and injectivity

• For s > 2r a_max, the quotient is injective on vertex balls of radius floor((s − 1)/(2a_max)), hence on each cube, and preserves complete upper stars.The deck group acts freely on vertices.Source: §A.4 · p. 34
• The congruence image is all of SL2(L_s), and the resulting generator images connect the entire decorated complex.The reduction is identified with determinant through the matrix realization modulo I_s.Source: §A.4 · p. 34
• A nonidentity word of length ℓ has coefficient degrees at most a_maxℓ, giving a lower bound ℓ ≥ s/a_max for separation.The bound follows by multiplying normalized representatives and applying scalar reduction.Source: §A.4 · pp. 34–35
• When s > 2r a_max, injective vertex balls contain every cube through their center, so the quotient map is injective on cubes.The descended product cubes are the decorated cubes of (13).Source: §A.4 · p. 35

A.5 The spectral assertion in each direction

• The level and transverse-stabilizer conditions are congruence conditions, allowing the congruence displacement estimate to rule out vertex stabilizers and edge inversions.This verifies the conditions needed to apply Lemma A.3 to the tree quotient.Source: §A.5 · p. 35
• The graph is obtained from the tree T_νj by quotienting by the subgroup fixing the transverse face pointwise.The proof realizes the required transverse-face moves using arithmetic approximation with prescribed reduction at ℓ_s.Source: §A.5 · p. 35
• The parallel-face graph is connected, bipartite, and (q_j + 1)-regular, and its spectrum satisfies the spectral bound in Proposition 4.3.It is the quotient by Λ or its bipartite double cover, which adds only negatives of the original eigenvalues.Source: §A.5 · p. 35

B Inner generation for Reed-Solomon codes

• If H ≥ K_r d, every codeword of B_h supported on Z_E is a sum of line-codewords on coordinate lines contained in Z_E.Here E is nonzero, has coordinate degrees at most d, and K_r depends only on r.Source: B Inner generation for Reed-Solomon… · p. 36

B.1 Parity and interpolation

• The parity test characterizes membership in B_h through vanishing against every polynomial whose coordinate degrees are below the corresponding redundancies.Multiplication by a polynomial of coordinate degrees e_i < h_i sends B_h into B_{h−e}.Source: §B.1 · p. 36
• Interpolation with bound w means every function on W is the restriction of a polynomial with each coordinate degree below w.At w ≤ min_i n_i, it is equivalent to B(w,...,w) ∩ F^W = 0; inner generation at redundancy w implies interpolation on Z_E \ U_E.Source: §B.1 · p. 36
• Unions of sets admitting interpolation inherit an interpolation bound given by the sum of their bounds, and interpolation on a set passes to subsets.A point outside an interpolating set can be separated from it by a polynomial with coordinate degrees at most the bound.Source: §B.1 · p. 36

B.2 Two gradient lemmas

• The derivative-isolation lemma constructs, for each variable on which F depends, a polynomial with the same grid zero set and an isolated nonzero partial derivative.The constructed polynomial has smaller total degree than F and no larger coordinate degrees.Source: §B.2 · p. 37
• If H ≥ mB, no m coordinate-degree-at-most-B polynomials vanishing on supp M have independent gradients at a point of that support.The proof forms a determinant polynomial and applies the parity test.Source: §B.2 · p. 37

• For H >= 2mR, the code space Bh ∩ FZ is generated by free-direction line codes in the cylinders.Source: §B.3 · p. 38
• When every ti >= 2mR - 1, H1(X, I(t)) vanishes, so every global section of OY(t) extends to X.Source: §B.3 · p. 39
• The proof interpolates annihilating functionals on cylinder bases, glues their sections across intersections, and extends the result globally.Pairwise agreement is lifted using component ideals, and the global extension yields a functional annihilating Bh.Source: §B.3 · p. 39

B.4 Proof of inner generation

• Induction on the number of directions bounds interpolation on zero-set bases, including sets annihilated by lower-variable polynomials.Source: §B.4 · pp. 39–40
• Multiplying a codeword by a polynomial whose grid zero set is UE reduces support restriction to proving the product vanishes.The proof obtains a contradiction from linearly independent gradients, forcing the product to vanish and the original support into UE.Source: §B.4 · pp. 40–41
• Once support lies on UE, Theorem B.6 decomposes the codeword into line-codewords on complete coordinate lines.Source: §B.4 · p. 41

C Some cohomological facts and useful lemma

• On X=(P1)^m, line bundles with every ti >= -1 have no higher cohomology, and finite resolutions by such acyclic sheaves preserve vanishing.Source: C Some cohomological facts and useful… · p. 41