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Asymptotically Good Quantum Locally Testable Codes

William Gay, Fernando Granha Jeronimo

arXiv:2609.20780v1quant-ph

TL;DR

Earlier locally testable-code constructions left losses in rate or query complexity, motivating codes with constant rate, relative distance, query complexity, and soundness. This paper constructs binary CSS codes from cubical complexes and compatible cosheaf data. The resulting family has constant rate and relative distance, with LDPC checks and constant-soundness quantum local testability.

  • Problem

    Earlier locally testable-code constructions left losses in rate or query complexity, motivating a construction with constant rate, relative distance, query complexity, and soundness.

    Source: Introduction · p. 4

  • Method

    The construction forms CSS codes from regular cubical complexes carrying cosheaves with compatible base-code encoders.

    Source: §8.2 · p. 34

  • Results

    The binary CSS family has constant rate, constant relative distance, LDPC checks, and constant-soundness quantum local testability.

    Source: Main Results · p. 6

  • Takeaways & Limitations

    The construction achieves quantum local testability alongside asymptotically good LDPC parameters.

    Source: Main Results · p. 6

    The parameter theorem requires λL<1 for its positive constant and assumes a spectral bound on the parallel-face graphs.

    Source: §6.3 · p. 25, §8.2 · p. 34

Abstract

from arXiv · show

We construct explicit families of asymptotically good quantum locally testable codes over qubits. More precseily, we construct explicit quantum LDPC CSS codes over qubits having constant rate, constant relative distance and constant weight local testers with constant soundness.

Introduction

The paper develops quantum locally testable codes by combining sparse quantum-code constructions with local testing and higher-dimensional expansion. It proves an explicit binary CSS family with constant rate, relative distance, and soundness, while outlining the geometric and local-code ingredients.

Introduction

  • Earlier quantum LDPC families achieved good rate and distance, while quantum local testability remained an additional requirement with distinct parameter tradeoffs.The paper builds on DLV’s higher-dimensional cubical construction and local-to-global expansion analysis.Source: Introduction · pp. 5–6

Main Results

  • A binary CSS family has constant rate, relative distance, LDPC checks, and constant-soundness quantum local testability.The paper also states a general-dimensional construction and a product-expansion theorem for Reed–Solomon codes on unequal evaluation sets.Source: Main Results · p. 6

Tools

  • Product expansion controls the cost of decomposing arrays into local codewords, and projective Reed–Solomon symmetries align local coordinates with cubical incidence data.Ramanujan cubical complexes supply the geometric spectral input used in the coding analysis.Source: Tools · pp. 6–7

Proof Overview

  • The construction combines cosheaf incidence maps and local product expansion with face-walk mixing to obtain soundness and distance estimates.Dual base codes transfer estimates to homology; the abstract criterion assembles the resulting bounds.Source: Proof Overview · pp. 7–8
  • The realization addresses noncommuting geometric labels by choosing base codes that respect coordinate changes, making the two incidence compositions around a square agree.Arithmetic quotients provide uniform spectral bounds, while paired polynomial degrees and Frobenius reindexing construct equivariant incidence maps.Source: Proof Overview · p. 8

1 Preliminaries

The preliminaries specify classical codes, product expansion, and the CSS framework used to define quantum local testability. They also fix the soundness and sparsity quantities measured by the paper.

1 Preliminaries

  • The preliminaries fix coordinate-space and weight notation, including Hamming weight and block weight for vector-valued coordinates.They also distinguish the vector-space dual from the dual code.Source: §1 · p. 9

1.1 Classical Codes and Product Expansion

  • Classical codes are represented by injective encoders and parity checks, with dual codes and coordinate lines used to formulate product expansion.The section also defines universal product expansion over field extensions and collective expansion for vector-valued entries.Source: §1.1 · p. 9

1.2 CSS Codes and Local Testability

  • A CSS code is the common +1 eigenspace of its X- and Z-checks, and its soundness is measured against the distance observable.Check weight is the maximum row weight, while qubit incidence is the maximum combined column weight across the two check matrices.Source: §1.2 · p. 10

2 Construction

The construction uses regular cubical complexes with compatible local code data to form a square-zero complex and associated CSS codes. Expansion and dimension conditions yield bounded sparsity and asymptotically good, locally testable families.

2.1 Cubical Complexes

  • Regular cubical complexes encode directional degrees and incidence patterns; their parallel-face graphs are required to satisfy connectedness and spectral bounds.These bounds control the relevant face-walk expansion.Source: §2.1 · pp. 12–13

2.2 Cosheaves and Compatible Base Codes

  • Compatible base-code data assign local encoders to faces and require identities that coordinate those encoders with incidence maps.The cosheaf supplies coordinate spaces on faces and compatible linear maps between them.Source: §2.2 · p. 14

2.3 Cosheaf Complexes and Their CSS Codes

  • The cosheaf complex produces CSS codes whose check matrices commute, with encoded-qubit count determined by the cohomology dimension.The stated check-weight and qubit-incidence bounds follow from bounded stalk dimensions and the cubical incidence structure.Source: §2.3 · p. 15

2.4 Main Results

  • The resulting binary CSS families have positive constant rate, relative distance, and soundness, with bounded check weight and qubit incidence.The main theorem states that lengths grow without bound and all constants are independent of the family index.Source: §2.4 · p. 16

3 Rate and Dimension

This section bounds cohomology dimensions through directional elimination and Euler-characteristic estimates. Its bounds yield a positive rate lower bound when the stated parameter conditions hold.

3 Rate and Dimension

  • The rate-and-dimension analysis assumes compatible base codes with 0<m_i,b<n_i and imposes no expansion hypothesis..Source: §3 · p. 16

3.1 Directional Differentials and Partial-Star Quotients

  • In a chart, quotienting by the directional images leaves the partial-star quotient, and adding a new direction gives an injective map with that quotient as cokernel.The quotient imposes exactly the combined directional relations.Source: §3.1 · p. 17

3.2 Cohomology-Preserving Directional Elimination

  • Eliminating directions produces a complex C_P with the same cohomology as C, by successively removing acyclic subcomplexes.At each step, the injective directional map supports a quotient that retains the remaining incidence maps.Source: §3.2 · pp. 17–18

3.3 Euler Characteristic and the Rate Bound

  • For every k, the cohomology dimension h_k is at least A_0Γ_k, while each h_j is at most A_0B_j.The proposition also gives M_j=A_0e_{t−j}(S).Source: §3.3 · p. 18
  • For t≥2, 1≤k<t, and 0<τ≤(8t2^t)^−1, the lemma establishes G_{t,k}(τ)>1/2; the corollary also gives Γ_k≥G_{t,k}(τ) and M_j≤A_0E_{t,k,j}(τ) for every j.The corollary further gives M_k≥A_0[2(1−τ)]^(t−k).Source: §3.3 · p. 19

4 Local Filling from Product Expansion

The section derives local filling inequalities from universal product expansion, with constants independent of local lengths. It extends the expansion property to vector-valued settings and constructs controlled primitives in tensor evaluation complexes.

4 Local Filling from Product Expansion

  • Universal product expansion of every nonempty chart subtuple yields filling inequalities on upward stars, with constants independent of local lengths.The theorem assumes a regular cubical complex with bounded ratio n+/n− and compatible base codes; its assertion also survives field extension and finite-dimensional tensoring.Source: §4 · p. 19

4.1 Vector-Valued Product Expansion

  • A universally ρ-product-expanding tuple retains the same collective product-expansion constant over field extensions and finite-dimensional coefficient spaces.The proof embeds the coefficient space into a rational-function extension while preserving support, then retracts the decomposition.Source: §4.1 · pp. 19–20

4.2 Tensor Evaluation Complexes and Local Exactness

  • The tensor evaluation complex is exact below its top degree, where its image is the sum of directional code spaces.Its dual complex is exact below the top degree as well, with the top kernel described by coordinate kernels.Source: §4.2 · p. 20

4.3 Controlled Primitives

  • Under universal product expansion and bounded length ratio, each b in im δ_u has a primitive y with controlled block weight.The controlled-primitive proposition supplies the local bound; the theorem proof transfers it to the actual upward star and extends it to coefficient extensions and passive spaces.Source: §4.3 · pp. 20–21

5 Mixing of Cubical Face Walks

This section constructs transported-face walks on regular expanding cubical complexes and establishes their incidence-measure properties. The mixing argument uses the spectral decomposition of parallel components and averaging between face dimensions.

5.2 Transported-Face Walks

  • The transported-face walk descends to a lower face, moves to a parallel face, then ascends, defining a reversible transition matrix on k-faces.Its transitions consist of paths through opposite lower-dimensional facets of a higher-dimensional face.Source: §5 · p. 21, §5.2 · p. 22

5.1 Parallel Components and Incidence Measures

  • The incidence measure has total mass one, and the averaging map preserves means and contracts the indicated weighted Euclidean norms.The associated operators preserve constants, and the transported-face walk is reversible for the uniform measure on k-faces.Source: §5.1 · p. 22, §5.2 · p. 22

5.3 The Mixing Estimate

  • The mixing argument separates the parallel-component eigenvectors with eigenvalues 1 and −1 from the remaining spectrum, whose norm is at most λ.It then applies the incidence averaging contraction to bound the averaged walk.Source: §5.3 · p. 23

6 Cosystolic Distance and Cocycle Expansion

The section relates local filling and geometric expansion to lower bounds on cosystolic distance and cocycle expansion. It develops the argument through local cominimality, support recursion, and syndrome repair.

6 Cosystolic Distance and Cocycle Expansion

  • Under local filling and a sufficiently small geometric expansion parameter, μ_j has a linear lower bound and ε_j has a positive lower bound.The global locally cominimal distance bound is stated under the additional condition λL<1 for a positive constant.Source: §6 · p. 23, §6.3 · pp. 25–26

6.1 Local Cominimality and Support Cancellation

  • For a locally cominimal cocycle, the local filling inequality bounds its support above each face by support in the two specified neighboring classes.The argument uses cocycle cancellation to assign each nonzero local coboundary an external support face.Source: §6.1 · p. 24

6.2 Neighbor Covering and the Support Recursion

  • Every locally cominimal cocycle has support satisfying the support-recursion inequality of Proposition 6.2.2.The proof combines the local support inequality with bounds on neighboring support classes.Source: §6.2 · p. 25

6.3 Global Locally Cominimal Distance

  • The global locally cominimal distance bound uses the condition λL<1 to ensure its constant c is positive.The corollary applies when the cosheaf satisfies Theorem 4.0.1's hypotheses and λL(t,k,ρ,K)<1.Source: §6.3 · pp. 25–26

6.4 Cosystolic Distance and Syndrome Repair

  • A minimum-weight representative of a nonzero cohomology class is locally cominimal, while repeated local syndrome corrections reduce integer syndrome weight until it vanishes.When the syndrome weight is below M, each correction has weight at most ℓ_j times the syndrome weight.Source: §6.4 · p. 26

7 Systolic distance and cycle expansion

The section constructs a kernel cosheaf with dual base codes and uses small-cycle filling to transfer cohomological distance and expansion to the original cosheaf, including in arbitrary stalk bases. It then applies a lemma and Theorem 7.0.1 to derive bounds and homological conclusions for F and K in specified degrees.

7 Systolic distance and cycle expansion

  • Theorem 7.0.1 transfers cohomological distance through the dual-code kernel cosheaf, and its assertions hold in arbitrary stalk bases, including nonorthogonal changes of basis.The anchored-array method transfers estimates from the auxiliary cosheaf K to the transpose complex of F.Source: §7 · p. 27

7.1 Anchored arrays and the kernel cosheaf

  • Anchored arrays realize the actual local top kernels as a cosheaf, with block weight equal to anchor weight and compatibility retained under changes of basis.The horizontal and vertical maps satisfy ∆2 = T2 = 0, ∆T = T∆, |∆a|a ≤tn|a|a, and |Ta|a ≤|a|a.Source: §7.1 · pp. 27–28

7.2 Local transpose exactness and dual base codes

  • The kernel cosheaf has locally exact transpose maps and compatible charts whose encoder images are the dual base codes in the actual edge coordinates.The construction preserves the geometric indexing across each star.Source: §7.2 · p. 28

7.3 Filling small transpose-kernel vectors

  • Small transpose-kernel cycles are boundaries, and an additional expansion hypothesis permits a primitive with controlled weight.For closed arrays, the filling argument uses H0 = 22tnt and gives a degree-zero chain z with |z|b ≤G0|a|a, where G0 = 2tnt.Source: §7.3 · pp. 30–31

7.4 Systolic distance and cycle expansion

  • Under the stated hypotheses, µk(F) ≥M/B, while small syndromes yield distance-to-kernel bounds and the corollary gives the stated four lower bounds for µk(F) and εk(F).The corollary’s bounds are µk(F) ≥cZk, µk(F) ≥cZt−k/B, εk(F) ≥uc, and εk(F) ≥vh; all three constants are positive.Source: §7.4 · pp. 32–33
  • The vertex-incidence bound gives ε_k(F) ≥ uc and ε_{t−k}(K) ≥ ud, using syndrome thresholds cZ_{k+1} and cZ_{t−k+1}.The constants are positive and at most one.Source: §7.4 · p. 33
  • Theorem 7.0.1, applied with M = cZ_{t−k+1} and ε = ud, gives two homological conclusions for degrees 2 ≤ k, t−k ≤ t−2.Source: §7.4 · p. 33

8 The abstract parameter theorem

The binary conversion turns block-complex estimates over a finite field into CSS code parameters and bounded-weight, bounded-incidence listed checks. Under the abstract theorem’s geometric, expansion, and growth hypotheses, it gives code families with constant rate, relative distance, soundness, and locality.

8.1 Binary conversion of block estimates

  • The resulting CSS codes have N = sD2, K = s(D2 − rank_F A − rank_F B), and distance at least min{µc, µh}.The conversion uses a binary basis for cochains and its trace-dual basis for dual vectors; it does not require a self-dual field basis.Source: §8.1 · pp. 33–34
  • When w_F bounds the nonzero entries per row and column of A and B, binary checks have weight at most sw_F and each qubit is incident to at most 2sw_F listed checks.The tester averages all s(D1 + D3) listed rows, including dependent and zero rows.Source: §8.1 · p. 34

8.2 Proof of the abstract parameter theorem

  • The abstract theorem assumes spectral expansion, weighted product expansion over field extensions, and vertex-parity masses tending to infinity.It applies to regular cubical complexes with compatible base-code encoders and fixed parameters in the stated middle-degree range.Source: §8.2 · pp. 34–35
  • The theorem yields CSS families with constant rate, relative distance, soundness, and locality, with block length growing with the vertex-parity mass.The proof gives d_ν ≥ cξZ_k and a positive soundness bound; the constants depend only on fixed parameters.Source: §8.2 · p. 36

9 Arithmetic realization of the cubical complexes

The arithmetic construction produces growing finite quotients of products of local trees with fixed local data and controlled expansion. Their local geometry and spectral properties supply cubical complexes for the abstract code theorem.

9 Arithmetic realization of the cubical complexes

  • Theorem 9.0.1 constructs finite quotients of a product of trees in a congruence tower, with free actions, growing parity mass, and fixed local data.At sufficiently deep levels, full-radius cubical balls agree with those in the product of trees.Source: §9 · pp. 36–37

9.1 Local trees and endpoint stabilizers

  • Each local tree is (q + 1)-regular, and its incident edges at a vertex correspond to the q + 1 lines in the residue-field plane.The local stabilizers act transitively on vertices of either type and on edges.Source: §9.1 · p. 37

9.2 Norm-one arithmetic quotients

  • Torsion-free norm-one congruence groups act freely on cells, giving finite quotient complexes whose directional parities descend.The construction starts from totally real fields and totally definite quaternion algebras split at the selected places.Source: §9.2 · pp. 38–39

9.3 Parallel components and Ramanujan bounds

  • Parallel-graph components are fibers of the inactive parities, and their normalized adjacency has nontrivial eigenvalues bounded by 2√q_j/(q_j + 1).The spectral conclusion uses the Ramanujan property for the indicated Hilbert cusp spaces.Source: §9.3 · p. 40

9.4 Congruence towers and complete upward stars

  • As congruence levels deepen, parity masses grow, while sufficiently deep quotients satisfy the intersection axiom and preserve entire upward stars.The quotient maps also identify every fixed-radius full cubical ball with its product-tree counterpart.Source: §9.4 · p. 41

10 Weighted incidence on unequal norm-one grids

This section develops weighted incidence bounds on unequal norm-one grids using polynomial closure, algebraic intersection estimates, Frobenius arguments, an anisotropic grid-preserving transform, and induction on the number of coordinates. The induction covers every branch, including constant-coordinate slices, and yields the assertion for every nonempty coordinate subtuple.

10.1 Norm-one grids and polynomial closure

  • Theorem 10.1.3 bounds incidence for closed subsets of norm-one grids under a dimension, power-ratio, and degree condition.The grids use coordinate sets of roots of unity; polynomial closure is defined using polynomials with bounded individual degrees, and the theorem also applies to nonempty coordinate subtuples.Source: §10.1 · pp. 42–43

10.2 Hilbert growth and weighted intersections

  • A degree-b hypersurface cuts an integral variety V in components of total degree at most b deg V, while curve intersections obey a weighted multiplicity bound of b deg C.The section also derives Hilbert-growth and genus controls for expected components of common zero loci.Source: §10.2 · p. 44

10.3 The two-variable Frobenius bound

  • For an irreducible curve of bidegree (a, b), the Frobenius construction intersects divisors of bidegrees (a, b) and (ca, b), with intersection number (c + 1)ab.This supplies the two-variable intersection estimate used in the grid incidence argument, including when the divisor is nonreduced.Source: §10.3 · p. 45

10.4 The anisotropic grid-preserving transform

  • The anisotropic transform preserves reduced components, dimensions, and grid zeros, while mapping degree-B polynomials to nonzero polynomials of degree at most eKB.For curves meeting the torus, it preserves the grid trace and relates coordinate pole degrees by a common positive integer; when at least two coordinates vary, the transformed curve differs from the original.Source: §10.4 · pp. 46–47

10.5 Induction on the number of coordinates

  • For 3 ≤ e ≤ t, if a coordinate is constant on C, then m ≤ m′ ≤ Km, ℓ ≤ θm′, and μ_A◦(C) ≤ K P_{e−1}(t,K) θ γ_e m deg C.Source: §10.5 · p. 48
  • Reduced components suffice for the argument; the intersection schemes need not be reduced.The induction establishes this in proving (10.21).Source: §10.5 · p. 49
  • cB_c ≤ b ≤ cB_c + L_e(t,K)m^θγ^{j/h} ensures H_V(b) > |Z ∩ V| for every tracked component V.The common choice uses x = (h!S_e(t,K))^{1/h}m^{θγ^{j/h}} and b = cB_c + ⌊x⌋ + 1.Source: §10.5 · p. 49
  • μ_A◦(C) ≤ P_e(t,K)θγ^em deg C holds on all branches, including constant-coordinate slices, simultaneously for every nonempty coordinate subtuple.The induction also controls terminal components and exit candidates by weighted Bézout degree bounds.Source: §10.5 · p. 50

11 Product expansion of Reed–Solomon codes

This section develops interpolation, regularity, and supported-decomposition tools for coordinate-flat arrangements and finite grids to prove uniform product expansion for suitable Reed–Solomon code tuples. It also establishes field-uniform variants, including for extension-field-valued words and actual-dual, permutation, multiplier, and nonempty-subtuple variants.

11.1 Affine and projective evaluation codes

  • Reed–Solomon codes have dimension u and distance n − u + 1; the projective evaluation construction has dimension u + 1, distance q + 1 − u, and a generalized Reed–Solomon dual.The projective dual has dimension q − u.Source: §11.1 · p. 51
  • Reed–Solomon tuples satisfying the stated size and dimension conditions have weighted product expansion at least ρ, uniformly under the listed subtuple and code transformations.The theorem assumes 1 ≤ d ≤ t, pairwise distinct m_i, max_i q_i/min_i q_i ≤ K, q_i = 2^m_i, and εn_i ≤ k_i ≤ (1−ε)n_i; the same ρ applies to nonempty subtuples, duals, coordinate permutations, and nonzero coordinate scalings.Source: §11.1 · p. 52

11.2 Point interpolation from weighted incidence

  • If point multiplicities on every integral curve are bounded by Λ times curve degree, then degree b interpolates on Z whenever b + a + 1 > aΛ; b = ⌈aΛ⌉ suffices.Under the same curve-incidence bound, O(d) separates ⌊δd⌋-jets for every 0 < δ < 1/Λ and all sufficiently large d.Source: §11.2 · pp. 52–53

11.3 Regularity and interpolation on coordinate-flat arrangements

  • For a union of coordinate flats through one point, a term order gives the stated monomial initial ideal and yields reg(Se/IYc) ≤ e + 1.The initial ideal consists of variables outside D and products z_Uz_V for incomparable U,V in D.Source: §11.3 · p. 55
  • A coordinate-flat arrangement whose fibers interpolate in individual degree R has saturated ideal regularity at most c_e(R + 1).The Segre coordinate ring surjects onto sections in equal multidegrees, and the ambient coordinate-flat scheme has regularity at most e.Source: §11.3 · pp. 54, 56
  • For coordinate-flat arrangements, reg IY ≤ ce(R + 1) yields restriction-surjective interpolation when each multidegree coordinate is at least b = cd(R + 1) − 1.The theorem assumes each indexing scheme interpolates in individual degree R.Source: §11.3 · p. 57
  • Compatible sections on finitely many coordinate flats glue to a unique section on their reduced union.Compatibility is required on every scheme-theoretic pairwise intersection.Source: §11.3 · p. 58

11.4 Line interpolation and supported decompositions

  • Under the theorem’s grid and incidence hypotheses, functions on a proper ℓ-closed T that restrict to RS(Si, ri) on every complete contained i-line extend to a polynomial with degXi P < ri.The theorem assumes ϵni ≤ ri ≤ ni for every i.Source: §11.4 · p. 60
  • Internal-line decompositions have cost at most d|T|, and both conclusions persist under independent nonzero coordinate multipliers.The proof bounds the number of disjoint parallel lines in direction i by |T|/ni.Source: §11.4 · pp. 61–62

11.5 Uniform product expansion and its variants

  • The uniform product-expansion proof fixes constants across coordinate subtuples, uses incidence bounds to meet Theorem 11.4.3’s premises, and reduces the desired expansion to (Di) using the Reed–Solomon dual identity.The proof uses locator closure over E0 with algebraic closure F2.Source: §11.5 · p. 62
  • The expansion bound is uniform over the field E, even when a word’s nonzero values lie outside E0.The support lies in the original finite grid, so its locator closure is computed using the evaluation matrix over E0; tensoring the finite-matrix identity with E preserves kernels, images, sums, and intersections.Source: §11.5 · pp. 62–63
  • Every nonzero word admits a directional decomposition with cost at most d(K/θ)^d times its support size.For the large-m range, the proof uses locator closure; the small-m range is handled directly without geometric interpolation.Source: §11.5 · p. 63
  • The same constant ρ applies to independent actual-dual, permutation, multiplier, and nonempty-subtuple variants.The constant is fixed before the field choices and subtuple choices; complementary dimensions remain in [ϵn_i, (1 − ϵ)n_i].Source: §11.5 · p. 63

12 Equivariant Reed–Solomon codes and compatible cosheaves

This section constructs equivariant diagrams from projective Reed–Solomon endpoint codes and combines them into compatible cosheaves, with endpoint rate, distance, and weighted product-expansion guarantees persisting at sufficiently deep tower levels. It also builds scalar-edge-representation full-star charts and, under a full upward-poset isomorphism condition, descends them to arithmetic quotients to support code construction and uniform expansion there.

12 Equivariant Reed–Solomon codes and compatible cosheaves

  • At every sufficiently deep tower level, a compatible cosheaf has endpoint rates in the stated intervals, relative distances at least ϵ for each code and its dual, and weighted product expansion at least ρ.The field E, endpoint dimensions, and ρ are independent of the tower index.Source: §12 · p. 64

12.1 Reciprocal projective evaluation representations

  • The invariant spaces U and V are monomial images of full projective Reed–Solomon codes, with relative distance at least ϵ and endpoint rates tending to 1/(2h + 1).For every nonempty J, the selected codes and duals also have weighted product expansion at least ρ(t, K, ϵ), preserved under independent monomial coordinate changes.Source: §12.1 · pp. 65–66

12.2 Induced encoders and diagrams on one tree

  • An I*-equivariant map from a K*-representation induces a well-defined, linear, K*-equivariant encoder whose evaluation on coset representatives is an isomorphism.The construction uses functions K* → W satisfying F(ki) = i^-1F(k), with coordinates indexed by left cosets K*/I*.Source: §12.2 · pp. 66–67

The pairing

  • Endpoint representations U and V share a one-dimensional I-module target under evaluation, yielding injective induced encoders whose images are the endpoint codes and a G-equivariant diagram on the tree.At each vertex, separate coordinate changes in incident edge stalks identify the aggregate incidence map with the corresponding encoder.Source: The pairing · pp. 67–68
  • The dual induced encoders define a G-equivariant diagram whose endpoint star codes are the orthogonal complements of the original star codes.When dim W = 1, the complementary endpoint dimension is q + 1 − dim U_b.Source: The pairing · p. 69

12.3 Tensor products and simultaneous full-star charts

  • Tensoring the tree diagrams’ stalks, incidence maps, and transports defines a G-equivariant cosheaf on the product of the trees.The cosheaf identities follow by tensoring the factorwise incidence and transport identities.Source: §12.3 · p. 69
  • For one-dimensional edge representations, injective scalar encoders give simultaneous incidence identities across the anchor face’s entire upward star, with coordinate maps confined to individual face stalks.Coordinates are chosen once per missing direction at its vertex; changing an incidence changes only the corresponding tensor factor.Source: §12.3 · pp. 69–70

12.4 Descent to the arithmetic quotients

  • The cosheaf and its scalar-edge charts descend when the group acts freely, preserves vertex types, has a finite quotient, and the quotient map preserves each entire upward coface poset.The descended charts retain their incidence identities and separate face blocks.Source: §12.4 · p. 70
  • The field and representation choices yield compatible cosheaf charts on sufficiently deep arithmetic quotients, with endpoint dimensions, rates, and relative distances as specified by Theorem 12.1.2.The construction uses endpoint representations over a common finite field and an external tensor product.Source: §12.4 · p. 71
  • The product-expansion assertion holds uniformly for the codes and their duals, with constant ρ(t, K, ϵ) independent of the congruence level ν.The representation and field choices precede the congruence tower.Source: §12.4 · p. 72

13 Simultaneous Realization and the Main Theorem

The section selects fixed parameters and realizes the required geometry and coefficient codes on one tower, establishing the hypotheses for the main construction. It then derives the main results and gives a fixed numerical instantiation with bounded check weights and qubit incidences.

13.1 Order of Parameter Selection

  • For every t ≥ 4 and 2 ≤ k ≤ t − 2, admissible parameter choices exist, with the product-expansion constant fixed before local lengths are chosen.The resulting local data remain fixed as the congruence-tower index grows.Source: §13.1 · p. 72

13.2 Geometry and Coefficients on the Same Tower

  • Theorem 13.2.1 realizes the geometry and coefficient conditions together on one tower, with endpoint proportions in [ϵ, τ] and spectral parameter λ_L < 1/2.The local field and dimensions stay fixed as V_ν grows, while universal product expansion and the remaining hypotheses of Theorem 2.4.1 hold simultaneously.Source: §13.2 · pp. 72–73

13.3 Proofs of the Main Results

  • Theorem 2.4.3 gives positive lower bounds on rate, relative distance, and soundness, with a fixed locality bound, for a family obtained from Theorem 2.4.2.The larger of the two fixed locality bounds is a valid choice of w.Source: §13.3 · p. 73

13.4 A Fixed Numerical Instantiation

  • The fixed numerical instantiation bounds both check weights and qubit incidences by 2217.The appendix verifies the parameter conditions for these choices.Source: §13.4 · p. 74

AI Disclosure

The authors describe the human contribution and the role of ChatGPT in the research. The appendices establish CSS stabilizer and syndrome facts, then give trace-dual coordinate identities.

AI Disclosure

  • The authors say the research direction was human, versions 5.6 and 6 Pro materially helped orchestrate some expander-code interactions, and ChatGPT 6 Pro served as an editorial assistant.The authors take responsibility for the results, proofs, and correctness.Source: AI Disclosure · p. 75

A.1 Stabilizers, logical operators, and syndrome spaces

  • Consistent syndromes are realized by Paulis and their eigenspaces form an orthogonal decomposition; logical distance is characterized through kernel-minus-image sets.The CSS codespace has dimension 2^K for K = N − rank H_X − rank H_Z.Source: §A.1 · pp. 79–80

A.2 Trace-dual coordinates

  • The trace pairing on the finite field is nondegenerate, so every ordered binary basis has a unique trace-dual basis.The resulting coordinate treatment distinguishes the algebraic dual map from an unpaired change of original coordinates.Source: §A.2 · p. 81

A.3 Proof of the binary conversion proposition

The binary-conversion proof transfers field-code dimension, distance, soundness, and sparsity bounds to binary CSS codes by controlling how field blocks expand into binary coordinates.

A.3 Proof of the binary conversion proposition

  • The binary logical distance is at least min{μ_c, μ_h}, and distance to ker H_X is at most sM_2 times distance to ker A.The dual-map argument supplies the corresponding conversion for the other check family.Source: §A.3 · pp. 81–82
  • On each syndrome eigenspace, the tester is bounded below by ε* s² M_2(D_1 + D_3) times the distance operator, yielding the proposition's soundness constant.The inequality extends to the full Hilbert space because the syndrome spaces decompose it orthogonally.Source: §A.3 · p. 82
  • The binary conversion gives dimension N = sD_2 and check weight and qubit incidence at most s w_F when field row and column weights are at most w_F.Each field-matrix entry becomes an s × s binary multiplication block.Source: §A.3 · pp. 81–82

B.1 Integral norm-one groups and principal levels

The appendix establishes finiteness and torsion properties for the norm-one groups and levels used in the construction, then collects regularity tools for coordinate-flat arrangements.

B.1 Integral norm-one groups and principal levels

  • For every finite place, C_v is compact and open in H_v; principal levels have finite index, deeper levels are normal in shallower ones, and their intersection is trivial.The global principal groups are kernels of reduction maps at primes dividing the level ideal.Source: §B.1 · p. 83

B.2 Compactness and finite norm-one quotients

  • A compact set C_H gives H(A_F) = H(F)C_H, and compactness with open stabilizers yields finitely many vertex and cell orbits.The compact representatives and finite orbit conclusion are used for the quotient action described in this subsection.Source: §B.2 · pp. 83–84

B.3 The torsion-free starting level

  • At the chosen split place, the starting level Γ_0 is torsion free and its action on the tree agrees with its projective image.The proof uses the odd, unramified prime condition to establish torsion-freeness and faithfulness of the action.Source: §B.3 · p. 85

C Regularity tools for coordinate-flat arrangements

  • For saturated ideals, regularity is characterized by cohomology vanishing and yields generator-degree and section-surjectivity bounds.The appendix also bounds quotient regularity through initial ideals, including a bound from the maximum degree of squarefree standard monomials.Source: C Regularity tools for coordinate-flat… · pp. 85, 87

D Verification of the Fixed Constants

The appendix verifies the fixed parameter choices and records explicit bounds on code parameters and derived constants.

D Verification of the Fixed Constants

  • For ρ > ρ*, the stated maximum of L(4, 2, ρ, 23) and Q(4, 2, ρ, 23) is bounded by 264ρ−6.The appendix derives this estimate after bounding the incidence recurrence and related quantities.Source: D Verification of the Fixed Constants · p. 87
  • n < 2r+4, s < 268, and 0 < c* = 2^-214 < (2Q)^-1.These bounds follow from the Section 13.4 choices and the stated spectral premise.Source: D Verification of the Fixed Constants · p. 88
  • The exact local data give n = 2r+3 + 1 < 2r+4 and s = lcm_{0≤a≤3} 2(r + a) < 268.The upper bound on s is used for code-parameter estimates; the field degree remains the indicated least common multiple.Source: D Verification of the Fixed Constants · p. 88
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