PaperlayerView exact revision on arXiv

Source-linked AI summary

Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes

Yiming Li, Zimu Li, Zi-Wen Liu

arXiv:2604.01874v2quant-phcs.ITmath-phmath.ITmath.MP

TL;DR

Achieving product expansion in three or more dimensions presents a fundamental difficulty, including cases where annihilating values on full lines requires degree n. The paper develops a general cohomological framework for constructing transversal logical multi-controlled-Z gates, and proves qLDPC gate results with stated parameters and soundness.

  • Problem

    In three and higher dimensions, product expansion is difficult because annihilating nonzero values on full lines can require degree n, which cannot be realized as a codeword for small m.

    Source: §4.2 · p. 21

  • Method

    The paper develops a general algebraic framework using cohomological invariant forms on cell-complex codes, including sheaf codes, to construct transversal multi-controlled-Z gates.

    Source: §5.3 · p. 38, §6 · pp. 42–43

  • Results

    Theorem 1.1 gives transversal logical Ct/2−1Z gates with parameters [[N, Θ(N), Θ(N/(log N)t−1)]] and soundness Θ(1/(log N)t−1); a corollary gives CCZ codes with parameters [[N, Θ(N), Θ(N/(log N)2)]].

    Source: §5.3 · p. 42

  • Takeaways & Limitations

    For codes defined from cell complexes, with or without sheaves, cohomological invariant forms provide a flexible way to construct fault-tolerant multi-controlled-Z gates.

    Source: §6 · p. 43

    The proved logical action is nontrivial, but stronger lower bounds on the number of parallelizable logical gates remain open.

    Source: §6 · p. 43

Abstract

from arXiv · show

We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,\Theta(N),\tilde\Theta(N)]\!]$ quantum low-density parity-check (qLDPC) codes with soundness $\tilde\Theta(1)$, combining nearly optimal code parameters with fault-tolerant non-Clifford gates on qLDPC and quantum locally testable codes for the first time. Remarkably, our proofs proceed through highly general algebraic arguments. Building on insights from [Li et al.,~arXiv:2603.25831], we develop a general covering space framework for constructing and computing a rich family of cohomological invariant forms on sheaf codes that induce transversal logical multi-controlled-$Z$. To certify their nontriviality, we further demonstrate the existence of two-way product-expanding punctured Reed--Solomon codes, which is striking in light of the many negative examples for the product expansion behavior of ordinary Reed--Solomon codes. This approach directly overcomes the previous obstruction to realizing nontrivial logical operations while simultaneously preserving the code parameters. The claimed almost-good code results follow immediately as examples.

2 Preliminaries

The preliminaries set up cell complexes and sheaves as combinatorial structures, then connect their chain and cochain complexes to CSS codes and logical multi-controlled-Z gates. They also define product expansion and extendability tools for the coding constructions.

2.1 Cell complexes and Alexandrov topology

  • Cell complexes are treated as cell posets with an Alexandrov topology, where continuity is equivalent to order preservation.The paper focuses on finite regular cell complexes and their combinatorial structure.Source: §2.1 · pp. 4–5
  • Sparse cell-complex families have uniformly bounded cell incidences, and the paper restricts incidence numbers to 0 or ±1 for simplicity.Closures of cells in regular complexes are homeomorphic to closed balls.Source: §2.1 · pp. 4–5

2.2 Sheaves and their operations

  • A sheaf assigns vector spaces to cells and compatible linear maps along the order, with pullback and tensor products providing its basic operations.The paper works over finite-dimensional vector spaces over Fq, with Fq of characteristic 2.Source: §2.2 · p. 5

2.3 Chain complexes and CSS codes

  • A CSS code is built from consecutive cochain-complex terms, with logical operators represented by homology and cohomology and distance and soundness defined from these spaces.The paper constructs these complexes from sheaves on cell complexes.Source: §2.3 · pp. 5–6

2.4 Logical multi-controlled-Z gates

  • Cohomological invariant forms are unchanged by adding coboundaries and induce logical multi-controlled-Z gates; sparse forms yield constant-depth circuits.The physical diagonal unitary is defined from the form, and its sparse implementation acts across code blocks with constant per-qubit participation.Source: §2.4 · p. 7

2.5 Product-expanding classical codes and extendability

  • Product expansion concerns decompositions of dual tensor product codewords, while extendability asks whether locally valid line restrictions extend to a global tensor-product codeword.The preliminaries also define epsilon-closed sets and the Schur product of classical codes.Source: §2.5 · pp. 7–8

3 Cohomological invariant forms on sheaf codes

The section extends cup and cap products to cell complexes and uses them to construct cohomological invariant forms. These forms induce transversal logical multi-controlled-Z gates on almost-good qLDPC codes and qLTCs.

3.1 Cup and cap products on sheaved simplicial complexes

  • Ref. [9] introduced cup and cap products on sheaved simplicial complexes; its cap product realizes Poincaré duality maps used to estimate logical transversal CZ gates on good qLDPC codes.Source: §3.1 · pp. 9–10

3.2 Barycentric subdivision of cell complexes

  • Barycentric subdivision converts cell posets into simplicial complexes, where vertices represent cells and simplices represent strictly increasing chains.Source: §3.2 · p. 11
  • For finite regular cell complexes, an approximate inverse to subdivision exists and is a chain map satisfying the carrier and left-inverse conditions.Source: §3.2 · pp. 12, 14
  • The sheaf-valued subdivision map and approximate inverse remain chain maps, extending the chain-level construction to sheaf codes.Source: §3.2 · pp. 14–15

3.3 General definition of cup and cap products

  • The generalized products are defined even when X is not simplicial, although their chain-level definitions depend on the approximate inverse; their cohomology-level definitions do not.Source: §3.3 · pp. 15–16
  • On sparse cell complexes, subdivision preserves sparsity, and the subdivision operators and cup and cap products are sparse.Source: §3.3 · p. 16
  • Cup and cap products build cohomological invariant forms whose sparse tensors induce constant-depth logical CCZ circuits and, as shown later, nontrivial multi-controlled-Z gates.The Leibniz rule makes the forms invariant under changing cocycle representatives by coboundaries.Source: §3.3 · p. 17

4 Existence of product-expanding punctured Reed–Solomon codes

This section develops product-expanding punctured Reed–Solomon codes by constructing evaluation sets and proving independence and extendability results in two and three dimensions. It uses extendability of closed sets to establish product expansion and transfer extendability to duals, code–dual combinations, Schur products, and diagonally scaled codes.

4 Existence of product-expanding punctured Reed–Solomon codes

  • The construction represents punctured Reed–Solomon codes by polynomial evaluations, with products corresponding to tensor products of Vandermonde matrices.The proofs consider t = 2, 3, using codes of common dimension m = min_i{m_i}.Source: §4 · p. 17

4.1 Two-dimensional preliminaries

  • For any finite field F_q with q > 2n^2, there exist evaluation sets {y_1, ..., y_n} and {z_1, ..., z_n} making the C_2 ⊗ C_3 evaluation map surjective on F_q^M for every M ⊆ [n]^2 satisfying (4.2).Source: §4.1 · p. 17
  • In two dimensions, sufficiently small-ε closed sets are extendable for the punctured RS codes constructed in Lemma 4.2.Sparse-point data can be adjusted independently while preserving codewords on full lines.Source: §4.1 · pp. 19–21

4.2 Three- and higher-dimensional extendability

  • In three dimensions, full-line configurations can make the coordinate index sets too large and require degree n to annihilate, obstructing codewords when m is small.The section addresses this difficulty with a weaker result that excludes full lines.Source: §4.2 · p. 21
  • For q > 2n^3, evaluation sets make the C1 ⊗ C2 ⊗ C3 evaluation map surjective on every subset M satisfying condition (4.36).Equivalently, the corresponding tensors are independent; condition (4.36) rules out full lines.Source: §4.2 · p. 21
  • For q > 23n^3, evaluation sets can be chosen so a polynomial vanishes on an ε-closed M, including its full lines, except at any specified point in M′ = M \ L(M).This point-isolation property is established for every (a, b, c) in M′.Source: §4.2 · p. 22
  • Cross sections, L(M), M′, and sectional graphs of full lines are ε-closed; Lcross(M) is 2ε-closed and Lcross(M) ∪ Ldense(M) is 3ε-closed.Here M′ = M \ L(M), and Lcross(M) consists of intersections of two lines in the sparse part.Source: §4.2 · p. 25
  • Under the stronger condition (4.75), any local data on the full lines of an ε-closed M extends to a global tensor in C1 ⊗ C2 ⊗ C3.The result applies to evaluation sets obtained in Lemma 4.5.Source: §4.2 · p. 26
  • The extension proof handles dense full-line regions by interpolation, then treats sparse lines by peeling off high-order powers before constructing the global tensor.In three dimensions, sparse lines are not localized in smaller slices, so the information-set technique fails; the proof combines the interpolation and point-isolation constructions instead.Source: §4.2 · pp. 26–28
  • Every ε-closed set satisfying condition (4.75) is extendable for the punctured Reed–Solomon codes C1, C2, C3, yielding product expansion.The proof first extends full lines and then independently reassigns residual values on sparse points.Source: §4.2 · pp. 30–31

4.3 Extendability of dual codes

  • Invertible diagonal scaling preserves extendability: a set is extendable for the original codes if and only if it is extendable for their diagonally scaled versions.The equivalence follows by applying the invertible coordinatewise map, and its inverse, to local and global codewords.Source: §4.3 · p. 31
  • If every ε-closed set is extendable for a tensor product of punctured Reed–Solomon codes, extendability also holds for the tensor product of their duals and for mixtures of codes and duals.The resulting mixed products are product-expanding with at least the same ρ as the product of the dual codes.Source: §4.3 · p. 31
  • Corollary 4.11 extends extendability to r-fold Schur products under its stated rate condition, with each factor freely chosen as a code or its dual.The proof uses that the r-fold product has rate at least that of the original code and establishes the needed rate bound for extendability.Source: §4.3 · p. 32

5 Transversal non-Clifford gates on almost-good qLDPC codes and qLTCs

The section develops algebraic constructions for nontrivial transversal logical multi-controlled-Z gates on almost-good qLDPC codes and qLTCs. It uses covering spaces to transfer cohomological products from HGP codes to sheaf codes, and combines cohomological invariant forms with two-way product-expanding punctured Reed–Solomon codes.

5 Transversal non-Clifford gates on almost-good qLDPC codes and qLTCs

  • The construction establishes nontrivial transversal multi-controlled-Z gates on almost-good qLDPC codes and qLTCs, answering Conjecture 1.2 in Ref. [9] affirmatively.It combines cup and cap products with two-way product-expanding punctured Reed–Solomon codes to preserve code parameters and obtain nontrivial products.Source: §5 · p. 32, §5.3 · p. 39

5.1 Covering spaces and covering maps

  • Odd-sheeted covering maps and transfer maps are compatible with sheaf-valued cup and cap products, allowing cohomological products to be related across the cover.The framework also generalizes the induction scheme for qLDPC codes on cubical complexes to arbitrary cell complexes [10].Source: §5.1 · pp. 32, 36, 37

5.2 Geometry of the cell complexes of almost-good qLDPC codes and qLTCs

  • The construction uses an odd-order finite abelian group to build cell complexes that cover the t-fold hypergraph product of a double cover of an expander graph.The base graph is an n-regular Cayley expander, and the trivial-group case yields the t-fold hypergraph product.Source: §5.2 · pp. 37–38

5.3 Proof details

  • Because the covering degree is odd, projection and transfer are injective, so nontrivial cup products on the HGP construction induce nontrivial products on the sheaf-code cover and nontrivial gates.The construction’s complex and sheaf agree with those used in Ref. for almost-good qLDPC codes and qLTCs.Source: §5.3 · p. 39
  • Low-rate two-way product-expanding punctured Reed–Solomon codes remain two-way product-expanding after replacing two codes with their self-product orthogonal complements.Specifically, (C1 ∗ C1)^⊥, (C2 ∗ C2)^⊥, and C3 are two-way product-expanding.Source: §5.3 · p. 40
  • The cohomological invariant form induces a nontrivial constant-depth logical CCZ gate among the three corresponding CSS codes.Source: §5.3 · p. 41
  • There exist [[N, Θ(N), Θ(N/(log N)^2)]] qLDPC codes supporting nontrivial transversal logical CCZ gates.The proof obtains transversality from a constant-depth logical CCZ construction with at most constant-factor parameter loss.Source: §5.3 · pp. 41–42
  • For t ≥ 1, the invariant form yields a nontrivial transversal logical C^(t−1)Z gate on qLDPC codes with parameters [[N, Θ(N), Θ(N/(log N)^(t−1))]].This construction uses t punctured Reed–Solomon codes satisfying Corollary 4.11.Source: §5.3 · p. 42
  • For even t, the construction also gives a nontrivial transversal logical C^(t/2−1)Z gate with soundness Θ(1/(log N)^(t−1)).The codes have parameters [[N, Θ(N), Θ(N/(log N)^(t−1))]].Source: §5.3 · p. 42

6 Discussion and outlook

The discussion frames the gate results as consequences of general algebraic reasoning and extends the framework to codes defined from cell complexes. It identifies open questions in logical-gate properties and code rates, alongside further directions for the framework.

6 Discussion and outlook

  • Cohomological invariant forms systematically construct fault-tolerant multi-controlled-Z gates for codes defined from cell complexes, with or without sheaf structures.The discussion expects the framework to apply to future good qLTCs built from cell complexes and sheaves, which would likewise admit transversal logical multi-controlled-Z gates.Source: §6 · p. 43
  • Only nontrivial logical Cr−1Z action is proved; better lower bounds on kCr−1Z and more intricate properties such as addressability and parallelizability remain open.The authors expect addressing those properties to require significantly refined analysis and ideas.Source: §6 · p. 43
  • One code block is known to have only a constant number of logical qubits, and sharper rate estimates may require refined sheaf-code rate analysis that permits a broader choice of local codes.The discussion connects this bottleneck to Ref..Source: §6 · p. 43
  • Recent Tanner-code results break the low-local-rate barrier to nontrivial global rate, but their graph node degree and local code length grow sublinearly, so the construction is non-LDPC.The discussion calls for further study of the low-local-rate regime and extension to high-dimensional sheaf codes; explicit logical representatives would also clarify cohomological invariants such as cup products [10].Source: §6 · p. 43
  • The constructions may enable further improvements in asymptotic spacetime overhead for quantum fault tolerance, which the authors leave to future work.Source: §6 · p. 43
Loading 2604.01874v2…