Transversal-\(CCZ\) Reed-Solomon cubical sheaf code

Transversal-\(CCZ\) Reed-Solomon cubical sheaf code [1]

Description

One of three degree-two Reed-Solomon cubical sheaf codes on a common six-dimensional complex, which together admit \(\Theta(N)\) addressable and parallel transversal logical \(CCZ\) gates for block length \(N\) [1]. Block \(a\in\{1,2,3\}\) uses lower-degree projective Reed-Solomon local codes in directions \(2a-1\) and \(2a\) than in the other four [1; Sec. 3.1].

Rate

Constant rate in all three blocks [1; Sec. 3.4].

Magic

Magic-state yield parameter \(\gamma=\log(M/k)/\log d=o(1)\) for \(CCZ\)-state distillation, where \(k=\Omega(N)\) logical \(CCZ\) gates use \(M=O(N)\) physical ones and \(d=\Theta(N)\) [1; Sec. 3.5].

Transversal and Permutation-Based Gates

Cup-product logical \(CCZ\) gates on any subset of \(m=\Omega(N)\) logical triples within a logical subspace, so the logical trilinear form has subrank at least \(m\) [1; Sec. 3.3]. After a fixed repetition encoding, each addressed product of logical \(CCZ\) gates is one layer of disjoint physical \(CCZ\) gates [1; Sec. 3.4].

Decoding

Constant-depth single-shot decoder based on parallel local flips for the construction of an earlier arXiv version of Ref. [1][2; Lemmas 6.8 and 6.11].

Fault Tolerance

Fault-tolerant quantum computation against locally stochastic noise with strictly logarithmic time and constant space overheads for sufficiently wide circuits, for an earlier arXiv version of Ref. [1][2; Thm. 3.12 and Cor. 3.13]. A width-\(W\), depth-\(D\) circuit runs in depth \(O(D\log(WD/\varepsilon))\), and only stabilizer resource states require separate preparation [2].

Cousins

Good QLDPC code— The transversal-\(CCZ\) Reed-Solomon cubical sheaf construction yields good QLDPC codes with \(\Theta(N)\) addressable and parallel transversal logical \(CCZ\) gates [1; Thm. 1.1].
Single-shot code— In the construction of an earlier arXiv version of Ref. [1], transversal-\(CCZ\) Reed-Solomon cubical sheaf codes admit a constant-depth single-shot decoder based on parallel local flips [2; Lemmas 6.8 and 6.11].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum

Dinur-Lin-Vidick (DLV) code

Reed-Solomon cubical sheaf codeQLTC Hamiltonian-based QECC Quantum

Parents

Reed-Solomon cubical sheaf code

Transversal-\(CCZ\) Reed-Solomon cubical sheaf codes are degree-two Reed-Solomon cubical sheaf codes on six-dimensional complexes, taken in triples whose blocks use lower-degree local codes in distinct pairs of directions [1; Sec. 3.1]. The cup product of their cochains yields a linear number of addressable transversal logical \(CCZ\) gates [1; Thm. 1.1].

Transversal-\(CCZ\) Reed-Solomon cubical sheaf code

References

[1]
Y. Li, Z. Li, Z. Han, and Z.-W. Liu, “Parallelizable and addressable transversal non-Clifford gates on good quantum LDPC codes”, (2026) arXiv:2609.26691
[2]
Z. Han and Z.-W. Liu, “Purely-logarithmic-time- and constant-space-overhead fault-tolerant quantum computation”, (2026) arXiv:2609.28461

Page edit log

Victor V. Albert (2026-11-08) — most recent

Cite as:

“Transversal-\(CCZ\) Reed-Solomon cubical sheaf code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/ccz_rs_cubical_sheaf, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/other_product/ccz_rs_cubical_sheaf.yml.