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La-cross code[1]

Description

Code constructed using a hypergraph product of two copies of a cyclic LDPC code. The construction uses cyclic LDPC codes with generating polynomials \(1+x+x^k\) for some \(k\). Using a length-\(n\) seed code yields an \([[2n^2,2k^2]]\) family for periodic boundary conditions and an \([[(n-k)^2+n^2,k^2]]\) family for open boundary conditions.

Cousins

  • Quasi-cyclic LDPC (QC-LDPC) code— La-cross codes are constructed using a hypergraph product of a cyclic LDPC code with itself.
  • Cyclic Hypergraph Product Code— The La-cross code is a reduced block length, full-rank cyclic HGP code with generator polynomials of the form \(1+x+x^k\)
  • Long-range enhanced surface code (LRESC)— La-cross codes yield LRESCs for \(k=2\). La-cross codes have a number of long-range stabilizers that scales linearly with code size, while the number of LRESC long-range stabilizers can be tuned to scale between the square-root of the size and linearly in the size.
  • Kitaev surface code— La-cross codes with periodic (open) boundary conditions reduce to the toric (planar surface) code at \(k=1\).

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Higher-dimensional homological product code
Homological product codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Hypergraph product (HGP) codeCSS QLDPC Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based Qubit QECC Quantum
Parents
Hypergraph product (HGP) code
La-cross codes are constructed using a hypergraph product of a cyclic LDPC code with itself.
Cyclic quantum codeQECC Quantum
La-cross code

References

[1]
L. Pecorari, S. Jandura, G. K. Brennen, and G. Pupillo, “High-rate quantum LDPC codes for long-range-connected neutral atom registers”, Nature Communications 16, (2025) arXiv:2404.13010 DOI
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Zoo Code ID: lacross

Cite as:
“La-cross code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/lacross, arXiv:2606.11484
BibTeX:
@incollection{eczoo_lacross,
title={La-cross code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/lacross}
}
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Permanent link:
https://errorcorrectionzoo.org/c/lacross

Cite as:

“La-cross code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/lacross, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/balanced_product/tensor/singlesector/hypergraph/lacross.yml.