La-cross code[1]
Description
La-cross codes are hypergraph products of two copies of a classical seed code with generating polynomial \(1+x+x^k\) for some \(k\) [1]. They are high-rate quantum LDPC codes with moderate-range, weight-six checks suited to neutral-atom registers.
A square circulant seed of length \(n\) gives a periodic-boundary code with parameters \([[2n^2,2k^2]]\). A rectangular full-rank seed \(H\in\mathbb{F}_2^{(n-k)\times n}\) instead gives an open-boundary code with parameters \([[(n-k)^2+n^2,k^2]]\).
Cousins
- Quasi-cyclic LDPC (QC-LDPC) code— Periodic-boundary La-cross codes are hypergraph products of classical quasi-cyclic (cyclic) LDPC seed codes [1].
- Cyclic hypergraph product code— Periodic-boundary La-cross codes are CxC codes built from the seed polynomial \(1+x+x^k\); the open-boundary La-cross family instead uses a rectangular full-rank seed [1].
- 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.
- Tile quantum code— Open-boundary La-cross codes are instances of tile codes whose stabilizers are induced by univariate polynomials [2].
- Kitaev surface code— La-cross codes with periodic (open) boundary conditions reduce to the toric (planar surface) code at \(k=1\).
- Multivariate multicycle (MM) code— La-cross codes with periodic boundary conditions are \(t=2\) MM codes over \(\mathbb{Z}_n\times\mathbb{Z}_n\); their open-boundary versions fall outside the periodic group-algebra construction [1,3].
Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Homological product codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Hypergraph product (HGP) codeCSS QLDPC Qubit Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Parents
La-cross codes are hypergraph products of two identical classical seed codes with generating polynomial \(1+x+x^k\) [1].
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
- [2]
- N. P. Breuckmann, S. H. Choe, J. N. Eberhardt, F. R. F. Pereira, and V. Steffan, “Logical Operators and Derived Automorphisms of Tile Codes”, (2025) arXiv:2511.14589
- [3]
- F. A. Mian, O. Gwilliam, and S. Krastanov, “Multivariate Multicycle Codes for Complete Single-Shot Decoding”, (2026) arXiv:2601.18879
Page edit log
- Victor V. Albert (2026-08-19) — most recent
- Victor V. Albert (2026-06-08)
- Victor V. Albert (2024-04-22)
Cite as:
“La-cross code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/lacross, arXiv:2606.11484