Layer code

Layer code[1]

Description

Member of a family of 3D lattice CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface code according to the Tanner graph of a QLDPC code (or a more general qubit stabilizer code). Geometric locality is maintained because, instead of being concatenated, each pair of parallel surface-code squares are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.

Rate

Code parameters on a cube, of order \((10,40,4)\) , achieve the 3D BPT bound when asymptotically good QLDPC codes are used in the construction.

Decoding

Decoders against stochastic and adversarial noise [2].

Cousins

Good QLDPC code— Layer code parameters, of order \((10,40,4)\) , achieve the BPT bound in 3D when asymptotically good QLDPC codes are used in the construction.
Concatenated qubit code— Each pair of surface-code squares in a layer code are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.
Self-correcting quantum code— The energy barrier of excitations for layer codes constructed using asymptotically good QLDPC codes scales as order \(\Theta{n^{1/3}}\) [1]. Layer codes are partially self-correcting quantum memories [2]. Layer codes constructed from random CSS codes have near-optimal scaling of code parameters and a polynomial energy barrier, exhibiting behavior consistent with partial self correction [2].
Kitaev surface code— Layer codes are combinations of constant-rate QLDPC codes with surface codes built using lattice surgery.

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

Parents

Qubit CSS code

Fracton stabilizer codeLattice stabilizer QLDPC Stabilizer Hamiltonian-based QECC Quantum

Layer codes are non-translation invariant 3D lattice stabilizer codes that can be viewed as fracton topological defect networks [1].

Abelian topological codeTopological Hamiltonian-based QECC Quantum

The Layer code realizes 2D layers of \(\mathbb{Z}_2\) gauge theory coupled along defects.

Layer code

References

[1]: D. J. Williamson and N. Baspin, “Layer Codes”, (2024) arXiv:2309.16503
[2]: S. Gu, L. Caha, S. H. Choe, Z. He, A. Kubica, and E. Tang, “Layer codes as partially self-correcting quantum memories”, (2025) arXiv:2510.06659

Page edit log

Victor V. Albert (2024-02-12) — most recent

Cite as:

“Layer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/layer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/fracton/layer.yml.