## Description

QLDPC code whose stabilizer generator matrix resembles the parity-check matrix of SC-LDPC codes. There exist CSS [1] and stabilizer constructions [2]. In either case, the stabilizer generator matrix is constructed by "spatially" coupling sub-matrix blocks in chain-like fashion (or, more generally, in grid-like fashion) to yield a band matrix. The sub-matrix blocks have to satisfy certain conditions amongst themselves so that the resulting band matrix is a stabilizer generator matrix. Matrices corresponding to translationally invariant chains are called time-variant, and otherwise are called time-invariant.

A finite-length chain is then capped by imposing either open boundary conditions (yielding non-tail-biting SC-QLDPC codes) or open boundary conditions (yielding tail-biting SC-QLDPC codes). Both constructions [1,2] are tail-biting.

In the stabilizer construction [2], the structure of the band matrix allows codes to be concisely defined in terms of characteristic polynomials, whose coefficients are the sub-matrix blocks and which resemble the Pauli-to-polynomial mapping associated with translationally invariant stabilizer codes. Some CSS code constructions can used to define sub-matrix blocks, yielding spatially coupled (i.e., translationally invariant) extensions of such codes.

For example, the \(3\times 3\) toric code can be expressed as an SC-QLDPC code with stabilizer generator matrix given in Figure I.

## Parents

- Qubit stabilizer code
- Translationally invariant stabilizer code — Stabilizer generator matrices of SC-QLDPC codes on infinite-length chains or grids define a class of translationally-invariant stabilizer codes.

## Children

- Hypergraph product (HGP) code — Hypergraph-product stabilizer generator matrices can be used as sub-matrices to define a 2D SC-QLDPC code [2].
- XYZ product code — XYZ-product stabilizer generator matrices can be used as sub-matrices to define a 2D SC-QLDPC code [2].

## Cousins

- Spatially coupled LDPC (SC-LDPC) code — SC-QLDPC code stabilizer-generator matrices have similar block form as the parity-check matrices of SC-LDPC codes.
- Quasi-cyclic code — Quasi-cyclic binary code parity-check matrices can be used as sub-matrices to define a 1D SC-QLDPC code [1].
- Generalized bicycle (GB) code — Qubit GB stabilizer generator matrices is equivalent to a 1D SC-QLDPC code, see [2; Remark 7].

## References

- [1]
- M. Hagiwara et al., “Spatially Coupled Quasi-Cyclic Quantum LDPC Codes”, (2011) arXiv:1102.3181
- [2]
- S. Yang and R. Calderbank, “Spatially-Coupled QDLPC Codes”, (2023) arXiv:2305.00137

## Page edit log

- Victor V. Albert (2023-05-10) — most recent

## Cite as:

“Quantum spatially coupled (SC-QLDPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/sc_qldpc