Quantum spatially coupled (SC-QLDPC) code[1,2]

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.