Spatially Coupled MacKay-Neal/Hsu-Anastasopoulos CSS Codes Achieve the Quantum-Erasure Hashing Bound by Seeded BP Decoding

Abstract: In classical sparse-graph coding, spatial coupling is a mechanism by which belief-propagation (BP) decoding attains the maximum-a-posteriori (MAP) or area-threshold performance of the uncoupled system. Since MacKay-Neal/Hsu-Anastasopoulos (MN/HA) punctured sparse ensembles achieve capacity under MAP decoding, it is natural to ask whether spatially coupled MN/HA-type Calderbank-Shor-Steane (CSS) codes can reach the hashing bound on the quantum erasure channel under seeded BP decoding. We answer this question at the density evolution (DE) level for hard-erasure CSS decoding. On an erased coordinate, the two binary Pauli components remain unresolved, equivalently the erased qubit is represented by the four Pauli possibilities. We first define the CSS ensemble through sparse punctured matrices and the corresponding dense parity-check matrices. For fixed finite Z-side, X-side, and check degrees, we then derive a five-message uncoupled DE recursion, decompose it into Z-side and X-side constituent systems, and define the two constituent potentials. Applying the coupled-vector potential method to the two constituents separately proves that seeded BP decoding on the resulting finite-degree factor graphs reaches the smaller of the Z-side degree ratio and the X-side complementary degree ratio. In the X/Z equal-rate specialization, where the Z-side and X-side constituent design rates are equal, this BP threshold is the hashing-bound channel parameter determined by the design rate. Thus the paper gives a DE-level proof that seeded BP decoding with finite-degree factor graphs achieves the hashing bound for the X/Z equal-rate family. Finite-length BP concentration, block-error convergence, and a finite-code realization of the ideal DE seed are separate questions.