Foliated quantum codes are a resource for fault-tolerant measurement-based
quantum error correction for quantum repeaters and for quantum computation.
They represent a general approach to integrating a range of possible quantum
error correcting codes into larger fault-tolerant networks. Here we present an
efficient heuristic decoding scheme for foliated quantum codes, based on
message passing between primal and dual code 'sheets'. We test this decoder on
two different families of sparse quantum error correcting code: turbo codes and
bicycle codes, and show reasonably high numerical performance thresholds. We
also present a construction schedule for building such code states.Comment: 23 pages, 15 figures, accepted for publication in Phys. Rev.
