A Factor-Graph Formulation of CSS Syndrome Decoding: Joint BP and Four-State BP
This is an incremental theoretical clarification for quantum error correction practitioners, showing equivalence of two decoding approaches.
The paper introduces joint belief propagation (joint BP) for CSS syndrome decoding, which retains local channel correlation between Pauli error components, and shows it is equivalent to four-state BP after relabeling and marginalization.
For CSS syndrome decoding, the two check matrices impose binary parity-check constraints on the two Pauli error components. The posterior can therefore be written as a binary factor graph with two Tanner graphs coupled by the local joint prior at each qubit. We call the sum-product algorithm on this factorization joint belief propagation (joint BP). Joint BP retains the local channel correlation between the two Pauli components. This note compares joint BP with the four-state Pauli-label factor graph used for four-state BP. The two algorithms are shown to have the same posterior weights, messages, and beliefs after relabeling the four local Pauli states and marginalizing the irrelevant binary component.