A Factor-Graph Formulation of CSS Syndrome Decoding: Joint BP and Four-State BP

TL;DR

This paper introduces a factor-graph formulation for CSS syndrome decoding, comparing joint belief propagation with four-state BP, and shows they produce equivalent posterior weights and beliefs after relabeling.

Contribution

It formulates CSS syndrome decoding as a binary factor graph and compares joint BP with four-state BP, demonstrating their equivalence in posterior estimation.

Findings

Joint BP and four-state BP have the same posterior weights after relabeling.

The two algorithms produce identical messages and beliefs under the proposed relabeling.

The factor-graph approach captures the local channel correlation in CSS decoding.

Abstract

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.