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

2026-05-06

AI Summary

This work addresses the challenge of modeling local channel correlations between Pauli-X and Pauli-Z error components in syndrome decoding for Calderbank–Shor–Steane (CSS) codes. The authors formulate the decoding problem as a factor graph over coupled binary Tanner graphs and propose a joint belief propagation (joint BP) algorithm. By introducing a four-state Pauli-label factor graph and leveraging message passing combined with probabilistic marginalization, they rigorously prove that joint BP is mathematically equivalent to conventional four-state BP in terms of posterior weights, message updates, and final beliefs. This equivalence not only clarifies the theoretical relationship between the two approaches but also demonstrates that joint BP explicitly preserves the local correlation between X and Z errors while maintaining identical decoding performance, thereby offering a novel perspective for correlation-aware decoding strategies.

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.

Problem

• CSS syndrome decoding
• factor graph
• belief propagation
• Pauli errors
• Tanner graph

Innovation

• factor graph
• CSS syndrome decoding
• joint belief propagation
• four-state BP
• quantum error correction