[Paper Review] Exploiting Degeneracy in Belief Propagation Decoding of Quantum Codes

Kao-Yueh Kuo, Ching–Yi Lai|arXiv (Cornell University)|Apr 28, 2021

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes Memory-Enhanced Belief Propagation (MBP), a quaternary belief propagation decoder with memory effects and inhibitory edge weights that exploit quantum code degeneracy to significantly improve decoding performance. MBP achieves error thresholds of 16% on surface codes and 17.5% on toric codes under depolarizing noise, outperforming conventional BP by leveraging memory and adaptive step-sizes to enhance convergence and error correction in highly-degenerate quantum stabilizer codes.

ABSTRACT

Quantum information needs to be protected by quantum error-correcting codes due to imperfect physical devices and operations. One would like to have an efficient and high-performance decoding procedure for the class of quantum stabilizer codes. A potential candidate is Pearl's belief propagation (BP), but its performance suffers from the many short cycles inherent in a quantum stabilizer code, especially highly-degenerate codes. A general impression exists that BP is not effective for topological codes. In this paper, we propose a decoding algorithm for quantum codes based on quaternary BP with additional memory effects (called MBP). This MBP is like a recursive neural network with inhibitions between neurons (edges with negative weights), which enhance the perception capability of a network. Moreover, MBP exploits the degeneracy of a quantum code so that the most probable error or its degenerate errors can be found with high probability. The decoding performance is significantly improved over the conventional BP for various quantum codes, including quantum bicycle, hypergraph-product, surface and toric codes. For MBP on the surface and toric codes over depolarizing errors, we observe error thresholds of 16% and 17.5%, respectively.

Motivation & Objective

• To address the poor performance of conventional belief propagation (BP) in decoding highly-degenerate quantum stabilizer codes due to short cycles and lack of degeneracy exploitation.
• To develop a low-complexity, high-performance decoding algorithm for quantum codes that outperforms standard BP, especially on topological codes like surface and toric codes.
• To integrate memory effects and inhibitory weights into BP to enhance convergence and perception, mimicking a recursive neural network with adaptive step-sizes.
• To achieve near-optimal decoding performance with complexity nearly linear in code length, enabling practical application to large-scale quantum codes.

Proposed method

• Proposes Memory-Enhanced Belief Propagation (MBP), a quaternary BP variant that models error probabilities using log-likelihood ratios (LLRs) and incorporates memory via time-varying edge weights.
• Introduces a generalized energy function combining parity-check satisfaction (J_S) and distance to initial channel statistics (J_D), framing BP as gradient descent on this function.
• Uses adaptive step-sizes scaled by 1/α and fixed inhibition strengths between nodes to stabilize message updates and prevent convergence to incorrect beliefs.
• Employs a recursive neural network-like structure with inhibitory edges (negative weights) to improve belief propagation dynamics and enhance detection of degenerate error patterns.
• Applies a scalar-based approach with iterative updates governed by equations (10)–(13), where edge weights are adjusted per iteration to improve convergence and error correction.
• Supports parallel decoding by grouping qubits into independent sets (e.g., 4-qubit groups in toric codes), enabling O(1) decoding time with full parallelism.

Experimental results

Research questions

• RQ1 Can memory effects and inhibitory weights in belief propagation significantly improve decoding performance for highly-degenerate quantum stabilizer codes?
• RQ2 How does exploiting code degeneracy through a quaternary BP framework enhance error correction beyond conventional BP?
• RQ3 What is the achievable error threshold of the proposed MBP decoder on surface and toric codes under depolarizing noise?
• RQ4 Can the proposed MBP achieve near-linear complexity while maintaining high performance, especially in the presence of short cycles in Tanner graphs?
• RQ5 How can adaptive step-sizes and edge weight dynamics be designed to improve convergence and reduce undetected error rates?

Key findings

• MBP achieves an error threshold of approximately 16% on surface codes under depolarizing noise, significantly outperforming conventional BP.
• For toric codes, MBP achieves a threshold of roughly 17.5%, demonstrating superior performance on topological codes.
• The average number of iterations for MBP is substantially lower than for conventional BP, indicating improved convergence behavior without increased complexity.
• The use of memory and inhibitory weights reduces the undetected error rate, especially as code size L increases, indicating better robustness.
• Parallel decoding using 4-qubit groups achieves a threshold of about 15.5% on surface codes, confirming scalability and practicality of the approach.
• The proposed method enables near-linear complexity decoding with O(Nj log log N) performance, making it suitable for large-scale quantum codes.