Quantum-enhanced belief propagation for LDPC decoding

TL;DR

The paper addresses LDPC decoding bottlenecks due to cycles in belief propagation by introducing quantum-enhanced belief propagation (QEBP), which uses QAOA as a pre-processing step to warm-start BP. By estimating per-bit error probabilities from QAOA and combining them with a channel model, QEBP achieves faster convergence and lower BLER than standard BP and QAOA-only syndrome decoding on small codes. A transfer-matrix approach is developed for the repetition code to optimize QAOA parameters and compare post-processing strategies, revealing trade-offs between BLER, rounds, and computational costs. The work demonstrates a promising hybrid quantum-classical approach with potential modular integration into decoders, while highlighting current hardware limitations and the need for scalable parameter optimization across decoding rounds.

Abstract

Decoding low-density parity-check codes is critical in many current technologies, such as fifth-generation (5G) wireless networks and satellite communications. The belief propagation algorithm allows for fast decoding due to the low density of these codes. However, there is scope for improvement to this algorithm both in terms of its computational cost when decoding large codes and its error-correcting abilities. Here, we introduce the quantum-enhanced belief propagation (QEBP) algorithm, in which the Quantum Approximate Optimization Algorithm (QAOA) acts as a pre-processing step to belief propagation. We perform exact simulations of syndrome decoding with QAOA, whose result guides the belief propagation algorithm, leading to faster convergence and a lower block error rate (BLER). In addition, through the repetition code, we study the possibility of having shared variational parameters between syndromes and, in this case, code lengths. We obtain a unique pair of variational parameters for level-1 QAOA by optimizing the probability of successful decoding through a transfer matrix method. Then, using these parameters, we compare the scaling of different QAOA post-processing techniques with code length.

Figures (13)

• Figure 1: Tanner graph representation of a linear code. On the left, the Tanner graph with check nodes $c_i$ and variable nodes $v_i$ and, on the right, the corresponding parity-check matrix.
• Figure 2: Binary symmetric channel with crossover probability $\varepsilon$.
• Figure 3: Message passing in the belief propagation algorithm. Here, $m_{f_1 \rightarrow v_2}$ is the message from factor $f_1$ to variable node $v_2$, $m_{v_1 \rightarrow f_1}$ is the message from variable node $v_1$ to factor $f_1$ and $b_1(v_1)$ is the belief of variable $v_1$.
• Figure 4: Decoding process for a single decoding round of belief propagation, quantum-enhanced belief propagation, and syndrome decoding with QAOA.
• Figure 5: Block error rate against signal-to-noise ratio $(E_0/N_b)$ for syndrome decoding with QAOA with $p=3$. We compare results using QAOA syndrome decoding, QAOA with codeword post-selection, and quantum-enhanced belief propagation decoding.

Theorems & Definitions (1)

• Lemma 1: Independence of Error Probability Under Symmetry