Beam search decoder for quantum LDPC codes

TL;DR

The paper addresses the need for fast, accurate decoding of quantum LDPC codes by introducing a beam search decoder that interleaves masked belief propagation with systematic branching on the least reliable qubits. By maintaining a beam of candidate decoding paths and using a reliability score based on the summed posterior LLRs, it achieves higher accuracy and lower tail latency than the traditional BP-OSD decoder, with tunable parameters to balance speed and precision. Across BB and HGP code families and circuit-level noise, the method demonstrates substantial improvements in logical error rate and 99.9th percentile runtime, including sub-millisecond decoding per syndrome extraction on a single CPU for realistic trapped-ion scenarios. These results suggest practical scalability for large quantum systems and potential generalization to XYZ-decoding and other quantum LDPC codes.

Abstract

We propose a decoder for quantum low density parity check (LDPC) codes based on a beam search heuristic guided by belief propagation (BP). Our beam search decoder applies to all quantum LDPC codes and achieves different speed-accuracy tradeoffs by tuning its parameters such as the beam width. We perform numerical simulations under circuit level noise for the $[[144, 12, 12]]$ bivariate bicycle (BB) code at noise rate $p=10^{-3}$ to estimate the logical error rate and the 99.9 percentile runtime and we compare with the BP-OSD decoder which has been the default quantum LDPC decoder for the past six years. A variant of our beam search decoder with a beam width of 64 achieves a $17\times$ reduction in logical error rate. With a beam width of 8, we reach the same logical error rate as BP-OSD with a $26.2\times$ reduction in the 99.9 percentile runtime. We identify the beam search decoder with beam width of 32 as a promising candidate for trapped ion architectures because it achieves a $5.6\times$ reduction in logical error rate with a 99.9 percentile runtime per syndrome extraction round below 1ms at $p=5 \times10^{-4}$. Remarkably, this is achieved in software on a single core, without any parallelization or specialized hardware (FPGA, ASIC), suggesting one might only need three 32-core CPUs to decode a trapped ion quantum computer with 1000 logical qubits.

Figures (4)

• Figure 1: Overview of the beam search decoder. (a) The decoder is initialized with a small number of BP iterations. Then, the least reliable error node (red) is selected and two branches are created corresponding to the two possible values of this node. After the first step, BP is replaced by a masked BP ignoring the previously fixed error nodes. (b) The beam search decoder repeats the following three steps: (i) branching over the least reliable error node (ii) running a masked BP and (iii) pruning to reduce the number of paths to the beam width (which is 2 in this figure). The decoder is terminated once a sufficient number of solutions is found or if a maximum number of repetitions is reached.
• Figure 2: Simulation results for the $[[144,12,12]]$ BB code under circuit-level noise. BP-OSD decoder is configured with 30 min-sum BP iterations followed by order-10 combination-sweep OSD.
• Figure 3: Simulation results for the $[[90,8,10]]$ BB code under circuit-level noise. The suffix _XYZ in the legend denotes XYZ-decoding, which utilizes both X and Z syndrome outcomes simultaneously for decoding.
• Figure 4: Simulation results for the $[[450,32,8]]$ HGP code aydin2025cyclic.