Improved belief propagation is sufficient for real-time decoding of quantum memory

TL;DR

Relay-BP introduces a real-time, FPGA-friendly decoder for quantum LDPC codes by augmenting Belief Propagation with disordered memory and relaying marginals across multiple DMem-BP legs. The approach mitigates BP oscillations and symmetry trapping, enabling multiple valid corrections and greatly lowering logical error rates on BB codes and competitively with matching-based decoders on surface codes. Memory strengths are problem-dependent, with negative values playing a crucial role, and a relay ensemble structure yields both accuracy gains and fast convergence within realistic iteration budgets. The results support real-time decoding viability for quantum memory in large-scale quantum processors, with potential FPGA/ASIC implementations.

Abstract

We introduce a new heuristic decoder, Relay-BP, targeting real-time quantum circuit decoding for large-scale quantum computers. Relay-BP achieves high accuracy across circuit-noise decoding problems: significantly outperforming BP+OSD+CS-10 for bivariate-bicycle codes and comparable to min-weight-matching for surface codes. As a lightweight message-passing decoder, Relay-BP is inherently parallel, enabling rapid low-footprint decoding with FPGA or ASIC real-time implementations, similar to standard BP. A core aspect of our decoder is its enhancement of the standard BP algorithm by incorporating disordered memory strengths. This dampens oscillations and breaks symmetries that trap traditional BP algorithms. By dynamically adjusting memory strengths in a relay approach, Relay-BP can consecutively encounter multiple valid corrections to improve decoding accuracy. We observe that a problem-dependent distribution of memory strengths that includes negative values is indispensable for good performance.

Figures (5)

• Figure 1: (a) In BP, the belief that each error occurred is updated over each iteration. However, some beliefs can oscillate (red, blue) instead of converging (green). (b) A memory term can dampen oscillations, but symmetric trapping sets may lead to convergence to uncertain beliefs (red, blue). (c) Disordered memory strengths can break symmetries, leading to decisive beliefs forming a valid solution (i.e. the syndrome is canceled). (d) Relay-BP chains together different DMem-BP runs, which can further aid convergence and provide multiple valid solutions without restarting. Solid lines indicate the weight of the proposed correction while dashed lines indicate the syndrome weight after the proposed correction.
• Figure 2: Circuit-noise decoding examples. (Top) Relay-BP-1 logical error rate heatmaps at $p = 3 \times 10^{-3}$ vs. memory strength intervals. Circles mark intervals used for decoding: $[-0.24, 0.66]$ (gross), $[-0.161, 0.815]$ (two-gross), $[-0.254, 0.985]$ (surface). The dashed line indicates a threshold above which negative $\gamma_j$ are present. (Bottom) Relay-BP outperforms BP+OSD+CS-10 on the gross and two-gross codes, and performs comparably to Matching on the surface code. Top panels share their width scale but have separate logical error rate heat maps; bottom panels share a logical error rate scale.
• Figure 3: Logical error rate vs. average number of BP iterations at $p = 3 \times 10^{-3}$. Relay-BP-$S$ curves are generated by varying the maximum number of legs $R$; BP and Mem-BP curves by varying their maximum iteration count. Relay-BP achieves substantially lower error rates than other decoders within the estimated 600-iteration real-time budget.
• Figure 4: The decoding graph visually represents the check matrix: circular ($\ocircle$) error nodes correspond to columns of $\mathbf{H}$, and square ($\square$) check nodes to its rows. Filled nodes denote value 1 and unfilled nodes value 0. The decoder relies solely on the syndrome (i.e., the values of the check nodes) to infer a candidate error. Check node $i$ has syndrome $\sigma_i = 1$ if it touches an odd number of error nodes with value one.
• Figure 5: Logical error rate vs. average number of BP iterations at $p = 3 \times 10^{-3}$, with XYZ-decoding. XYZ-Relay-BP-$S$ curves are generated by varying the maximum number of legs $R$; XYZ-BP and XYZ-Mem-BP curves by varying their maximum iteration count.