Neural Belief-Matching Decoding for Topological Quantum Error Correction Codes

Abstract

Quantum error correction (QEC) is critical for scalable fault-tolerant quantum computing. Topological codes, such as the toric code, offer hardware-efficient architectures but their Tanner graphs contain many girth-4 cycles that degrade the performance of belief-propagation (BP) decoding. For this reason, BP decoding is typically followed by a more complex second stage decoder such as minimum-weight perfect matching. These combined decoders achieve a remarkable performance, albeit at the cost of increased complexity. In this paper we propose two key improvements for the decoding of toric code. The first one is replacing the BP decoder by a neural BP decoder, giving rise to the neural belief-matching decoder which substantially decreases the average decoding complexity. The main drawback of this approach is the high cost associated with the training of the neural BP decoder. To address this issue, we impose a convolutional architecture on the neural BP decoder, enabling weight sharing across the spatially homogeneous structure of the code's factor graph. This design allows a model trained on a modest-size topological code to be directly transferred to much larger instances, preserving decoding quality while dramatically lowering the training burden. Our numerical experiments on toric-code lattices of various sizes demonstrate that this technique does not result in a noticeable loss in performance.

Figures (8)

• Figure 1: Illustration of the toric code. Edge qubits (dots) are measured by weight‑4 stabilizers $\boldsymbol{X}$-type vertex checks (blue) and $\boldsymbol{Z}$‑type plaquette checks (red). Light‑blue and light‑red non‑contractible strings denote the logical operators $\overline{\boldsymbol{X}}_{1,2}$ and $\overline{\boldsymbol{Z}}_{1,2}$ that wrap the two cycles of the torus.
• Figure 2: Construction of $\boldsymbol{G} _{\text{type}}^{\text{\tiny CN}}$ for $\boldsymbol{X}$‑stabilizers on a $d\!=\!4$ toric code. A $2\times2$ patch of $\boldsymbol{X}$‑stabilizers (and their edges) is tiled over the lattice, enforcing weight‑sharing and translation invariance; the $\boldsymbol{Z}$‑stabilizers are built analogously.
• Figure 3: Convolution in the convolutional NBP decoder: the check‑node weight tensor $\boldsymbol{W}_{CN}$ is filtered by $\boldsymbol{G} _{\text{type}}^{\text{\tiny CN}}$ to enforce weight-sharing, and average‑pooling over identical pivots produces the final tensor $\boldsymbol{W}_{CN}^{*}$.
• Figure 4: Weight‑reuse across code distances: the tensors $\boldsymbol{W}_{d\times d}^{n^{*}}$ for check nodes, variable nodes, and LLRs trained on a $d\!=\!4$ toric code are tiled to construct decoders for larger $d$, eliminating the need for costly retraining.
• Figure 5: versus physical error rate for toric codes ($d\!=\!4,6,8,10$) using belief‑matching, , and RNBP‑matching decoders. results are taken from miao2025quaternary.