An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise

TL;DR

The paper tackles the challenge of real-time decoding for quantum LDPC codes under circuit-level noise by introducing an almost-linear-time pipeline: an initial belief-propagation stage on the full detector graph, a sparsified-second BP stage guided by a transfer matrix, and an orders-based Kruskal-derived post-processing step (OTF). This BP+BP+OTF framework, together with a sparsification strategy that preserves the detector-to-syndrome mapping, achieves decoding performance comparable to state-of-the-art methods like BP+OSD and MWPM while offering substantial runtime advantages. Empirical results on bivariate bicycle codes and rotated surface codes demonstrate strong logical error suppression and significant speedups, with convergence guarantees for surface codes via a virtual node augmentation. The work emphasizes hardware-friendly design, enabling real-time decoding prospects on ASIC/FPGA platforms and suggesting broader applicability to other QEC codes and decoder families through the sparsified-detector approach.

Abstract

Fault-tolerant quantum computers must be designed in conjunction with classical co-processors that decode quantum error correction measurement information in real-time. In this work, we introduce the belief propagation plus ordered Tanner forest (BP+OTF) algorithm as an almost-linear time decoder for quantum low-density parity-check codes. The OTF post-processing stage removes qubits from the decoding graph until it has a tree-like structure. Provided that the resultant loop-free OTF graph supports a subset of qubits that can generate the syndrome, BP decoding is then guaranteed to converge. To enhance performance under circuit-level noise, we introduce a technique for sparsifying detector error models. This method uses a transfer matrix to map soft information from the full detector graph to the sparsified graph, preserving critical error propagation information from the syndrome extraction circuit. Our BP+OTF implementation first applies standard BP to the full detector graph, followed by BP+OTF post-processing on the sparsified graph. Numerical simulations show that the BP+OTF decoder achieves similar logical error suppression compared to state-of-the-art inversion-based and matching decoders for bivariate bicycle and surface codes, respectively, while maintaining almost-linear runtime complexity across all stages.

Figures (8)

• Figure 1: Top: the Tanner graph of a classical cyclic code altogether with its parity check matrix $H$. Grey circles denote bits and green circles parity checks. Bottom: the ordered Tanner forest generated by the OTF algorithm and its corresponding parity check matrix $H_{otf}$. Note that the third bit has been removed from the graph. This ensures that the $H_{otf}$ matrix is cycle free.
• Figure 2: Logical error rate per syndrome extraction round as a function of the physical error rate for bivariate bicycle codes. Each code is simulated over a number of syndrome rounds equal to its distance. The shading highlights the region of estimated probabilities where the likelihood ratio is within a factor of $1,000$; similar to a confidence interval.
• Figure 3: Mean decoding time per round for BP+OSD-0 and BP+BP+OTF when decoding bivariate bicycle codes. For the BP+BP+OTF ensemble we record the time of the first instance that finishes decoding. a) Decoding at physical error rate $p=0.003$. b) Decoding at physical error rate $p=0.004$.
• Figure 4: Logical error rate per syndrome extraction round as a function of the physical error rate for rotated surface codes. Each code is simulated over a number of syndrome rounds equal to its distance using Stim's surface_code:rotated_memory_z experiment. The shading highlights the region of estimated probabilities where the likelihood ratio is within a factor of $1,000$; similar to a confidence interval.
• Figure 5: Process of considering a column in the parity check matrix for the ordered Tanner forest. Green circles represent the elements of the DYNAMIC_LIST, which are labelled. For a violet line connecting a node on the bottom $j$ with a bottom on the top $i$, DYNAMIC_LIST$\left[j\right]\left[0\right] = i$. The top figure indicates three trees with three different roots on the top. The three red-circled nodes belong to the non-trivial elements of the column that is being considered. On the middle image, the value of the three roots of each considered node is found within the DYNAMIC_LIST. Since all roots are different, we can incorporate the column, as it does not produce a loop. We choose the root of the tree on the left as the overall root and, in the bottom figure, we merge the three trees by changing the first element in the DYNAMIC_LIST of the elements 2 and 3 to be 1. Moreover, as the tree on the left and the one on the right had the same depth, the depth of the overall tree increases by 1 to be 4.

Theorems & Definitions (2)

• Definition 1: Sparsified Detector Matrix
• Definition 2: The transfer matrix