Even More Efficient Soft-Output Decoding with Extra-Cluster Growth and Early Stopping

TL;DR

This paper addresses the high overhead of soft-output calculations for cluster-based decoders in fault-tolerant quantum computing and introduces two hardware-friendly early-stopping techniques: the bounded cluster gap and the extra-cluster gap, including both w/o CG and w/ CG variants. The bounded gap uses bounded Dijkstra searches to terminate once the distance exceeds $\epsilon_{\max}$, reducing complexity toward $O(d^2\log d)$ in the low-error regime. The extra-cluster gap reuses the decoder's cluster-growth and, when needed, optionally builds a cluster graph to recover the exact distance, enabling near-hardware implementation for UF decoders. Numerical results show substantial reductions in visited nodes and favorable scaling with code distance, while retaining accuracy for decoder switching and multi-boundary QEC, thereby enabling real-time decoders with reliable soft outputs.

Abstract

In fault-tolerant quantum computing, soft outputs from real-time decoders play a crucial role in improving decoding accuracy, post-selecting magic states, and accelerating lattice surgery. A recent paper by Meister et al. [arXiv:2405.07433 (2024)] proposed an efficient method to evaluate soft outputs for cluster-based decoders, including the Union-Find (UF) decoder. However, in parallel computing environments, its computational complexity is comparable to or even surpasses that of the UF decoder itself, resulting in a substantial overhead. Furthermore, this method requires global information about the decoding graph, making it poorly suited for existing hardware implementations of the UF decoder on Field-Programmable Gate Arrays (FPGAs). In this paper, to alleviate these issues, we develop more efficient methods for evaluating high-quality soft outputs in cluster-based decoders by introducing several early-stopping techniques. Our central idea is that the precise value of a large soft output is often unnecessary in practice. Based on this insight, we introduce two types of novel soft-outputs: the bounded cluster gap and the extra-cluster gap. The former reduces the computational complexity of Meister's method by terminating the calculation at an early stage. Our numerical simulations show that this method achieves improved scaling with code distance $d$ compared to the original proposal. The latter, the extra-cluster gap, quantifies decoder reliability by performing a small, additional growth of the clusters obtained by the decoder. This approach offers the significant advantage of enabling soft-output computation without modifying the existing architecture of FPGA-implemented UF decoders. These techniques offer lower computational complexity and higher hardware compatibility, laying a crucial foundation for future real-time decoders with soft outputs.

Figures (10)

• Figure 1: Schematic illustrations of different methods for calculating soft outputs. ( a) A cluster-based decoder forms multiple clusters (blue circles) around each error syndrome (yellow star) to determine an initial error-correction path (black line) for a given decoding problem. ( b) The complementary gapgidney2025yokedsurfacecodes is the weight difference between the initial correction and the optimal correction for the complementary logical class (red dotted line). The gray area indicates the search space explored during the complementary decoding. ( c) The cluster gapmeister2024efficientsoftoutput is the shortest path distance between boundaries $b_1$ and $b_2$ (red line), calculated using Dijkstra's algorithm after all intra-cluster edge weights are set to zero. The gray area represents the region explored by the algorithm. ( d) The bounded cluster gap, one of the methods we introduce, modifies the cluster gap by terminating the Dijkstra's search early. This process stops once the path distance is guaranteed to exceed a predefined threshold $\epsilon_\mathrm{max}$, which significantly reduces the search area (gray). ( e) The extra-cluster gap, another method we propose, is determined by an additional growth of all clusters. The calculation is terminated if no single grown cluster connects boundaries $b_1$ and $b_2$ within a growth limit of $\epsilon_\mathrm{max}/2$. The gray area shows the region covered by this additional growth.
• Figure 2: ( Left) A connection between the two boundaries is formed and detected via the extra-cluster gap method. ( Right) A cluster graph is constructed using the distances between the colliding clusters to compute the precise shortest path.
• Figure 3: The number of nodes visited during Dijkstra's algorithm for the bounded cluster gap (solid lines) and the original cluster gap (dotted lines). The shaded areas represent the standard deviation. Each data point is an average over $5\cdot 10^6$ samples. Samples with no detection events are excluded from the analysis.
• Figure 4: Comparison of the fraction of samples with a soft output below the threshold $\epsilon_\mathrm{max}=20$ dB. The solid line represents the extra-cluster gap w/o CG, and the dotted line represents the cluster gap. The shaded areas indicate the standard error. Each data point is an average over $5\cdot 10^6$ samples.
• Figure 5: The maximum growth radius required for the UF decoder to complete its search. The shaded areas represent the standard deviation. Each data point is an average over $2\cdot 10^6$ samples.

Theorems & Definitions (12)

• Definition 1: Cluster Gap: $g_\mathrm{c}$
• Definition 2: Extra-Cluster Gap w/o CG: $g_\mathrm{ec}$
• Definition 3: Maximum Inter-Cluster Edge Weight on $P_\mathrm{c}$: $w_\mathrm{max}(P_\mathrm{c})$
• Theorem 1
• proof
• Theorem 2
• proof
• Definition 4: Extra-Cluster Gap w/ CG: $g_{\mathrm{eccg}}$
• Theorem 3
• proof