Efficient Post-Selection for General Quantum LDPC Codes

TL;DR

This work introduces cluster-based confidence metrics for decoding confidence in quantum LDPC codes, enabling efficient post-selection that generalizes beyond surface codes and avoids the exponential overhead of the traditional logical gap method. By leveraging clustering-based decoders (e.g., UF, BP+LSD, AC) to quantify error-cluster structure, the authors define two families of metrics, the cluster size norm fraction $Q_ ext{size}^{(\alpha)}$ and the cluster LLR norm fraction $Q_ ext{LLR}^{(\alpha)}$, and integrate them into both global and real-time post-selection strategies. Their extensive simulations across rotated surface codes, bivariate bicycle codes, and hypergraph product codes show orders-of-magnitude reductions in the logical error rate at modest abort rates, with real-time sliding-window decoding achieving performance comparable to or better than global post-selection while reducing overhead. The results demonstrate a practical, code-agnostic foundation for post-selection in fault-tolerant quantum computing with QLDPC codes, and highlight directions for theoretical grounding and broader noise-model testing to further enhance applicability.

Abstract

Post-selection strategies that discard low-confidence computational results can significantly improve the effective fidelity of quantum error correction at the cost of reduced acceptance rates, which can be particularly useful for offline resource state generation. Prior work has primarily relied on the "logical gap" metric with the minimum-weight perfect matching decoder, but this approach faces fundamental limitations including computational overhead that scales exponentially with the number of logical qubits and poor generalizability to arbitrary codes beyond surface codes. We develop post-selection strategies based on computationally efficient heuristic confidence metrics that leverage error cluster statistics (specifically, aggregated cluster sizes and log-likelihood ratios) from clustering-based decoders, which are applicable to arbitrary quantum low-density parity check (QLDPC) codes. We validate our method through extensive numerical simulations on surface codes, bivariate bicycle codes, and hypergraph product codes, demonstrating orders of magnitude reductions in logical error rates with moderate abort rates. For instance, applying our strategy to the [[144, 12, 12]] bivariate bicycle code achieves approximately three orders of magnitude reduction in the logical error rate with an abort rate of only 1% (19%) at a physical error rate of 0.1% (0.3%). Additionally, we integrate our approach with the sliding-window framework for real-time decoding, featuring early mid-circuit abort decisions that eliminate unnecessary overheads. Notably, its performance matches or even surpasses the original strategy for global decoding, while exhibiting favorable scaling in the number of rounds. Our approach provides a practical foundation for efficient post-selection in fault-tolerant quantum computing with QLDPC codes.

Figures (6)

• Figure 1: Overview of our heuristic cluster-based confidence metrics and post-selection strategies based on them.(a) Given detector outcomes, a clustering-based decoder constructs valid clusters and performs decoding within each cluster in parallel. As an example, we show a distance-9 surface code patch under bit-flip noise. Blue squares denote violated detectors (check nodes), red circles indicate fault nodes within clusters grown by the decoder, and circles marked with '$\times$' represent the final corrections obtained by decoding each cluster. Note that the depicted cluster configuration is provided solely for illustration; actual results may vary depending on the decoder. (b) Cluster statistics are then used to quantify decoding confidence. As the size and number of clusters grow, a larger portion of the logical operator’s support tends to overlap with clusters, thereby increasing the probability of logical failure. Based on this intuition, we define two metrics (the cluster size and LLR norm fractions) in Definitions \ref{['def:cluster_size_norm_fraction']} and \ref{['def:cluster_llr_norm_fraction']}, and introduce post-selection strategies that abort a trial when the metric value exceeds a chosen cutoff. (c) This approach is further extended to real-time decoding with the sliding-window framework. After decoding each window of $W$ rounds and committing its first $F$ rounds, clusters are constructed from committed errors within the latest $LF$ rounds (where $L$ denotes the lookback window size), and a metric is evaluated based on them. If the value exceeds the cutoff, the trial is immediately aborted, thereby avoiding unnecessary computation.
• Figure 2: Post-selection analysis of global decoding for rotated surface codes. (a) Logical error rates $p_\mathrm{log}$ are plotted against $p_\mathrm{abort}$ for Strategy \ref{['strategy:global']} based on the cluster LLR 2-norm fraction $Q_\mathrm{LLR}^{(2)}$, across different physical error rates $p \in \{0.001, 0.003, 0.005, 0.01\}$ and code distances $d \in \{5, 9, 13\}$. The values of $p_\mathrm{log}$ without post-selection ($p_\mathrm{abort} = 0$) are emphasized as filled circles. For $p=0.001$, an inset displaying the same data with a logarithmic scale on $p_\mathrm{abort}$ is included. Shaded regions indicate 95% confidence intervals. (b) Our strategy based on $Q_\mathrm{LLR}^{(2)}$ is compared with three other baseline strategies (the logical gap calculated by MWPM, correction weight, and detector density) for $d=13$ and $p=0.005$. (c) Various strategies are compared at a fixed abort rate $p_\mathrm{abort} = 0.3$. We consider four norm orders $\alpha \in \{0.5, 1, 2, \infty\}$ (specified above or below the markers) for the cluster size and LLR norm fractions. (d) Required abort rates $p_\mathrm{abort}$ to achieve target values of $p_\mathrm{log} \in \{10^{-3}, 10^{-6}\}$ are plotted against $p$ when using the metric $Q_\mathrm{LLR}^{(2)}$.
• Figure 3: Post-selection analysis of global decoding for bivariate bicycle codes. Two variants of bivariate bicycle codes are considered: $[[144, 12, 12]]$ and $[[72, 12, 6]]$. Shaded regions represent 95% confidence intervals. (a) Any-observable logical error rates $p_\mathrm{log}$ are plotted against the abort rate $p_\mathrm{abort}$ at $p \in \{0.001, 0.003, 0.005\}$ for Strategy \ref{['strategy:global']} based on $Q_\mathrm{LLR}^{(2)}$. For $p = 0.001$, a log-scale inset is included. (b), (c) Various strategies are compared for the $[[144, 12, 12]]$ BB code at $p = 0.003$. $p_\mathrm{abort}$ is fixed to 0.3 in (c). (d) Required $p_\mathrm{abort}$ to achieve target values of $p_\mathrm{log} \in \{10^{-3}, 10^{-6}\}$ are plotted against $p$.
• Figure 4: Post-selection analysis of global decoding for a hypergraph product code. We consider a $[[225, 9, 6]]$$(3, 4)$-regular hypergraph product code, defined from the product of a 12-variable classical LDPC codes with itself. (a) Any-observable logical error rates $p_\mathrm{log}$ are plotted against the abort rate $p_\mathrm{abort}$ at $p = 0.001$, comparing Strategy \ref{['strategy:global']} based on $Q_\mathrm{LLR}^{(2)}$ with the baseline metrics. Shaded regions represent 95% confidence intervals. (b) Various strategies are compared at a fixed $p_\mathrm{abort} = 0.3$.
• Figure 5: Analysis of real-time post-selection via Strategy \ref{['strategy:realtime']} for (a) the surface code with $d=13$ and (b) the $[[144, 12, 12]]$ bivariate bicycle code. Logical error rates $p_\mathrm{log}$ are plotted against the average time cost per accepted shots $\overline{T}_\mathrm{accepted}$, which represents the average time cost required to succeed when immediately retrying after each aborted attempt. The results are for the memory experiments with $T=d$ at $p \in \{0.003, 0.005\}$, decoded with the $(5, 1)$ or $(3, 1)$ sliding window method. The LLR 2-norm fraction $Q_\mathrm{LLR}^{(2)}$ is used for the metric and the parameter $L$ of the strategy varies across $\{1, 2, 3, 5, 7\}$. For comparison, the values for the global strategy (Strategy \ref{['strategy:global']}) and those for sliding window decoding without post-selection are presented additionally as dashed lines and '$\times$' marks, respectively. Shaded regions represent 95% confidence intervals.

Theorems & Definitions (2)

• Definition 1: Cluster size $\alpha$-norm fraction
• Definition 2: Cluster LLR $\alpha$-norm fraction