Cluster Decomposition for Improved Erasure Decoding of Quantum LDPC Codes

TL;DR

This work introduces a cluster decoder for erasure decoding of quantum LDPC codes, generalizing the VH approach by performing peeling on the Tanner graph and then decomposing the residual stopping set into clusters that are solved sequentially via Gaussian Elimination. By allowing clusters of unrestricted size, the method attains ML performance with lower complexity than solving the entire stopping set; constraining cluster sizes yields linear-time decoding and connects to VH as a special case. Empirical results on hypergraph product codes show near-ML performance in the low-erasure regime with modest cluster-size caps, while simulations on general quantum LDPC codes demonstrate ML-curve estimation with reduced complexity across erasure rates. Overall, the cluster decoder offers a flexible, code-structure-agnostic framework that generalizes VH, enabling scalable ML-like erasure decoding for a broad class of quantum LDPC codes.

Abstract

We introduce a new erasure decoder that applies to arbitrary quantum LDPC codes. Dubbed the cluster decoder, it generalizes the decomposition idea of Vertical-Horizontal (VH) decoding introduced by Connelly et al. in 2022. Like the VH decoder, the idea is to first run the peeling decoder and then post-process the resulting stopping set. The cluster decoder breaks the stopping set into a tree of clusters which can be solved sequentially via Gaussian Elimination (GE). By allowing clusters of unconstrained size, this decoder achieves maximum-likelihood (ML) performance with reduced complexity compared with full GE. When GE is applied only to clusters whose sizes are less than a constant, the performance is degraded but the complexity becomes linear in the block length. Our simulation results show that, for hypergraph product codes, the cluster decoder with constant cluster size achieves near-ML performance similar to VH decoding in the low-erasure-rate regime. For the general quantum LDPC codes we studied, the cluster decoder can be used to estimate the ML performance curve with reduced complexity over a wide range of erasure rates.

Figures (8)

• Figure 1: Example of decomposing a graph $\mathcal{G}$ into a cluster tree.
• Figure 2: Example of a Tanner graph and a variable-induced subgraph of a stopping set $\{v_2,v_3,v_6\}$.
• Figure 3: Example of the cluster decomposition of the variable-induced subgraph of a stopping set. On top, we show the subgraph $\mathcal{G}'$ of a stopping set $\{v_1,\ldots,v_8\}$ that has no dangling checks. Graph $\mathcal{G}'$ can be decomposed into five clusters $B_r=\{c_2,c_3,c_4,v_4,v_5,v_6\}$, $B_1=\{c_1,c_2,v_1,v_2\}$, $B_2=\{c_2,v_3\}$, $B_3=\{c_5,c_6,v_6,v_7\}$, and $B_4=\{c_6,v_8\}$. They are connected by three cut nodes $c_2$, $v_6$, and $c_6$, highlighted in blue. At the bottom, we show the cluster tree of $\mathcal{G}'$ rooted at its largest cluster.
• Figure 4: Performance of the [[1600,64]] hypergraph product code over erasures with various decoders. The plot shows the failure rates of the decoders for recovering a Pauli-$X$ error supported on the erasure pattern, up to code degeneracy. Error bars represent the 95% confidence intervals around the simulated data points.
• Figure 5: Performance of the [[2025,81]] hypergraph product code over erasures with various decoders. The plot shows the failure rates of the decoders for recovering a Pauli-$X$ error supported on the erasure pattern, up to code degeneracy. Error bars represent the 95% confidence intervals around the simulated data points.

Theorems & Definitions (7)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1