Linear Time Iterative Decoders for Hypergraph-Product and Lifted-Product Codes

TL;DR

This work tackles the decoding challenge for quantum LDPC codes, focusing on HP and LP constructions, by systematically characterizing trapping sets from both stabilizer structure and constituent classical codes. It develops TS-aware decoders derived from the decoders of the underlying classical LDPC codes and introduces a concise stabilizer-induced subgraph representation to study decoding dynamics. A TS-aware bit-flipping decoder and a decoder-diversity min-sum framework are proposed to mitigate both stabilizer-induced TSs and classical TSs, with encodings running in parallel to yield significant reductions in the error floor. The results demonstrate improved logical error rates for LP and HP codes, highlighting practical paths toward low-depth, fault-tolerant quantum computation via TS-aware, diversified decoders. The approach leverages existing classical LDPC decoding strategies, enabling scalable, low-complexity decoding for a broad family of HP and LP quantum LDPC codes, under the assumption of perfect syndrome measurements.

Abstract

Quantum low-density parity-check (QLDPC) codes with asymptotically non-zero rates are prominent candidates for achieving fault-tolerant quantum computation, primarily due to their syndrome-measurement circuit's low operational depth. Numerous studies advocate for the necessity of fast decoders to fully harness the capabilities of QLDPC codes, thus driving the focus towards designing low-complexity iterative decoders. However, empirical investigations indicate that such iterative decoders are susceptible to having a high error floor while decoding QLDPC codes. The main objective of this paper is to analyze the decoding failures of the \emph{hypergraph-product} and \emph{lifted-product} codes and to design decoders that mitigate these failures, thus achieving a reduced error floor. The suboptimal performance of these codes can predominantly be ascribed to two structural phenomena: (1) stabilizer-induced trapping sets, which are subgraphs formed by stabilizers, and (2) classical trapping sets, which originate from the classical codes utilized in the construction of hypergraph-product and lifted-product codes. The dynamics of stabilizer-induced trapping sets is examined and a straightforward modification of iterative decoders is proposed to circumvent these trapping sets. Moreover, this work proposes a systematic methodology for designing decoders that can circumvent classical trapping sets in both hypergraph product and lifted product codes, from decoders capable of avoiding their trapping set in the parent classical LDPC code. When decoders that can avoid stabilizer-induced trapping sets are run in parallel with those that can mitigate the effect of classical TS, the logical error rate improves significantly in the error-floor region.

Figures (22)

• Figure 1: The figure shows the implementation of $T$ gate using measurement and Clifford circuit given a $T$ state given by $\frac{1}{2}\left(\ket{0}+\exp{i\frac{\pi}{4}}\ket{1}\right)$.
• Figure 2: The figure illustrates lifting of the protograph to a code.
• Figure 3: The figure illustrates the process of obtaining two base graphs corresponding to a lifted-product code given the base graphs of two classical protograph LDPC codes.
• Figure 4: Subgraph $\mathscr{T}(c^2v^1)$ induced by the support of $X$-type check node $c^2v^1$ in $\mathscr{G}_\mathrm{Z}$. Note that $c^2v^1$ is not part of $\mathscr{T}(c^2v^1).$ The edges connected to $c^2v^1$ in $\mathscr{G}_{\mathrm{X}}$ are shown in red, whereas those in $\mathscr{T}(c^2v^1)$ are shown in black.
• Figure 5: The figure shows a different view of $T(c^2v^1)$ shown in Fig. 3. Note that the CC-type (or VV-type) variable nodes have non-intersecting set of neighboring checks and have identical neighborhoods, meaning when the errors are only on a subset of CC-type (or VV-type) variable nodes, each VV-type (or CC-type) variable node has the same number of unsatisfied checks as neighbors. In this figure, CC-type variable nodes $c^2c^1_1$ and $c^2c_2^1$, shown in black, are in error, causing their eight neighboring check nodes to be unsatisfied. As a result, each of the VV-type variable nodes is connected to two unsatisfied checks.

Theorems & Definitions (30)

• Example 1
• Example 2
• Definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1