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arXiv:2610.31877

Temporal Degeneracy Limits Message Passing Decoding of QLDPC Codes Under Repeated Noisy Syndrome Measurements

Dimitris Chytas, Nithin Raveendran, Bane Vasić

quant-phcs.ITSubmitted Oct 22, 2026• Announced Oct 26, 2026• 0 citations
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Abstract

Practical quantum computations can require logical error rates of $10^{-12}$ or lower. Quantum low-density parity-check (QLDPC) codes are promising candidates for reaching such rates with low qubit overhead, provided that their syndromes can be decoded efficiently. Repeated noisy syndrome measurements introduce temporal degeneracy into the space-time decoding graph. We characterize the temporal kernel, whose vectors have zero total data parity, and show that it is generated by elementary temporal codewords of weight $d_v+2$, where $d_v$ is the variable-node degree. For $(d_v,d_c)$-regular parity-check matrices with fixed degrees $d_v,d_c>2$ and girth greater than four, we derive counting bounds and construct degenerate error patterns of weight $(d_v+2)/2$ for even $d_v$ and $d_v+1$ for odd $d_v$, with multiplicity $Θ(nr)$ in the blocklength $n$ and number of measurement rounds $r$. Under the isolation assumption, these patterns cause local failures of deterministic isotropic message-passing decoders under a flooding schedule. These failures depend on the node degrees rather than the minimum distance; for the bivariate bicycle (BB) codes considered, their weight is equal to four. Monte carlo simulations and fixed-weight importance sampling quantify harmful low-weight degenerate error patterns for the min-sum (MS) decoder beyond the isolation assumption. Motivated by the identified symmetries, we propose space-time anisotropic min-sum decoding with syndrome-guided realizations. Its complexity is linear in the number of Tanner graph edges for fixed maximum number of iterations and number of decoders. Numerical results on BB codes show threshold-like behavior acrossthe tested code sizes and improved failure probabilities relative to MS decoding with ordered statistics decoding post-processing.

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Practical quantum computations can require logical error rates of 10−1210^{-12}10−12 or lower. Quantum low-density parity-check (QLDPC) codes are promising candidates for reaching such rates with low qubit overhead, provided that their syndromes can be decoded efficiently. Repeated noisy syndrome measurements introduce temporal degeneracy into the space-time decoding graph. We characterize the temporal kernel, whose vectors have zero total data parity, and show that it is generated by elementary temporal codewords of weight dv+2d_v+2dv​+2, where dvd_vdv​ is the variable-node degree. For (dv,dc)(d_v,d_c)(dv​,dc​)-regular parity-check matrices with fixed degrees dv,dc>2d_v,d_c>2dv​,dc​>2 and girth greater than four, we derive counting bounds and construct degenerate error patterns of weight (dv+2)/2(d_v+2)/2(dv​+2)/2 for even dvd_vdv​ and dv+1d_v+1dv​+1 for odd dvd_vdv​, with multiplicity Θ(nr)Θ(nr)Θ(nr) in the blocklength nnn and number of measurement rounds rrr. Under the isolation assumption, these patterns cause local failures of deterministic isotropic message-passing decoders under a flooding schedule. These failures depend on the node degrees rather than the minimum distance; for the bivariate bicycle (BB) codes considered, their weight is equal to four. Monte carlo simulations and fixed-weight importance sampling quantify harmful low-weight degenerate error patterns for the min-sum (MS) decoder beyond the isolation assumption. Motivated by the identified symmetries, we propose space-time anisotropic min-sum decoding with syndrome-guided realizations. Its complexity is linear in the number of Tanner graph edges for fixed maximum number of iterations and number of decoders. Numerical results on BB codes show threshold-like behavior acrossthe tested code sizes and improved failure probabilities relative to MS decoding with ordered statistics decoding post-processing.
Quantum low-density parity-check (QLDPC) codes provide non vanishing rates, distance scaling with the blocklength of the code, and facilitate fast iterative decoding because of their sparsity. However, in practice iterative decoding fails to exploit the distance of the code, because it cannot resolve the symmetries imposed by degeneracy. In this work, we provide a graph theoretic characterization of degeneracy for the family of generalized bicycle (GB) codes. This viewpoint shows that harmful degenerate error patterns persist whenever they remain related by automorphisms preserved by the decoder. Motivated by symmetry breaking via graph coloring, we compare three coloring approaches: no coloring, block-coloring, and edge-coloring. For GB codes, we show that edge-coloring can eliminate all automorphisms in low-weight stabilizer-induced subgraphs. We practically realize the coloring schemes as isotropic, block- anisotropic and edge-anisotropic min-sum (MS) decoding. Experimental results show that edge anisotropic min-sum decoding obtains improved performance over isotropic and block anisotropic decoding for several GB codes in a small number of iterations.
Quantum low-density parity-check codes are a promising approach to fault-tolerant quantum computation, offering potential advantages in rate and decoding efficiency. Quantum Margulis codes are a new class of QLDPC codes derived from Margulis' classical LDPC construction via the two-block group algebra framework. We show that quantum Margulis codes, unlike bivariate bicycle codes which require ordered statistics decoding for effective error correction, can be efficiently decoded using a standard min-sum decoder with linear complexity, when decoded under depolarizing noise. This is attributed to their Tanner graph structure, which does not exhibit group symmetry, thereby mitigating the well-known problem of error degeneracy in QLDPC decoding. To further enhance performance, we propose an algorithm for constructing 2BGA codes with controlled girth, ensuring a minimum girth of 6 or 8, and use it to generate several quantum Margulis codes of length 240 and 642. We validate our approach through numerical simulations, demonstrating that quantum Margulis codes behave significantly better than BB codes in the error floor region, under min-sum decoding.
Real-time quantum error correction requires classical decoders to process growing syndrome workloads with low and predictable latency. For quantum low-density parity-check (qLDPC) codes, iterative belief propagation (BP) repeatedly updates messages over sparse Tanner graphs, creating substantial memory-access and data-movement demands. We map normalized Min-Sum BP decoding of the [[144,12,12]] Bivariate Bicycle qLDPC code onto a DPU-based Processing-in-Memory (PIM) architecture. Within each DPU, 11 tasklets cooperatively decode one syndrome, while multiple DPUs process independent syndrome instances in parallel. Using uPIMulator and a data-qubit Pauli error model with ideal syndrome measurements, we compare throughput, per-syndrome processing time, logical error rate (LER), and single-syndrome tail latency against a 16-logical-CPU baseline. At a component-wise physical error probability of p=0.001 and one BP iteration, the projected aggregate kernel throughput of 2,560 DPUs reaches 1.071 x 10^7 decodes/s, compared with 1.22 x 10^6 decodes/s for the CPU, an 8.8x improvement. From two iterations onward, the measured LER remains below the physical error probability for every evaluated value of p. For one to five iterations, the maximum sampled serialized X+Z DPU compute latency remains below the 1 ms decoder-side reference for trapped-ion QEC, reaching approximately 0.873 ms at five iterations. These results show that near-memory processing can provide high aggregate throughput and sub-millisecond compute latency for qLDPC BP decoding under the evaluated conditions.
We design low-complexity, fault-tolerant decoders for quantum low-density parity-check (QLDPC) codes with the goal of reducing decoding latency. We target two major bottlenecks of decoding under the \emph{circuit-level} noise model: (i) post-processing via order-statistics decoding (OSD), and (ii) the large number of auxiliary variable nodes commonly introduced to represent CNOT-induced correlations during syndrome extraction. Our key observation is that propagating CNOT faults (\emph{hook errors}) create \emph{stabilizer-induced} trapping sets (TSs) that are intrinsic to hypergraph-product (HGP) and lifted-product (LP) constructions. Therefore, instead of modeling each such fault with an explicit correlation node and relying on OSD to clean up the resulting failures, we design message-passing decoders that resolve the corresponding \emph{stabilizer-induced} TSs directly. We obtain these decoders by deriving QLDPC decoders from decoders for the parent classical LDPC codes and using them collectively to correct broad families of \emph{stabilizer-induced} TSs. For CNOT faults that manifest primarily as syndrome errors, we show that their effect is equivalent to a data error together with syndrome-bit measurement errors. Consequently, given repeated measurements and a decoding graph that already includes nodes representing syndrome-bit errors, no distinct variable node is needed for each CNOT fault. Using a \emph{phenomenological} Tanner graph with nodes representing only data errors and syndrome-bit errors, simulations on LP codes show a significant improvement in logical error rate relative to BP+OSD. Moreover, the improved error correction relies on low-complexity finite-state iterative decoders, targeted for fast and low-latency FPGA implementation.
In this letter, we propose a novel iterative decoding algorithm that exploits the degenerate nature of three different families of quantum low-density parity-check codes, i.e., surface, toric, and row-degree-4 bicycle codes. Such families of codes share harmful trapping sets that constitute symmetric stabilizers, making it impossible for any parallel-scheduled iterative message-passing decoder to converge even for error patterns of weight as low as two. By biasing subsets of nodes in the symmetric stabilizers, the decoder is able to converge to a valid error pattern. Furthermore, the proposed decoder has low decoding complexity - linear in the code’s blocklength - and a fully parallel schedule, making it suitable for low-latency efficient implementation.
We introduce a sliding window decoder based on belief propagation (BP) with guided decimation for the purposes of decoding quantum low-density parity-check codes in the presence of circuit-level noise. Windowed decoding keeps the decoding complexity reasonable when, as is typically the case, repeated rounds of syndrome extraction are required to decode. Within each window, we employ several rounds of BP with decimation of the variable node that we expect to be the most likely to flip in each round, Furthermore, we employ ensemble decoding to keep both decimation options (guesses) open in a small number of chosen rounds. We term the resulting decoder BP with guided decimation guessing (GDG). Applied to bivariate bicycle codes, GDG achieves a similar logical error rate as BP with an additional OSD post-processing stage (BP+OSD) and combination-sweep of order 10. For a window size of three syndrome cycles, a multi-threaded CPU implementation of GDG achieves a worst-case decoding latency of 3ms per window for the [[144,12,12]] code.