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

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 $\Theta (nr)$ 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.