Dimitris Chytas

Michele Pacenti, Dimitris Chytas, Bane Vasic

Quantum low-density parity-check codes are a promising approach to fault-tolerant quantum computation, offering potential advantages in rate and decoding efficiency. In this work, we introduce quantum Margulis codes, 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 the code capacity noise model. 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.

Dimitris Chytas, Paul N. Fessatidis, Boulat A. Bash et al.

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.