We construct two families of non-binary entanglement assisted (EA) quasi-cyclic (QC) quantum low-density parity-check (QLDPC) codes over arbitrary finite fields, each possessing a precisely determined code rate. The first family is derived from a pair of non-binary classical QC-LDPC codes, designed such that the unassisted portion of the overall Tanner graph of the resulting EA-QC-QLDPC code is free of 4-cycles. The second family, on the other hand, is constructed from a single non-binary classical QC-LDPC code whose Tanner graph itself is 4-cycle-free. In developing the codes belonging to the first family, we employ a \emph{single Bell pair} to establish entanglement between the transmitter and the receiver, thereby minimizing the required entanglement resources. Furthermore, these constructions demonstrate that careful graph-based design can effectively balance error-correction performance with entanglement consumption, providing a practical approach for realizing efficient non-binary EA-QC-QLDPC codes.
For any positive integer
g≥2, we derive general condition for the existence of a
2g-cycle in the Tanner graph of two-dimensional (
2-D) classical quasi-cyclic (QC) low-density parity-check (LDPC) codes. Depending on whether
p is an odd prime or a composite number, we construct two distinct families of
2-D classical QC-LDPC codes with girth
>4 by stacking
p×p×p tensors. Furthermore, using generalized Behrend sequences, we propose an additional family of
2-D classical QC-LDPC codes with girth
>6, constructed via a similar tensor-stacking approach. All the proposed
2-D classical QC-LDPC codes exhibit an erasure correction capability of at least
p×p. Based on the constructed
2-D classical QC-LDPC codes, we derive two families of
2-D entanglement-assisted (EA) quantum low-density parity-check (QLDPC) codes. The first family of
2-D EA-QLDPC codes is obtained from a pair of
2-D classical QC-LDPC codes and is designed such that the unassisted part of the Tanner graph of the resulting EA-QLDPC code is free of
4-cycles, while requiring only a single ebit to be shared across the quantum transceiver. The second family is constructed from a single
2-D classical QC-LDPC code whose Tanner graph is free from
4-cycles. Moreover, the constructed EA-QLDPC codes inherit an erasure correction capability of
p×p, as the underlying classical codes possess the same erasure correction property.
We provide code constructions along with code properties for a few families of entanglement-assisted (EA) Calderbank-Shor-Steane (CSS) codes derived from quasi-cyclic (QC) low-density parity check (LDPC) codes. These codes require only a minimum number of Bell pairs to be shared between the transmitter and the receiver. The Tanner graph of the proposed EA QC LDPC code has girth
>4. We evaluate the performance of these codes using both random and Markovian noise models to assess random and burst error performance using a modified-version of the min-sum algorithm.
Quantum low-density parity-check (QLDPC) codes are among the most promising candidates for future quantum error correction schemes. However, a limited number of short to moderate-length QLDPC codes have been designed and their decoding performance is sub-optimal with a quaternary belief propagation (BP) decoder due to unavoidable short cycles in their Tanner graphs. In this paper, we propose a novel joint code and decoder design for QLDPC codes. The constructed codes have a minimum distance of about the square root of the block length. In addition, it is, to the best of our knowledge, the first QLDPC code family where BP decoding is not impaired by short cycles of length 4. This is achieved by using an ensemble BP decoder mitigating the influence of assembled short cycles. We outline two code construction methods based on classical quasi-cyclic codes and finite geometry codes. Numerical results demonstrate outstanding decoding performance over depolarizing channels.
Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.
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.