Chapter 9 - Quantum low-density parity-check codes

Abstract

This chapter is devoted to quantum low-density parity-check (LDPC) codes, which have many advantages over other classes of quantum codes thanks to the sparseness of their quantum-check matrices. Both semirandom and structured quantum LDPC codes are described. Key advantages of structured quantum LDPC codes over other codes include (i) a regular structure in corresponding parity-check (H-) matrices, leading to low-complexity encoders/decoders, and (ii) sparse H-matrices that require a small number of interactions per qubit to determine error location. The chapter begins by introducing classical LDPC codes, their design, and decoding algorithms. Further, the dual-containing quantum LDPC codes are described. The next section is devoted to entanglement-assisted quantum LDPC codes. Further, the probabilistic sum-product algorithm is described based on a quantum-check matrix instead of a classical parity-check matrix. Note that encoders for dual-containing and entanglement-assisted quantum LDPC codes can be implemented based on standard form or conjugation methods (described in Chapter 8). Since there is no difference in the encoder implementation of quantum LDPC codes compared with other classes of quantum block-codes described in Chapter 8, we omit a discussion on encoder implementation and concentrate instead on design and decoding algorithms for quantum LDPC codes. Quantum spatially coupled LDPC codes are briefly described in the last section.