Chapter 10 - Quantum LDPC Codes

Abstract

This chapter is devoted to quantum low-density parity-check (LDPC) codes, which have many advantages compared to 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 compared to other codes include: (1) regular structure in corresponding parity-check (H-) matrices leads to low complexity encoders/decoders, and (2) their sparse H-matrices require a small number of interactions per qubit to determine the error location. The chapter begins with the introduction of classical LDPC codes in Section 10.1, their design and decoding algorithms. Furthermore, dual-containing quantum LDPC codes are described in Section 10.2. The next section (Section 10.3) is devoted to entanglement-assisted quantum LDPC codes. Furthermore, in Section 10.4, the probabilistic sum-product algorithm based on the quantum-check matrix instead of the classical parity-check matrix is described. Notice that encoders for dual-containing and entanglement-assisted quantum LDPC codes can be implemented based on either the standard form method or the conjugation method (described in Chapter 9). Since there is no difference in encoder implementation of quantum LDPC codes compared to other classes of quantum block codes described in Chapter 9 we omit discussion on encoder implementation and concentrate instead on designing and decoding algorithms for quantum LDPC codes. Quantum spatially coupled LDPC codes are briefly introduced in Section 10.5.