Efficient Quantum Stabilizer Codes: LDPC and LDPC-Convolutional Constructions

Existing quantum stabilizer codes constructed from the classic binary codes exclusively belong to the special subclass of Calderbank-Shor-Steane (CSS) codes. This paper fills in the gap by proposing five systematic constructions for non-CSS stabilizer codes, the first four of which are based on classic binary quasi-cyclic low-density parity-check (LDPC) codes and last on classic binary LDPC-convolutional codes. These new constructions exploit structured sparse graphs, make essential use of simple and powerful coding techniques including concatenation, rotation and scrambling, and generate rich classes of codes with a wide range of lengths and rates. We derive the sufficient, and in some cases also the necessary, conditions for each construction to satisfy the general symplectic inner product (SIP) condition, and develop practical decoder algorithms for these codes. The resulting codes are the first classes of non-CSS quantum LDPC codes and non-CSS quantum convolutional codes rooted from classic binary codes (rather than codes in GF(4)), and some of them perform as well as or better than the existing quantum codes.