Factor-graph representations of stabilizer quantum codes

Abstract

We study normal factor graph (NFG) representations of stabilizer quantum error-correction codes (QECCs), in particular NFG representations of the stabilizer label code and the normalizer label code associated with a stabilizer QECC. The structure of the NFGs we are using is such that the (symplectic) self-orthogonality constraint that stabilizer label codes have to satisfy can be proven rather straightforwardly by applying certain NFG reformulations. We show that a variety of well-known stabilizer QECCs can be expressed in this framework: (tail-biting) convolutional stabilizer QECCs, the toric stabilizer QECCs by Kitaev, and a class of stabilizer QECCs that was recently introduced by Tillich and Zemor. Our approach not only gives new insights into these stabilizer QECCs, but will ultimately help to formulate new classes of stabilizer QECCs and low-complexity (approximate) decoding algorithms.