Normal Factor Graph Representations of Stabilizer Quantum Codes with Applications

Abstract

We develop normal factor graph (NFG) representations of stabilizer quantum error-correcting codes and entanglement-assisted quantum error-correcting codes. We introduce two complementary representations. Quantum-code NFGs describe stabilizer and normalizer label codes and provide a graphical framework for establishing self-orthogonality, constructing codes from component codes, and identifying encoding and decoding circuits. Quantum-probability NFGs represent the quantum-mechanical probabilities associated with encoding, noise, and measurement. We also introduce a doubleedge formulation that makes the relationship between these two representations explicit. We apply the framework to convolutional and tail-biting convolutional stabilizer codes, as well as to two classes of low-density parity-check stabilizer codes. Finally, we extend the quantum-probability representation to entanglementassisted codes.