Compiling Circuits with Belief Propagation

Traditionally, in the quantum context, belief propagation has most often been used for decoding. However, its computational efficiency and versatility make it an attractive approach for other applications as well. We explore how belief propagation, and message passing in general, can be related to the quantum circuit compilation of CNOT circuits from parity check matrices. Compiling a circuit from such a matrix involves extracting a sequence of CNOT gates that implements a quantum unitary corresponding to the given parity check matrix. Current methods for this task typically rely on matrix decomposition algorithms such as Gauss-Jordan elimination, and belief propagation may offer a computational advantage over these approaches. The key challenge lies in constructing an appropriate factor graph based on the structure of the parity check matrix and in adapting the belief propagation algorithm to the compilation problem.