Decomposition of Clifford Gates

In fault-tolerant quantum computation and quan-tum error-correction one is interested on Pauli matrices that commute with a circuit/unitary. This information is encoded by the support (Pllaha et al., 2020) of the given circuit/unitary. We provide a fast algorithm that decomposes any Clifford gate as a minimal product of Clifford transvections. The algorithm can be directly used for computing the support of any given Clifford gate. To achieve this goal, we exploit the structure of the symplectic group with a novel graphical approach.