Synthesis of Logical Clifford Operators via Symplectic Geometry

Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in \mathbb{C}-{N\times N} as a 2m\times 2m binary sym-plectic matrix, where N=2-{m}. We show that for an \!\!\!\![\!\!\![\ {m, m-k}\ ]\!\!\!]\!\!\!\! stabilizer code every logical Clifford operator has 2-{k(k+1)/2} symplectic solutions, and we enumerate them efficiently using symplectic transvections. The desired circuits are then obtained by writing each of the solutions as a product of elementary symplectic matrices. For a given operator, our assembly of all of its physical realizations enables optimization over them with respect to a suitable metric. Our method of circuit synthesis can be applied to any stabilizer code, and this paper provides a proof of concept synthesis of universal Clifford gates for the well-known \!\!\!\![\!\!\![\ 6,4,2\ ]\!\!\!]\!\!\!\! code. Programs implementing our algorithms can be found at https://github.com/nrenga/symplectic-arxiv18a.