Decomposition of Clifford Gates

Abstract

In fault-tolerant quantum computation and quantum error-correction one is interested on Pauli matrices that commute with a circuit/unitary. We provide a fast algorithm that decomposes any Clifford gate as a $\text{minimal}$ product of Clifford transvections. The algorithm can be directly used for finding all Pauli matrices that commute with any given Clifford gate. To achieve this goal, we exploit the structure of the symplectic group with a novel graphical approach.

References in corpus (8)

• Quantum Error Correction and Orthogonal Geometry
• Quantum error correction with only two extra qubits
• Flag fault-tolerant error correction with arbitrary distance codes
• Flag fault-tolerant error correction for any stabilizer code
• Fault-tolerant magic state preparation with flag qubits
• Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations
• Un-Weyl-ing the Clifford Hierarchy
• Synthesis of Logical Clifford Operators via Symplectic Geometry