Classification of the Subgroups of the Two-Qubit Clifford Group

Abstract

We perform a complete classification of all 56 subgroups of the two-qubit Clifford group containing the two-qubit Pauli group. We provide generators for these groups using gates familiar to the quantum information community and we reference these groups against the group libraries provided in GAP. We also list several families of groups in higher levels of the two-qubit Clifford hierarchy.