Classification of the Subgroups of the Two-Qubit Clifford Group

TL;DR

This paper provides a comprehensive classification of all 56 subgroups of the two-qubit Clifford group, including generators and references, aiding quantum information research.

Contribution

It offers the first complete classification of two-qubit Clifford subgroups, with explicit generators and hierarchy listings, enhancing understanding of quantum symmetries.

Findings

Complete list of 56 subgroups with generators

Identification of subgroup families in the Clifford hierarchy

Reference to group libraries in GAP for further research

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.