On Groups in the Qubit Clifford Hierarchy

TL;DR

This paper classifies groups formed from elements in the qubit Clifford Hierarchy, providing a canonical form and exploring the structure of such groups, including non-diagonal examples, with implications for transversal gates.

Contribution

It offers a canonical form for semi-Clifford elements, classifies groups in the hierarchy, and extends understanding beyond diagonal gate groups, with potential implications for quantum error correction.

Findings

Classified groups formed from semi-Clifford elements in the hierarchy

Identified non-diagonal generalized symmetric groups within the hierarchy

Discussed restrictions on transversal gates based on group structure

Abstract

Here we study the unitary groups that can be constructed using elements from the qubit Clifford Hierarchy. We first provide a necessary and sufficient canonical form that semi-Clifford and generalized semi-Clifford elements must satisfy to be in the Clifford Hierarchy. Then we classify the groups that can be formed from such elements. Up to Clifford conjugation, we classify all such groups that can be constructed using generalized semi-Clifford elements in the Clifford Hierarchy. We discuss a possible minor exception to this classification in the appendix. This may not be a full classification of all groups in the qubit Clifford Hierarchy as it is not currently known if all elements in the Clifford Hierarchy must be generalized semi-Clifford. In addition to the diagonal gate groups found by Cui et al., we show that many non-isomorphic (to the diagonal gate groups) generalized symmetric groups are also contained in the Clifford Hierarchy. Finally, as an application of this classification, we examine restrictions on transversal gates given by the structure of the groups enumerated herein which may be of independent interest.