Climbing the Clifford Hierarchy

TL;DR

This paper investigates the structure of the Clifford Hierarchy in quantum computation, specifically characterizing Clifford gates whose square roots ascend to the third level, enhancing understanding of quantum gate hierarchies.

Contribution

It provides a complete characterization of Clifford gates with square roots that reach the third level of the hierarchy, advancing the theoretical understanding of quantum gate structures.

Findings

Characterization of Clifford gates with square roots in the third level

Insights into the structure of the Clifford Hierarchy

Enhanced understanding of quantum gate control and hierarchy climbing

Abstract

The Clifford Hierarchy has been a central topic in quantum computation due to its strong connections with fault-tolerant quantum computation, magic state distillation, and more. Nevertheless, only sections of the hierarchy are fully understood, such as diagonal gates and third level gates. The diagonal part of the hierarchy can be climbed by taking square roots and adding controls. Similarly, square roots of Pauli gates (first level) are Clifford gates (climb to the second level). Based on this theme, we study gates whose square roots climb to the next level. In particular, we fully characterize Clifford gates whose square roots climb to the third level.