Diagonal gates in the Clifford hierarchy

TL;DR

This paper characterizes diagonal gates within the Clifford hierarchy for prime-dimensional qudits, revealing their structure as roots of unity raised to polynomial functions and classifying their hierarchy level.

Contribution

It provides a complete characterization of diagonal gates in the Clifford hierarchy for prime-dimensional qudits, clarifying their algebraic structure and hierarchy level.

Findings

Diagonal gates are roots of unity raised to polynomial functions.

Hierarchy level depends on polynomial degree and root order.

Complete classification for prime-dimensional qudits.

Abstract

The Clifford hierarchy is a set of gates that appears in the theory of fault-tolerant quantum computation, but its precise structure remains elusive. We give a complete characterization of the diagonal gates in the Clifford hierarchy for prime-dimensional qudits. They turn out to be $p^{m}$-th roots of unity raised to polynomial functions of the basis state to which they are applied, and we determine which level of the Clifford hierarchy a given gate sits in based on $m$ and the degree of the polynomial.