The Clifford hierarchy for one qubit or qudit

TL;DR

This paper completely characterizes the Clifford hierarchy for one qubit or qudit, providing explicit decompositions and formulas that facilitate understanding and implementing fault-tolerant quantum gates.

Contribution

It offers a complete solution to identifying and decomposing hierarchy gates for one qubit or qudit, including efficient teleportation protocols.

Findings

Every hierarchy gate can be uniquely decomposed into three simple gates.

Formulas for the size of each hierarchy level are derived.

All hierarchy gates can be expressed in a form suitable for efficient teleportation.

Abstract

The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The groups of Pauli and Clifford gates constitute the first and second 'levels', respectively. Non-Clifford gates from the third level or higher, such as the $T$ gate, are necessary for achieving fault-tolerant universal quantum computation. Since it was defined twenty-five years ago by Gottesman-Chuang, two questions have been studied by numerous researchers. First, precisely which gates constitute the Clifford hierarchy? Second, which subset of the hierarchy gates admit efficient gate teleportation protocols? We completely solve both questions in the practically-relevant case of the Clifford hierarchy for gates of one qubit or one qudit of prime dimension. We express every such hierarchy gate uniquely as a product of three simple gates, yielding also a formula for the size of every level. These results are a consequence of our finding that all such hierarchy gates can be expressed in a certain form that guarantees efficient gate teleportation. Our decomposition of Clifford gates as a unique product of three elementary Clifford gates is of broad applicability.