Abstract

The Clifford hierarchy defines those quantum gates which can be performed fault-tolerantly via gate teleportation. Semi-Clifford gates are those gates which can be performed using a more efficient protocol. Despite their importance, the structures of the Clifford hierarchy and of the semi-Clifford subset remain poorly understood, especially in the qudit case. While every gate in the one-qubit Clifford hierarchy was shown to be semi-Clifford, the analogous result for qudits was only known to hold for third-level gates.

Using the technique of conjugate pairs and the action of the symplectic group on the Pauli basis, we show that for any odd prime dimension, the single-qudit Clifford hierarchy coincides with the semi-Clifford subset. This allows us to give a normal form for all Clifford hierarchy gates, expressing each one uniquely as product of three simple gates, as well as a precise count of the number of gates at each level of the hierarchy. We thereby completely classify the single-qudit Clifford hierarchy. This is joint work with Nadish de Silva.