Qudit versions of the qubit "pi-over-eight" gate

Understand

When visualised as an operation on the Bloch sphere, the qubit "pi-over-eight" gate corresponds to one-eighth of a complete rotation about the vertical axis.

• This simple gate often plays an important role in quantum information theory, typically in situations for which Pauli and Clifford gates are insufficient.
• Most notably, when it supplements the set of Clifford gates then universal quantum computation can be achieved.
• The "pi-over-eight" gate is the simplest example of an operation from the third level of the Clifford hierarchy (i.e., it maps Pauli operations to Clifford operations under conjugation).