An Ideal Characterization of the Clifford Operators

The Clifford operators are an important and well-studied subset of quantum operations, in both the qubit and higher-dimensional qudit cases. While there are many ways to characterize this set, this paper aims to provide an ideal characterization, in the sense that it has the same characterization in every finite dimension, is characterized by a minimal set of gates, is constructive, and does not make any assumptions about non-Clifford operations or resources (such as the use of ancillas or the ability to make measurements). While most characterizations satisfy some of these properties, this appears to be the first characterization satisfying all of the above. As an application, we use these results to briefly analyze characterizations of Clifford embeddings, that is, the action of logical Clifford operations acting on qunits embedded in higher-dimensional qudits, inside the qudit Clifford framework.