  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.
