  This paper introduces a novel abstraction for programming quantum operations,
specifically projective Cliffords, as functions over the qudit Pauli group. We
define a categorical semantics for projective Cliffords based on Pauli
encodings in terms of $\mathbb{Z}_d$-linear maps. We then introduce a type
system and lambda calculus for both $\mathbb{Z}_d$-linear maps and projective
Cliffords, and prove that these type systems have a sound denotational
semantics in terms of the relevant categories. Finally, we explore what it
means to program with projective Cliffords through a number of examples and
programming constructions.
