  We investigate the algebraic reasoning of quantum programs inspired by the
success of classical program analysis based on Kleene algebra. One prominent
example of such is the famous Kleene Algebra with Tests (KAT), which has
furnished both theoretical insights and practical tools. The succinctness of
algebraic reasoning would be especially desirable for scalable analysis of
quantum programs, given the involvement of exponential-size matrices in most of
the existing methods. A few key features of KAT including the idempotent law
and the nice properties of classical tests, however, fail to hold in the
context of quantum programs due to their unique quantum features, especially in
branching. We propose Non-idempotent Kleene Algebra (NKA) as a natural
alternative and identify complete and sound semantic models for NKA as well as
their quantum interpretations. In light of applications of KAT, we demonstrate
algebraic proofs in NKA of quantum compiler optimization and the normal form of
quantum while-programs. Moreover, we extend NKA with Tests (i.e., NKAT), where
tests model quantum predicates following effect algebra, and illustrate how to
encode propositional quantum Hoare logic as NKAT theorems.
