



01692nas a2200133   4500008004100000245007800041210006900119260001500188520126500203100001801468700002001486700001501506856003701521       2021                        eng d00aAlgebraic Reasoning of Quantum Programs via Non-Idempotent Kleene Algebra0 aAlgebraic Reasoning of Quantum Programs via NonIdempotent Kleene  c10/13/20213 a<p>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 the Non-idempotent Kleena Algebra (NKA) as a natural alternative and identify complete and sound semantic models for NKA as well as their appropriate quantum interpretations. In light of applications of KAT, we are able to 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 the rules of effect algebra, and illustrate how to encode propositional quantum Hoare logic as NKAT theorems.</p>1 aPeng, Yuxiang1 aYing, Mingsheng1 aWu, Xiaodi  uhttps://arxiv.org/abs/2110.07018