Algebraic Reasoning of Quantum Programs via Non-idempotent Kleene Algebra

TL;DR

This paper introduces Non-idempotent Kleene Algebra (NKA) as a new algebraic framework for reasoning about quantum programs, overcoming limitations of classical Kleene algebra with tests by accommodating quantum features.

Contribution

It proposes NKA and NKAT, complete semantic models, and demonstrates their application in quantum compiler optimization, program normalization, and quantum Hoare logic.

Findings

NKA provides a sound and complete algebraic semantics for quantum programs.

Algebraic proofs of quantum compiler optimization are achieved using NKA.

NKAT encodes propositional quantum Hoare logic as algebraic theorems.

Abstract

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.