A Unifying Coalgebraic Semantics Framework for Quantum Systems

Abstract

As a quantum counterpart of labeled transition system (LTS), quantum labeled transition system (QLTS) is a powerful formalism for modeling quantum programs or protocols, and gives a categorical understanding for quantum computation. With the help of quantum branching monad, QLTS provides a framework extending some ideas in non-deterministic or probabilistic systems to quantum systems. On the other hand, quantum finite automata (QFA) emerged as a very elegant and simple model for resolving some quantum computational problems. In this paper, we propose the notion of reactive quantum system (RQS), a variant of QLTS capturing reactive system behavior, and develop a coalgebraic semantics for QLTS, RQS and QFA by an endofunctor on the category of convex sets, which has a final coalgebra. Such a coalgebraic semantics provides a unifying abstract interpretation for QLTS, RQS and QFA. The notions of bisimulation and simulation can be employed to compare the behavior of different types of quantum systems and judge whether a coalgebra can be behaviorally simulated by another.