(Un)decidable problems about reachability of quantum systems

Abstract

We study the reachability problem of a quantum system modeled by a quantum automaton, namely, a set of processes each of which is formalized as a quantum unitary transformation. The reachable sets are chosen to be boolean combinations of (closed) subspaces of the state Hilbert space of the quantum system. Four different reachability properties are considered: eventually reachable, globally reachable, ultimately forever reachable, and infinitely often reachable. The main result of this paper is that all of the four reachability properties are undecidable in general; however, the last three become decidable if the reachable sets are boolean combinations without negation. © 2014 Springer-Verlag.