(Un)decidable Problems about Reachability of Quantum Systems

TL;DR

This paper investigates the decidability of various reachability problems in quantum automata, showing that most are undecidable in general but become decidable under certain restrictions on the reachable sets.

Contribution

It establishes the undecidability of four reachability properties in quantum automata and identifies conditions under which three of these properties become decidable.

Findings

All four reachability properties are undecidable in general.

The last three reachability properties are decidable when reachable sets lack negation.

Provides a clear boundary between decidable and undecidable cases in quantum system reachability.

Abstract

We study the reachability problem of a quantum system modelled by a quantum automaton. The reachable sets are chosen to be boolean combinations of (closed) subspaces of the state 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.