Quantum temporal logic and reachability problems of matrix semigroups

We study the reachability problems of a quantum finite automaton. More precisely, we introduce quantum temporal logic (QTL) that specifies the time-dependent behavior of quantum finite automaton by presenting the time dependence of events temporal operators ◊ (eventually) and □ (always) and employing the projections on subspaces as atomic propositions. The satisfiability of QTL formulae corresponds to various reachability problems of matrix semigroups. We prove that the satisfiability problems for $\square {\vee }_i^m{p}_i$, $◊\square {\vee }_i^m{p}_i$ and $\square ◊{\vee }_i^m{p}_i$ with atomic propositions $p_i$ are decidable. This result solves the open problem of Li and Ying 2014. Notably, the decidability of $\square ◊p$ can be interpreted as a generalization of Skolem-Mahler-Lech's celebrated theorem based on additive number theory. This paper's last part shows how the quantum finite automaton can model the general concurrent quantum programs, which may involve an arbitrary classical control flow.