Model checking QCTL plus on quantum Markov chains

Verifying temporal properties of quantum systems, including quantum Markov chains (QMCs), has attracted an increasing interest in the last decade. Typically, the properties are specified by quantum computation tree logic (QCTL), in which reachability analysis plays a central role. However, safety as the dual problem is known little. Motivated by this, we propose a more expressive logic - QCTL(+) (QCTL plus), which extends QCTL by allowing the conjunction in path formulas and the negation in the top level of path formulas. The former can be adopted to express conditional events, and the latter can express safety. To deal with conjunction, we present a product construction of classical states in the QMC and the tri-valued truths of atomic path formulas; to deal with negation, we develop an algebraic approach to compute the safety of the bottom strongly connected component subspaces with respect to a super-operator under some necessary and sufficient convergence conditions. Thereby we conditionally decide QCTL(+) formulas over QMCs; without the convergence conditions the safety problem still remains open. The complexity of our method is provided in terms of the size of both the input QMC and the QCTL+ formula. (C) 2022 Elsevier B.V. All rights reserved.