In this paper, we study the problem of model-checking quantum pushdown systems from a computational complexity point of view. We arrive at the following equally important, interesting new results: We first extend the notions of the { \it probabilistic pushdown systems} and { \it Markov chains} to their quantum counterparts, i.e., { \em quantum pushdown system (qPDS)} and { \em quantum Markov chains}, and prove a necessary and sufficient condition for a qPDS to be well formed, also presenting a method to extend the local transition function of a well-formed qPDS to a unitary local time evolution operator. Next, we investigate the question of whether it is necessary to define a quantum analogue of { \it probabilistic computational tree logic} to describe the probabilistic and branching-time properties of the { \it quantum Markov chain}. We study its model-checking question and show that model-checking of { \it stateless quantum pushdown systems (qBPA)} against { \it probabilistic computational tree logic (PCTL)} is generally undecidable, i.e., there exists no algorithm for model-checking { \it stateless quantum pushdown systems (qBPA)} against { \it probabilistic computational tree logic}. We then study in which case there exists an algorithm for model-checking { \it stateless quantum pushdown systems} and show that the problem of model-checking { \it stateless quantum pushdown systems (qBPA)} against { \it bounded probabilistic computational tree logic} (bPCTL) is decidable, and further show that this problem is in $\mathit{ NP}$-hard. Our reduction is from the { \it bounded Post Correspondence Problem} for the first time, a well-known $\mathit{ NP}$-complete problem.
Categories: cs.LO cs.CC