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.

翻译：本文从计算复杂度的角度研究了量子下推系统的模型检验问题，并得到了以下同等重要且有趣的新结果：首先，我们将概率下推系统和马尔可夫链的概念扩展至其量子对应物，即量子下推系统（qPDS）和量子马尔可夫链，并证明了qPDS良构的充要条件，同时提出了一种将良构qPDS的局部转移函数扩展为酉局部时间演化算子的方法。其次，我们探讨了是否有必要定义概率计算树逻辑的量子类比，以描述量子马尔可夫链的概率和分支时间特性。我们研究了其模型检验问题，并表明无状态量子下推系统（qBPA）对概率计算树逻辑（PCTL）的模型检验通常是不可判定的，即不存在用于无状态量子下推系统对概率计算树逻辑进行模型检验的算法。随后，我们研究了在何种情况下存在无状态量子下推系统的模型检验算法，并证明无状态量子下推系统对有界概率计算树逻辑（bPCTL）的模型检验问题是可判定的，且进一步表明该问题属于$\mathit{NP}$-难问题。我们的归约首次采用了著名的$\mathit{NP}$-完全问题——有界波斯特对应问题。