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 analogues, 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 generalized 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 generalized stateless quantum pushdown systems} 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} 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. Our above results advance the field of model-checking quantum systems significantly, since all of the above important and interesting results on model-checking quantum pushdown systems were completely unknown previously.

翻译：本文从计算复杂性角度研究量子下推系统的模型检验问题，获得了以下同等重要且新颖的结果：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，即量子下推系统和量子马尔可夫链，并证明量子下推系统良构的充要条件，同时提出将良构量子下推系统的局部转移函数扩展为幺正局部时间演化算子的方法。其次，我们探讨是否需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性与分支时间性质。我们研究其模型检验问题，证明广义无状态量子下推系统对概率计算树逻辑的模型检验通常不可判定，即不存在针对该问题的通用算法。随后，我们研究在何种情况下存在无状态量子下推系统的模型检验算法，证明无状态量子下推系统对有界概率计算树逻辑的模型检验是可判定的，并进一步证明该问题属于NP难问题。我们首次通过有界波斯特对应问题——一个著名的NP完全问题——进行归约证明。上述成果显著推进了量子系统模型检验领域的发展，因为此前关于量子下推系统模型检验的所有重要结论均属未知。