Computational Complexity of Model-Checking Quantum Pushdown Systems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914544101621760 |
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| author | Lin, Deren Lin, Tianrong |
| author_facet | Lin, Deren Lin, Tianrong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_18439 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computational Complexity of Model-Checking Quantum Pushdown Systems Lin, Deren Lin, Tianrong Logic in Computer Science Computational Complexity 68Q15, 03B25 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. |
| title | Computational Complexity of Model-Checking Quantum Pushdown Systems |
| topic | Logic in Computer Science Computational Complexity 68Q15, 03B25 |
| url | https://arxiv.org/abs/2506.18439 |