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Measuring the constrained reachability in quantum Markov chains

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Abstract

Constrained reachability is a kind of quantitative path property, which is generally specified by multiphase until formulas originated in continuous stochastic logic. In this paper, through proposing a positive operator valued measure on the set of infinite paths, we develop an exact method to solve the constrained reachability problem for quantum Markov chains. The convergence rate of the reachability is also obtained. We then analyse the complexity of the proposed method, which turns out to be in polynomial-time w.r.t. the size of the classical state space and the dimension of the accompanied Hilbert space. Finally, our method is implemented and applied to a simple quantum protocol.

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Notes

  1. Throughout this paper, we specify the complexity \({\mathcal {O}}(n^6)\) in terms of arithmetic operations, rather than \({\mathcal {O}}(n^4)\) in terms of field operations (e. g. based on the Jordan decomposition described in [20]). The latter requires the assumption that the characteristic polynomial can be factorised into linear factors, which is not simply in \({\mathcal {O}}(n)\) arithmetic operations. So the two complexity is consistent.

  2. In fact, the complexity is \({\mathcal {O}}(\Vert {\mathscr {Q}}\Vert ^6)\) in the size \(\Vert {\mathscr {Q}}\Vert \) of the input QMC \({\mathscr {Q}}\), which is already \({\mathcal {O}}(|S|^2 n^4)\), since there are |S| classical states, \(|S|^2\) super-operators Q[s, t] (\(s,t \in S\)) attached to transitions, each super-operator Q[s, t] is given by at most \(n^2\) Kraus operators \({\mathbf {F}}_{s,t,k}\), and each Kraus operator \({\mathbf {F}}_{s,t,k}\) has plainly \(n^2\) entries.

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Authors and Affiliations

  1. Shanghai Key Lab of Trustworthy Computing, East China Normal University, Shanghai, China

    Ming Xu

  2. Institute of Intelligent Software, Guangzhou, China

    Cheng-Chao Huang

  3. Centre for Quantum Software and Information, University of Technology Sydney, Sydney, Australia

    Yuan Feng

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  1. Ming Xu
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  2. Cheng-Chao Huang
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  3. Yuan Feng
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Correspondence to Yuan Feng.

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Ming Xu was supported by the National Natural Science Foundation of China (Grant No. 11871221) and the Fundamental Research Funds for the Central Universities. Cheng-Chao Huang was supported by the Guangdong Science and Technology Department (Grant Nos. 2019A1515011689, 2018B010107004). Yuan Feng was supported by the National Key R&D Program of China (Grant No. 2018YFA0306704) and the Australian Research Council (Grant No. DP180100691)

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Xu, M., Huang, CC. & Feng, Y. Measuring the constrained reachability in quantum Markov chains. Acta Informatica 58, 653–674 (2021). https://doi.org/10.1007/s00236-020-00392-5

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