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A new definition of hitting time and an embedded Markov chain in continuous-time quantum walks

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Abstract

We present a new probabilistic definition for the hitting time of a continuous-time quantum walk into a marked set of nodes, when measurements with respect to the canonical basis are performed according to the jump times of a Poisson process. Furthermore, we derive a formula for the calculation of the mean hitting time, based on our novel definition of hitting time, Wald’s theorem and the stochastic process that models our quantum measurement outcomes. This stochastic process results in a Markov chain, whose transition matrix contains the expected values of the squared norm of the entries of a random unitary matrix, and it can be thought as a way to embed a Markov chain in a continuous-time quantum walk.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

MARO acknowledges the financial support of Centro de Investigación en Matemáticas. SEVA acknowledges the financial support provided by Tecnologico de Monterrey, Escuela de Ingenieria y Ciencias and Consejo Nacional de Ciencia y Tecnología (CONACyT) [SNI No. 41594]. SEVA acknowledges the unconditional support of his family. All authors gratefully acknowledge the support of IBM via the IBM Cloud Credit Award (No. 1027LA) granted to our project Quantum Walks over Genotype Networks.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Guanajuato, Guanajuato, Mexico

    Miguel A. Ruiz-Ortiz & Ehyter M. Martín-González

  2. Department of Integrative Biology, University of South Florida, Tampa, FL, USA

    Diego Santiago-Alarcon

  3. Escuela de Ingenieria y Ciencias, Tecnologico de Monterrey, Ave. Eugenio Garza Sada 2501, 64849, Monterrey, N.L., Mexico

    Salvador E. Venegas-Andraca

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  1. Miguel A. Ruiz-Ortiz
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  2. Ehyter M. Martín-González
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  3. Diego Santiago-Alarcon
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  4. Salvador E. Venegas-Andraca
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Correspondence to Salvador E. Venegas-Andraca.

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Ruiz-Ortiz, M.A., Martín-González, E.M., Santiago-Alarcon, D. et al. A new definition of hitting time and an embedded Markov chain in continuous-time quantum walks. Quantum Inf Process 22, 224 (2023). https://doi.org/10.1007/s11128-023-03972-9

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