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Exact simulation of coined quantum walks with the continuous-time model

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Abstract

The connection between coined and continuous-time quantum walk models has been addressed in a number of papers. In most of those studies, the continuous-time model is derived from coined quantum walks by employing dimensional reduction and taking appropriate limits. In this work, we produce the evolution of a coined quantum walk on a generic graph using a continuous-time quantum walk on a larger graph. In addition to expanding the underlying structure, we also have to switch on and off edges during the continuous-time evolution to accommodate the alternation between the shift and coin operators from the coined model. In one particular case, the connection is very natural, and the continuous-time quantum walk that simulates the coined quantum walk is driven by the graph Laplacian on the dynamically changing expanded graph.

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Notes

  1. Let \(n_t\) be a Poisson random variable and denote the expected value by \(\langle \,\cdot \,\rangle \). For the continuous-time transition matrix \(M_{\mathrm{CT}}(t)\) and the discrete-time transition matrix \(M_{\mathrm{DT}}(k)\) of the RW, we have

    $$\begin{aligned} M_{\mathrm{CT}}(t) = \langle M_{\mathrm{DT}}(n_t) \rangle = \sum _{k=0}^\infty P(n_t=k)M_{\mathrm{DT}}(k) = \text {exp}[t(M_{\mathrm{DT}}-I)] . \end{aligned}$$

    .

  2. In order to write down a matrix representation of C or S, one first has to decide how to list the basis elements \(|v,j\rangle \). Arranging them with respect to v yields a matrix representation of C that is in block diagonal form. The representation of S can be made block diagonal by ordering in the j-component.

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Acknowledgements

P.P. would like to thank CNPq for its financial support (Grant Nos. 400216/2014-0 and 150726/2015-5). R.P. acknowledges financial support from Faperj (Grant No. E-26/102.350/2013) and CNPq (Grant Nos. 303406/2015-1 and 474143/2013-9).

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  1. National Laboratory of Scientific Computing, Petrópolis, RJ, Brazil

    Pascal Philipp & Renato Portugal

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  1. Pascal Philipp
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  2. Renato Portugal
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Correspondence to Pascal Philipp.

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Philipp, P., Portugal, R. Exact simulation of coined quantum walks with the continuous-time model. Quantum Inf Process 16, 14 (2017). https://doi.org/10.1007/s11128-016-1475-9

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