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Quantum walks via quantum cellular automata

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Abstract

Very much as its classical counterpart, quantum cellular automata are expected to be a great tool for simulating complex quantum systems. Here we introduce a partitioned model of quantum cellular automata and show how it can simulate, with the same amount of resources (in terms of effective Hilbert space dimension), various models of quantum walks. All the algorithms developed within quantum walk models are thus directly inherited by the quantum cellular automata. The latter, however, has its structure based on local interactions between qubits, and as such it can be more suitable for present (and future) experimental implementations.

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Notes

  1. Note that the space \(\mathscr {H}_G\) is not given by the tensor product of a space associated to the edges and another space associated to the vertices.

  2. Note that since the evolution for t time steps is given by \(U^t\), then the operator X can be absorbed in the coin operator by suitably applying corrections on the initial and final state: \(U^t =X \cdot (S_I\cdot CX)^t X^{-1}\).

  3. Here again the action of the third tiling can be absorbed in the action of the first one, plus modifications in the initial and final state. We, however, present the translation with three tilings for clarity reasons.

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Acknowledgements

We would like to thank Osvaldo J. Farias for various discussions on the topic of quantum cellular automata. We acknowledge financial support from the National Institute for Science and Technology of Quantum Information (INCT-IQ/CNPq, Brazil).

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Authors and Affiliations

  1. Brazilian Center for Research in Physics-CBPF, Rua Dr. Xavier Sigaud, 150, Urca, Rio de Janeiro, RJ, Brazil

    Pedro C. S. Costa & Fernando de Melo

  2. National Laboratory of Scientific Computing-LNCC, Av. Getúlio Vargas, 333, Quitandinha, Petrópolis, RJ, Brazil

    Renato Portugal

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  1. Pedro C. S. Costa
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Correspondence to Pedro C. S. Costa.

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Costa, P.C.S., Portugal, R. & de Melo, F. Quantum walks via quantum cellular automata . Quantum Inf Process 17, 226 (2018). https://doi.org/10.1007/s11128-018-1983-x

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