Document Zbl 1316.81066

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H.</a>; <a href="/authors/werner.reinhard-f" title="Author Profile">Werner, R. F.</a></div> <h2 class="title"> <strong>Asymptotic evolution of quantum walks with random coin.</strong> <i>(English)</i> <a class="label nowrap" href="/1316.81066">Zbl 1316.81066</a> </h2> <div class="source"> <a href="/serials/134" title="Journal Profile">J. Math. Phys.</a> <a href="/?q=in%3A346403" title="Articles in this Issue">52, No. 4, 042201, 36 p. (2011)</a>. </div> <div class="abstract">Summary: We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.{<br class="zbmathjax-paragraph">©2011 American Institute of Physics}</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6459813">Cited in <strong>45</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A81S25" title="MSC2020">81S25</a> </td> <td class="space"> Quantum stochastic calculus </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81Q35" title="MSC2020">81Q35</a> </td> <td class="space"> Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60G50" title="MSC2020">60G50</a> </td> <td class="space"> Sums of independent random variables; random walks </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60J10" title="MSC2020">60J10</a> </td> <td class="space"> Markov chains (discrete-time Markov processes on discrete state spaces) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A62E20" title="MSC2020">62E20</a> </td> <td class="space"> Asymptotic distribution theory in statistics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adiffusion+matrix">diffusion matrix</a>; <a href="/?q=ut%3Aasymptotically+Gaussian+state">asymptotically Gaussian state</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1316.81066" data-ciurl="/ci/06459813" data-biburl="/bibtex/06459813.bib" data-amsurl="/amsrefs/06459813.bib" data-xmlurl="/xml/06459813.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06459813.pdf" title="Zbl 1316.81066 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1063/1.3575568" aria-label="DOI for “Asymptotic evolution of quantum walks with random coin”" title="10.1063/1.3575568">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1009.2019"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ambainis, A.; Bach, E.; Nayak, A.; Vishwanath, A.; Watrous, J., Proc. $33^{rd}$ ACM Stoc (2001)</td> </tr><tr> <td>[2]</td> <td class="space">Meyer, D. 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