Document Zbl 1530.81040

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Appl. Algebra Geom.</a> <a href="/?q=in%3A501196" title="Articles in this Issue">7, No. 3, 567-584 (2023)</a>. </div> <div class="abstract">Summary: The adjacency matrix of a graph $G$ is the Hamiltonian for a continuous-time quantum walk on the vertices of $G$. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues – what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7727506">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A81P68" title="MSC2020">81P68</a> </td> <td class="space"> Quantum computation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60K50" title="MSC2020">60K50</a> </td> <td class="space"> Anomalous diffusion models (subdiffusion, superdiffusion, continuous-time random walks, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A82C70" title="MSC2020">82C70</a> </td> <td class="space"> Transport processes in time-dependent statistical mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37A25" title="MSC2020">37A25</a> </td> <td class="space"> Ergodicity, mixing, rates of mixing </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C69" title="MSC2020">05C69</a> </td> <td class="space"> Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35P15" title="MSC2020">35P15</a> </td> <td class="space"> Estimates of eigenvalues in context of PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A70H05" title="MSC2020">70H05</a> </td> <td class="space"> Hamilton’s equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C81" title="MSC2020">05C81</a> </td> <td class="space"> Random walks on graphs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A42A75" title="MSC2020">42A75</a> </td> <td class="space"> Classical almost periodic functions, mean periodic functions </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acontinuous-time+quantum+walk">continuous-time quantum walk</a>; <a href="/?q=ut%3Apretty+good+state+transfer">pretty good state transfer</a>; <a href="/?q=ut%3Aaverage+mixing+matrix">average mixing matrix</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 1530.81040" data-ciurl="/ci/07727506" data-biburl="/bibtex/07727506.bib" data-amsurl="/amsrefs/07727506.bib" data-xmlurl="/xml/07727506.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07727506.pdf" title="Zbl 1530.81040 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/10.1137/22M1521262" aria-label="DOI for “Irrational quantum walks”" title="10.1137/22M1521262">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2208.08971"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">Adamczak, W., Andrew, K., Bergen, L., Ethier, D., Hernberg, P., Lin, J., and Tamon, C., Non-uniform mixing of quantum walk on cycles, Int. 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Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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