Quantum walks on embeddings


We introduce a new type of discrete quantum walks, called vertex-face walks, based on orientable embeddings. We first establish a spectral correspondence between the transition matrix U and the vertex-face incidence structure. Using the incidence graph, we derive a formula for the principal logarithm of \(U^2\), and find conditions for its underlying digraph to be an oriented graph. In particular, we show this happens if the vertex-face incidence structure forms a partial geometric design. We also explore properties of vertex-face walks on the covers of a graph. Finally, we study a non-classical behavior of vertex-face walks.

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Correspondence to Hanmeng Zhan.

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Zhan, H. Quantum walks on embeddings. J Algebr Comb (2020). https://doi.org/10.1007/s10801-020-00958-z

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  • Quantum walks
  • Graph embeddings
  • Incidence matrices
  • Graph spectra

Mathematics Subject Classification

  • 05C50
  • 05E99