We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. One can numerically observe that our proposing quantum walks possess linear spreading and localization as in the case of the Grover walk on lattices. Moreover, our numerical simulation suggests that localization of our quantum walks reflect not only topological but also geometric structures. On the other hand, our proposing quantum walk contains an intrinsic problem concerning exhibition of nontrivial behavior, which is not seen in typical quantum walks such as Grover walks on graphs.

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We discuss the description of eigenspace of a quantum walk model U with an associating linear operator T in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of U without the spectral decomposition of T. Arguments in this direction reveal the eigenspaces of U characterized by the generalized kernels of linear operators given by T.

We show that discrete-time quantum walks on the line, Z, behave as "the quantum tunneling." In particular, quantum walkers can tunnel through a double-well with the transmission probability 1 under a mild condition. This is a property of quantum walks which cannot be seen on classical random walks, and is different from both linear spreadings and localizations.