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Quantum walks and coarse structure

Banff International Research Station for Mathematical Innovation and Discovery

Locality is the key property that ties a quantum walk to the underlying lattice. By this we mean that matrix elements of the unitary one-step operator become small between distant sites. The extreme case of this is a nearest neighbour walk, where matrix elements become zero for distance >1. We are interested here in the notions of locality for infinite lattices, allowing some decay of matrix elements, and focusing on the large scale propagation behaviour. The mathematical notion for a locality structure, which takes due note of composition properties, is called a coarse structure. I will describe this and show how even in one dimension there are different natural choices, which subtly differ even in the translation invariant case. I will then go to higher dimension, and discuss the relation to compactifications, C*-algebras, and their classification via K-theory.

Mathematics; Quantum theory; Mathematical physics

video/mp4

Author affiliation: Leibniz Universität Hannover

2019-10-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/71978

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/