# Localization, Diffusion, Topological Phases in 2-D quantum walks

## János Asbóth

### (MTA Wigner RCP SZFI)

Time: Mon Jan 19 10:00:00 2015

Location: Building 1, Auditorium

The hallmark property of Quantum Walks is that they spread ballistically, <x^2> \propto t^2, faster than the diffusive, <x^2> \propto t spreading of random walks. This gives them a "quantum speedup", which can be exploited in quantum algorithms. In the presence of time-independent disorder, however, 1- and 2-dimensional (1D and 2D) quantum walks are expected to suffer Anderson localization, whereby their spread stops completely, <x^2> \to \const, except for a few exceptions. This has recently been seen in simulations of a simple quantum walk, the 2D split-step Hadamard walk [1].

We have shown [2] that the conclusions drawn about the 2D split-step Hadamard walk are in fact incorrect: over longer timescales, instead of localization, it spreads diffusively. The reason is that this quantum walk is an iterative dynamical system tuned to a critical point at a topological phase transition. In the talk, I will explain what we mean by that.

[1]: J. Svozilik, R. D. J. Leon-Montiel, and J. P. Torres, PRA 86, 052327 (2012). [2]: J. M. Edge and J. K. Asboth, submitted to PRB, http://arxiv.org/abs/1411.7691