Stefan Falkner

Weak limit of the three-state quantum walk on the line

We revisit the one dimensional discrete time quantum walk with 3 states and the Grover coin. We derive analytic expressions for observed the localization, an long time approximation for the probability density function (PDF). We also connect the time averaged approximation to the PDF found by Inui et. al. to a spatial average of the walk. We show that this smooth approximation constitutes a proper PDF that predicts moments of the real PDF accurately.

One-dimensional coinless quantum walks

A coinless, discrete-time quantum walk possesses a Hilbert space whose dimension is smaller compared to the widely-studied coined walk. Coined walks require the direct product of the site basis with the coin space, coinless walks operate purely in the site basis, which is clearly minimal. These coinless quantum walks have received considerable attention recently because they have evolution operators that can be obtained by a graphical method based on lattice tessellations and they have been shown to be as efficient as the best known coined walks when used as a quantum search algorithm. We argue that both formulations in their most general form are equivalent. In particular, we demonstrate how to transform the one-dimensional version of the coinless quantum walk into an equivalent extended coined version for a specific family of evolution operators. We present some of its basic, asymptotic features for the one-dimensional lattice with some examples of tessellations, and analyze the mixing time and limiting probability distributions on cycles.

Renormalization and scaling in quantum walks

We show how to extract the scaling behavior of quantum walks using the renormalization group (RG). We introduce the method by efficiently reproducing well-known results on the one-dimensional lattice. As a nontrivial model, we apply this method to the dual Sierpinski gasket and obtain its exact, closed system of RG-recursions. Numerical iteration suggests that under rescaling the system length,

Relation between random walks and quantum walks

L^{\prime}=2L

Renormalization group for quantum walks

, characteristic times rescale as

Finite-size corrections for ground states of Edwards-Anderson spin glasses

t^{\prime}=2^{d_{w}}t

Quantum Ultra-Walks: Walks on a Line with Spatial Disorder

with the exact walk exponent

Scaling of Quantum Walks on Complex Networks

d_{w}=\log_{2}\sqrt{5}=1.1609\ldots

On the Relation between Random Walks and Quantum Walks

. Despite the lack of translational invariance, this is very close to the ballistic spreading,

Renormalization for Quantum Walks

d_{w}=1