Continuous-time quantum walks on directed bipartite graphs

Abstract

This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter α. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes N , time t and the model parameter α. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerical simulations of quantum walks on circulant graphs show complex interference phenomena and how complete suppression of transport is achieved near α = π 2. By proving two mirror symmetries around α = 0 and π 2 we show that these quantum walks have a period of π in α. We show that undirected edges lose their effect on the quantum walk at α = π 2 and present non-bipartite graphs that exhibit suppression of transport. Finally, we analytically compute the Hamiltonians of quantum walks on the directed ring graph.

Short-time behavior of continuous-time quantum walks on graphs

“…Similar features for designing (non-)preferred directions or even generating dark states by tuning the hopping were already studied in Refs. [29,61,62]; our methods could provide a more systematic treatment of this. Another possible application of our results comes from the observation that the actual measurement of the short time asymptotics resulting in the distance of the nodes can be interpreted as a distance oracle.…”

Complex networks from classical to quantum

“…Network topology has further been shown as a means to direct transport by adding complex numbers-while maintaining Hermiticity-to the networks adjacency matrix in 'chiral quantum walks' [65,71,72] (note that chiral walks were realized experimentally in [12]). Open system walks which mix stochastic and quantum effects in 'open' evolutions [64] have aided in the study of quantum effects in biological exciton transport (again, modeled as a quantum walk) and developments in a quantum version of Google's PageRank [8][9][10] has been seen, providing a practical solution to overcome the degeneracy issues affecting the classical version and enhancing node ranking in large networks.…”

Handbook of Network Analysis [KONECT -- the Koblenz Network Collection]

“…These matrices appear in the modeling of quantum walks, as used for instance by Tödtli et al (2016).…”