Abstract
Quantum walks on graphs can model physical processes and serve as efficient tools in quantum information theory. Once we admit random variations in the connectivity of the underlying graph, we arrive at the problem of percolation, where the long-time behaviour appears untreatable with direct numerical methods. We develop novel analytic methods based on the theory of random unitary operations which help us to determine explicitly the asymptotic dynamics of quantum walks on two-dimensional finite integer lattices with percolation. Based on this theory, we find new unexpected features of percolated walks like asymptotic position inhomogeneity or special directional symmetry breaking.
Highlights
The dynamics of particles is one of the central problems of physics
A novel elementary model of motion has been introduced for quantum particles: quantum walks [1, 2, 3] have been defined in analogy to classical random walks
In a previous paper [38] we introduced a general formalism describing the evolution of quantum walks on graphs with dynamically occurring defects – percolation graphs [39, 40, 41]
Summary
The dynamics of particles is one of the central problems of physics. The rules formulated for the motion of particles reflect our knowledge at an elementary level. When focusing on the so called discrete quantum walks we can study the ideal situation and analyze the effect of different perturbations They can lead to diffusion like behaviour representing the final state in a transition from quantum ballistic motion to classical diffusion [26]. In this paper we improve significantly our method which, in certain cases, leads to a considerably more efficient way to construct the attractor space and the asymptotic dynamics and is suited for dynamically percolated quantum walks on integer lattices. In striking contrast to the one-dimensional case, here a nonuniform position distribution can be obtained asymptotically Another new effect is the breaking of the directional symmetry: Should we rotate the initial state and the underlying graph by 90 degrees in a certain unpercolated walk, the obtained position distribution will be rotated.
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