Decoherence in two-dimensional quantum walks

Abstract

We analyze the decoherence in quantum walks in two-dimensional lattices generated by brokenlink-type noise. In this type of decoherence, the links of the lattice are randomly broken with some given constant probability. We obtain the evolution equation for a quantum walker moving on 2-D lattices subject to this noise, and we point out how to generalize for lattices in more dimensions. In the non-symmetric case, when the probability to break links in one direction is different from the probability in the perpendicular direction, we have obtained a non-trivial result. If one fixes the link-breaking probability in one direction, and gradually increases the probability in the other direction from 0 to 1, the decoherence initially increases until it reaches a maximum value, and then it decreases. This means that, in some cases, one can increase the noise level and still obtain more coherence. Physically, this can be explained as a transition from a decoherent 2-D walk to a coherent 1-D walk.

Implementation of a spatial two-dimensional quantum random walk with tunable decoherence

“…The shapes obtained in Fig. 3 correspond to the probability distributions of a two-dimensional Grover walk [19,25]. The walker propagates with ballistic speed, characterized by a quadratic dependence of the variance V (n) with the step index, that is, V (n) ≈ n 2 .…”

“…Along these lines, two-dimensional QRWs provide a powerful tool for modeling complex quantum information and energy transport systems [19,20]. Their realization presents a challenge because of the need for four-level coin operation [21][22][23].…”

The QWalk simulator of quantum walks

“…It is also possible to simulate a unitary noise [16] which randomly opens links of the lattice. This noise model has been studied for the quantum walk on a line [9] and on a plane [8].…”

“…The technique of broken links for quantum walks was developed originally by Romanelli et al [9] and later generalized for the two-dimensional case by Oliveira et al [8].…”

“…If we apply the evolution operator U a to state (1) and include functions L 1 and L 2 as in [8], we obtain the evolution equation…”