Continuous-time quantum walks in the presence of a quadratic perturbation

Abstract

We address the properties of continuous-time quantum walks with Hamiltonians of the form H = L + λL 2 , with L the Laplacian matrix of the underlying graph and the perturbation λL 2 motivated by its potential use to introduce next-nearest-neighbor hopping. We consider cycle, complete, and star graphs as paradigmatic models with low and high connectivity and/or symmetry. First, we investigate the dynamics of an initially localized walker. Then we devote attention to estimating the perturbation parameter λ using only a snapshot of the walker dynamics. Our analysis shows that a walker on a cycle graph spreads ballistically independently of the perturbation, whereas on complete and star graphs one observes perturbation-dependent revivals and strong localization phenomena. Concerning the estimation of the perturbation, we determine the walker preparations and the simple graphs that maximize the quantum Fisher information. We also assess the performance of position measurement, which turns out to be optimal, or nearly optimal, in several situations of interest. Besides fundamental interest, our study may find applications in designing enhanced algorithms on graphs.

Floquet engineering of continuous-time quantum walks: towards the simulation of complex and next-to-nearest neighbor couplings

“…We also leave open the question of whether real NNN couplings can be effectively implemented in quantum walks via Floquet engineering. In principle, this could be used to simulate quadratic perturbations to the usual quantum walk Hamiltonian [55] or quantum walks on other structures such as complex networks [22].…”

Decoherence and classicalization of continuous-time quantum walks on graphs

“…The parameter γ ≥ 0 is referred to as the decoherence rate, and L is the graph Laplacian. From an operational point of view, intrinsic decoherence corresponds to a randomized quadratic perturbation [32].…”

Universality of the fully connected vertex in Laplacian continuous-time quantum walk problems

“…By fully connected vertex we mean a vertex which is adjacent (connected) to all the other vertices of the graph, as shown in figure 1. The dynamics of the central vertex of the star graph and that of any vertex of the complete graph are equivalent, showing periodic perfect revivals and strong localization on the initial vertex [15], even in the presence of a perturbation λL 2 [16]. The spatial search of a marked vertex on the complete graph or on the star graph, when the target is the central vertex, are equivalent [17], and the same qualitative results are observed even in the presence of weak random telegraph noise [18].…”