Mixing-time and large-decoherence in continuous-time quantum walks on one-dimension regular networks

Abstract

In this paper, we study mixing and large decoherence in continuous-time quantum walks on one dimensional regular networks, which are constructed by connecting each node to its 2l nearest neighbors(l on either side). In our investigation, the nodes of network are represented by a set of identical tunnel-coupled quantum dots in which decoherence is induced by continuous monitoring of each quantum dot with nearby point contact detector. To formulate the decoherent CTQWs, we use Gurvitz model and then calculate probability distribution and the bounds of instantaneous and average mixing times. We show that the mixing times are linearly proportional to the decoherence rate. Moreover, adding links to cycle network, in appearance of large decoherence, decreases the mixing times.

Time evolution of continuous-time quantum walks on dynamical percolation graphs

“…In figure 2, we demonstrate that equation (22) gives a very good approximation for the time evolution with finite τ values if we take an average over the trajectories, i.e. we use the superoperator formalism.…”

“…The jumps in the function can be explained by the oscillatory behavior of the P 1 (t ) function. This figure shows that for every T time and ε > 0 parameter there exist a τ step-size, for which the relative error of the (22) (or the (17)) formula is smaller than ε. It means that our results can be extended for an arbitrary large time interval, provided we choose a small enough step-size.…”

“…We take the [0, T ], T = 10 interval, and divide it into S parts, therefore the step-size is τ = T/S. In each step we calculate the relative error of the P 1 (t ) probability both with formula (22) and from the simulation. If the relative difference of these two values becomes bigger than or equal to an ε parameter we stop the simulation.…”

“…transport in disordered media [20], similar to percolation in a classical system. Even small changes in the underlying graph can lead to quite dramatic effects for the dynamics of the walk [21,22].…”