Section: New Results

Relations between quantum walks, open quantum walks, and lifted walks: the cycle graph

Participants: Alain Sarlette

The convergence time of a random walk on a graph towards its stationary distribution is an important indication of the efficiency of random algorithms based on it. Quantum random walks have been shown to allow quadratically accelerated convergence for large graphs, at least in some cases. The famous Grover search algorithm has been shown to actually fit this framework in an abstracted setting (it is doing the opposite of a random walk: converging from the uniform distribution towards a particular identified element). Yet also with classical dynamics, simple mechanisms have been proposed which allow to quadratically accelerate the convergence with respect to a standard random walk. Some basic principles have been conjectured to cause this acceleration, basically transforming a diffusion-like behavior into a more transport-like behavior, but with remaining trail. We are working towards formally characterizing the effect of these principles, and extracting similar principles in the quantum walks. This should help identify key effects to be protected in the associated quantum algorithms. We currently have worked out the equivalence of all these accelerating settings for the simplest example of the cycle graph. Quantum coherences turn out to play no major role and a classical feedback structure can be identified. We are now working towards other graphs, where the convergence effect of quantum coherences might be hidden in propagating classical information. This work has been presented at [35] .