Section: New Results

Simulation of quantum walks and fast mixing with classical processes

Participants: A. Sarlette

This is the final result of a line of work where we show that the mixing behavior of quantum walks on graphs can always be simulated by a classical "lifted Markov chain". This implies that quantum walks must satisfy a conductance bound on mixing speed, like classical Markov chains. Also current efficient quantum walk constructions are linked to classical processes that provide the same convergence speed. This excludes a simple characterization of quantum walk advantages in terms of bare mixing speed, as has been done by some previous authors comparing just to simple Markov chains. The question of efficient design of walks on graphs, on the basis of local graph queries and for specific applications, is thus brought back to the center of the focus for quantum walks. This collaborative work with F. Ticozzi (U. of Padova) has been published in [11].

As a follow-up on this work, we have developed algorithms in the latter sense: quantum walks on the basis of local design and which do speed up some applications. These last results have been presented as posters at conferences and will hopefully be part of next year's publications.