Quantum walks based on an interferometric analogy

There are presently two models for quantum walks on graphs. The “coined” walk uses discrete-time steps, and contains, besides the particle making the walk, a second quantum system, the coin, that determines the direction in which the particle will move. The continuous walk operates with continuous time. Here a third model for quantum walks is proposed, which is based on an analogy to optical interferometers. It is a discrete-time model, and the unitary operator that advances the walk one step depends only on the local structure of the graph on which the walk is taking place. This type of walk also allows us to introduce elements, such as phase shifters, that have no counterpart in classical random walks. Several examples are discussed.