Figure 6

(a) Beams propagating through a regular grid of beam splitters achieve a walk with spatial encoding. The powers of beams emerging from a grid of depth N correspond to the probability distribution of the walker and coin (encoded in beam location and direction, respectively) after a walk of N steps. (b) Optical scheme for the generation of robust coherent superpositions of position states of the walker. The scheme consists of the preparation of the initial coin state \(|\phi\rangle_{c} = \frac {1}{\sqrt{2}} (|0\rangle+i|1\rangle)\) through a beam splitter at \(\theta= \pi/4\) where the output arms are then recombined to allow the walker to start the walk from position 0. The evolution is then performed with a biased coin, beam splitters with transmission ratio θ, and a detection stage. The latter projects the coin onto a desired state through an array of beam splitters at \(\theta= \pi/4\) and collecting the output from one side of arms. The phases \(\phi _{n,k}\) where n is the step and k the sub-step index, indicate how noise can be inserted in the system.