Experimental observations of 1D quantum walks in a limited region

The quantum walk (QW) is the quantum counterpart of the classical walk (CW) and is widely used in universal quantum computations. QWs provide exponential acceleration in hitting times and polynomial acceleration in searching times. Taking advantage of the integration and stability of waveguide structures, large-scale QWs can be implemented on chips. Here, we simulate both CWs and QWs in limited regions to qualitatively analyze the boundary conditions. Subsequently, we used a silicon chip to achieve 21-step QWs in a limited region and experimentally measured the photon number distributions by utilizing the fiber-coupling platform. Then we simulated the theoretical result by using the splitting ratio of the directional couplers which was experimentally obtained by measuring an individual directional coupler and the phase difference of the boundary waveguides which was fit from the experimental results. We also simulated the photon number distributions that are caused by the parameters of the waveguides, such as the splitting ratio, the phase difference, and the size of the walking region.