Properties of long quantum walks in one and two dimensions

摘要: The quantum walk (QW) is the term given to a family of algorithms governing evolution discrete system and as such has founding role in study computation. We contribute investigation QW phenomena by performing detailed numerical discrete-time walks. In one dimension (1D), we compute structure probability distribution, which not smooth curve but shows oscillatory features on all length scales. By analyzing walks up N = 1,000,000 steps, discuss scaling characteristics limiting forms both real Fourier space. 2D, with view ready experimental realization, consider two types QW, based four-faced coin other sequential flipping single two-faced coin. Both QWs may be generated using coins, first case are completely unentangled second maximally entangled. draw our 1D results characterize properties walks, demonstrating maximal speed-up emerging semi-classical behavior entangled QW.