One-dimensional quantum walk with unitary noise
Daniel Shapira, Ofer Biham, Anthony J. Bracken, Michelle Hackett · Physical Review A · 2003
The effect of unitary noise on the discrete one-dimensional quantum walk is studied using computer simulations. For the noiseless quantum walk, starting at the origin $(n=0)$ at time $t=0,$ the position distribution ${P}_{t}(n)$ at time t is very different from the Gaussian distribution obtained for the classical random walk. Furthermore, its standard deviation, $\ensuremath{\sigma}(t)$ scales as $\ensuremath{\sigma}(t)\ensuremath{\sim}t,$ unlike the classical random walk for which $\ensuremath{\sigma}(t)\ensuremath{\sim}\sqrt{t}.$ It is shown that when the quantum walk is exposed to unitary noise, it exhibits a crossover from quantum behavior for short times to classical-like behavior for long times. The crossover time is found to be $T\ensuremath{\sim}{\ensuremath{\alpha}}^{\ensuremath{-}2},$ where $\ensuremath{\alpha}$ is the standard deviation of the noise.