One-dimensional quantum walk with unitary noise

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 $\mathrm{}(n=0\mathrm{})\mathrm{}$ at time $t=0,$ the position distribution $P_t\mathrm{}\left(n\right)\mathrm{}$ at time t is very different from the Gaussian distribution obtained for the classical random walk. Furthermore, its standard deviation, $\sigma \mathrm{}\left(t\right)\mathrm{}$ scales as $\sigma \mathrm{}\left(t\right)\mathrm{}\sim t,$ unlike the classical random walk for which $\sigma \mathrm{}\left(t\right)\mathrm{}\sim \sqrt{t}.$ 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 $T\sim {\alpha }^{-2},$ where $\alpha$ is the standard deviation of the noise.