Quantum and random walks as universal generators of probability distributions

Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without losing coherence as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.

Figure 1

Probability functions at $t=30$. The red dots were obtained by averaging 10 000 simulated trajectories of an inhomogeneous time-dependent random walk. The solid black line corresponds to the probability function of a quantum walk with a Hadamard coin. Only even locations are shown as the probability is zero for odd sites.

Figure 2

Probability functions at $t=30$. The red dots were obtained by averaging 10 000 simulated trajectories of an inhomogeneous time-dependent random walk. The solid black line corresponds to the probability function of a symmetric Hadamard walk. Only even locations are shown as the probability is zero for odd sites.