Quantum-driven sampling of the almost-uniform distribution via quantum walks

We investigate the use of discrete-time quantum walks to sample from an almost-uniform distribution, in the absence of any external source of randomness. Integers are encoded on the vertices of a cycle graph, and a quantum walker evolves for a fixed number of steps before its position is measured and recorded. The walker is then reset to the measured site, and the procedure is iterated to generate the sequence of outcomes. We show that when the number of nodes of the graph is odd, i.e. the condition of the ergodic theorem for classical random walks on finite groups are satisfied, the marginal distributions converges to the uniform distribution. Although correlations between successive outcomes are unavoidable, they can be significantly reduced by a suitable choice of the evolution time.