### Abstract

Quantum versions of random walks on the line and cycle show a quadratic improvement in their spreading rate and mixing times respectively. The addition of decoherence to the quantum walk produces a more uniform distribution on the line, and even faster mixing on the cycle by removing the need for time-averaging to obtain a uniform distribution. By calculating the entanglement between the coin and the position of the quantum walker, the optimal decoherence rates are found to be such that all the entanglement is just removed by the time the final measurement is made.

This requires only O(log T) random bits for a quantum walk of T steps.