To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk.
For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.
We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.