  Inspired by the classical phenomenon of random walk, the concept of quantum
walk has emerged recently as a powerful platform for the dynamical simulation
of complex quantum systems, entanglement production and universal quantum
computation. Such a wide perspective motivates a renewing search for efficient,
scalable and stable implementations of this quantum process. Photonic
approaches have hitherto mainly focused on multi-path schemes, requiring
interferometric stability and a number of optical elements that scales
quadratically with the number of steps. Here we report the experimental
realization of a quantum walk taking place in the orbital angular momentum
space of light, both for a single photon and for two simultaneous
indistinguishable photons. The whole process develops in a single light beam,
with no need of interferometers, and requires optical resources scaling
linearly with the number of steps. Our demonstration introduces a novel
versatile photonic platform for implementing quantum simulations, based on
exploiting the transverse modes of a single light beam as quantum degrees of
freedom.
