  Single photon interference due to passage through a periodic grating is
considered in a novel proposal for processing D-dimensional quantum systems
(quDits) encoded in the spatial degrees of freedom of light. We show that free
space propagation naturally implements basic single quDit gates by means of the
Talbot effect: an intricate time-space carpet of light in the near field
diffraction regime. Adding a diagonal phase gate, we show that a complete set
of single quDit gates can be implemented. We then introduce a
spatially-dependent beam splitter that allows implementation of controlled
operations between two quDits. A new form of universal quantum information
processing can then be implemented with linear optics and ancilla photons.
Though we consider photons, our scheme should be directly applicable to a
number of other physical systems. Interpretation of the Talbot effect as a
quantum logic operation provides a beautiful and interesting way to visualize
quantum computation through wave propagation and interference.
