  We discuss the use of the transverse spatial degrees of freedom of photons
propagating in the paraxial approximation for continuous variable information
processing. Given the wide variety of linear optical devices available, a
diverse range of operations can be performed on the spatial degrees of freedom
of single photons. Here we show how to implement a set of continuous quantum
logic gates which allow for universal quantum computation. In contrast with the
usual quadratures of the electromagnetic field, the entire set of single photon
gates for spatial degrees of freedom does not require optical nonlinearity and,
in principle, can be performed with a single device: the spatial light
modulator. Nevertheless, nonlinear optical processes, such as four-wave mixing,
are needed in the implementation of two-photon gates. The efficiency of these
gates is at present very low, however small scale investigations of continuous
variable quantum computation are within the reach of current technology. In
this regard, we show how novel cluster states for one-way quantum computing can
be produced using spontaneous parametric down conversion.
