Continuous-variable quantum computation with spatial degrees of freedom of photons

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.