  Random unitary matrices find a number of applications in quantum information
science, and are central to the recently defined boson sampling algorithm for
photons in linear optics. We describe an operationally simple method to
directly implement Haar random unitary matrices in optical circuits, with no
requirement for prior or explicit matrix calculations. Our physically-motivated
and compact representation directly maps independent probability density
functions for parameters in Haar random unitary matrices, to optical circuit
components. We go on to extend the results to the case of random unitaries for
qubits.
