  We present two hybrid linear-optical architectures that simultaneously
exploit spatial and temporal degrees of freedom of light to effect arbitrary
discrete unitary transformations. Our architectures combine the benefits of
spatial implementations of linear optics, namely low loss and parallel
operation, with those of temporal implementations, namely modest resource
requirements and access to transformations of potentially unbounded size. We
arrive at our architectures by devising and employing decompositions of large
discrete unitary transformations into smaller ones, decompositions we expect to
have broad utility beyond spatio-temporal linear optics. We show that hybrid
architectures promise important advantages over both spatial-only and
temporal-only architectures.
