Hybrid spatiotemporal architectures for universal linear optics

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 spatiotemporal linear optics. We show that hybrid architectures promise important advantages over both spatial-only and temporal-only architectures.