Embedding matrices in programmable photonic networks with flexible depth and width

We show that programmable photonic circuit architectures composed of alternating mixing layers and active layers offer a high degree of flexibility. This alternating configuration enables the systematic tailoring of both the network's depth (number of layers) and width (size of each layer) without compromising computational capabilities. From a mathematical perspective, our approach can be viewed as embedding an arbitrary target matrix into a higher-dimensional matrix, which can then be represented with fewer layers and a larger number of active elements. We derive a general relation for the width and depth of a network that guarantees representing all N × N complex-valued matrix operations. Remarkably, we show that just two such active layers-interleaved with passive mixing layers-are sufficient to universally implement arbitrary matrix transformations. This result promises a more adaptable and scalable route to photonic matrix processors.