Optimal linear-optical interferometers for unitary t-designs in a multi-photon Fock space

We investigate the construction of unitary t-designs in a multi-photon Fock space on programmable linear-optical interferometers. Using the frame potential as the optimization objective, we identify architectures that realize approximate unitary 1- and 2-designs with considerably fewer controllable parameters than required for universal linear optics. For example, we find numerical evidence suggesting that for unitary 1-designs, the required number of programmable parameters scales empirically as 2d with the number of modes d, instead of O\(d²\). We benchmark the resulting ensembles using single-photon transition probabilities and average channel fidelities. Furthermore, as a practical result, we develop a photon-number estimation protocol for multi-photon Fock states that leverages these optimized 1- and 2-designs. Our findings demonstrate that shallow interferometers can accurately reproduce Haar-averaged first moments, while second-moment-sensitive observables provide a considerably stricter test for approximate unitary 2-designs.