Reconfigurable devices capable to implement any unitary operation with a given fidelity are crucial for photonic universal quantum computation, optical neural networks, and boson sampling. Here, we address the problems of approximating with a given infidelity any unitary operation and any pure state using multi-port interferometers, which are of current interest due to the recent availability of multi-core fiber integrated multi-port interferometers. We show that any pure state, in any dimension , can be prepared with infidelity with ~layers of -dimensional Fourier transforms and ~layers of configurable phase shifters. In contrast, the schemes in [Phys. Rev. Lett. \textbf{73}, 58 (1994) and Optica \textbf{3}, 1460 (2016)], require optical depth . We also present numerical evidence that ~layers of -dimensional Fourier transforms and ~layers of configurable phase shifters can produce any unitary with infidelity , while the scheme in [Phys. Rev. Lett. \textbf{124}, 010501 (2020)] only achieves an infidelity in the order of for block-diagonal unitary transformations