Commentary on the decomposition of universal multiport interferometers: how it works in practice

The decomposition of multiport interferometers is a fundamental tool in quantum optics and computing. This note aims to serve as a concise reference for performing the decomposition according to the most common design approaches, offering a self-contained treatment with essential mathematical details and practical working examples. Specifically, we provide a clear, self-consistent presentation of all the mathematics required to determine the gate parameters and apply the necessary phase corrections to achieve the desired structure.