Optimal approximation to unitary quantum operators with linear optics

Linear optical systems acting on photon number states produce many interesting evolutions, but cannot give all the allowed quantum operations input state. Using Toponogov's theorem from differential geometry, we propose an iterative method that, for any arbitrary operator $U$ $n$ photons in $m$ modes, returns $\widetilde{U}$ which can be implemented with linear optics. The approximation is locally optimal and converges. resulting translated into experimental setup using previous results.