Minimum optical depth multiport interferometers for approximating arbitrary unitary operations and pure states

Reconfigurable devices which can implement arbitrary unitary operations are crucial for photonic quantum computation, optical neural networks, and boson sampling. Here, we address the problem of, using multiport interferometers, approximating with a given infidelity any pure-state preparation and any unitary operation. By means of numerical calculations, we show that pure states, in any dimension $d$, can be prepared with infidelity $\ensuremath{\le}{10}^{\ensuremath{-}15}$ with three layers of $d$-dimensional Fourier transforms and three layers of configurable phase shifters. We also present numerical evidence that $d+1$ layers of $d$-dimensional Fourier transforms and $d+2$ layers of configurable phase shifters can produce any unitary operation with infidelity $\ensuremath{\le}{10}^{\ensuremath{-}14}$. The conclusions are achieved by numerical simulations in the range from $d=3$ to $d=10$. These results are interesting in light of the recent availability of multicore fiber-integrated multiport interferometers.