Optimal approximation to unitary quantum operators with linear optics

TL;DR

This paper introduces an iterative, locally optimal method based on differential geometry to approximate any quantum operator with linear optical systems, enabling practical implementation of complex quantum evolutions.

Contribution

It proposes a novel iterative approximation technique for quantum operators using linear optics, leveraging Toponogov's theorem for convergence and optimality.

Findings

Method converges to a locally optimal approximation.

Approximate operators can be translated into experimental setups.

Applicable to any quantum operator on photon modes.

Abstract

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