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 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 $$\widetilde{U}$$ U ~ which can be implemented with linear optics. The approximation method is locally optimal and converges. The resulting operator $$\widetilde{U}$$ U ~ can be translated into an experimental optical setup using previous results.

The displacement function and the return map

This chapter studies the displacement function dᵧ on X that is associated with a semisimple element γ‎ ∈ G. If φ‎″, t ∈ R denotes the geodesic flow on the total space X of the tangent bundle of X, the critical set X(γ‎) ⊂ X of dᵧ can be easily related to the fixed point set Fᵧ ⊂ X of the symplectic transformation γ‎⁻¹φ‎₁ of X. The chapter studies the nondegeneracy of γ‎⁻¹φ‎₁ − 1 along Fᵧ. More fundamentally, this chapter gives important quantitative estimates on how much φ‎ ½ differs from φ‎ ˗½γ‎ away from Fᵧ. These quantitative estimates are based on Toponogov's theorem.