Riemannian optimization of photonic quantum circuits in phase and Fock space

Understand

We propose a framework to design and optimize generic photonic quantum circuits composed of Gaussian objects (pure and mixed Gaussian states, Gaussian unitaries, Gaussian channels, Gaussian measurements) as well as non-Gaussian effects such as photon-number-resolving measurements.

• In this framework, we parametrize a phase space representation of Gaussian objects using elements of the symplectic group (or the unitary or orthogonal group in special cases), and then we transform it into the Fock representation using a single linear recurrence relation that computes the Fock amplitudes of any Gaussian object recursively.
• We also compute the gradient of the Fock amplitudes with respect to phase space parameters by differentiating through the recurrence relation.
• We can then use Riemannian optimization on the symplectic group to optimize M-mode Gaussian objects, avoiding the need to commit to particular realizations in terms of fundamental gates.