The recursive representation of Gaussian quantum mechanics

We introduce a unified and differentiable Fock space representation of pure and mixed Gaussian states, Gaussian unitaries and Gaussian channels in terms of a single linear recurrence relation that can generate their Fock space amplitudes recursively. Due to its recursive and differentiable nature, it makes for a simple and fast computational implementation that enables calculating the gradient of the Fock amplitudes with respect to parametrizations. To show the flexibility and the generality of the gradient calculation, we show how to optimize $M$-mode Gaussian objects (pure and mixed states, unitaries, and channels) without the need to express them using fundamental components, by performing an optimization directly on the manifold of the symplectic group (or the orthogonal group for $M$-mode interferometers). We also find the composition rule of Gaussian operations expressed in the recurrent form, which allows us to obtain the correct global phase when composing Gaussian operations, and therefore extend our model to states that can be written as linear combinations of Gaussians. We implemented all of these methods in the freely available open-source library MrMustard.