  We exploit insights into the geometry of bosonic and fermionic Gaussian
states to develop an efficient local optimization algorithm to extremize
arbitrary functions on these families of states. The method is based on notions
of gradient descent attuned to the local geometry which also allows for the
implementation of local constraints. The natural group action of the symplectic
and orthogonal group enables us to compute the geometric gradient efficiently.
While our parametrization of states is based on covariance matrices and linear
complex structures, we provide compact formulas to easily convert from and to
other parametrization of Gaussian states, such as wave functions for pure
Gaussian states, quasiprobability distributions and Bogoliubov transformations.
We review applications ranging from approximating ground states to computing
circuit complexity and the entanglement of purification that have both been
employed in the context of holography. Finally, we use the presented methods to
collect numerical and analytical evidence for the conjecture that Gaussian
purifications are sufficient to compute the entanglement of purification of
arbitrary mixed Gaussian states.
