  Fermionic Gaussian states are eigenstates of quadratic Hamiltonians and
widely used in quantum many-body problems. We propose a highly efficient
algorithm that converts fermionic Gaussian states to matrix product states. It
can be formulated for finite-size systems without translation invariance, but
becomes particularly appealing when applied to infinite systems with
translation invariance. If the ground states of a topologically ordered system
on infinite cylinders are expressed as matrix product states, then the fixed
points of the transfer matrix can be harnessed to filter out the anyon
eigenbasis, also known as minimally entangled states. This allows for efficient
computation of universal properties such as entanglement spectrum and modular
matrices. The potential of our method is demonstrated by numerical calculations
in two chiral spin liquids that have the same topological orders as the bosonic
Laughlin and Moore-Read states, respectively. The anyon eigenbasis for the
first one has been worked out before and serves as a useful benchmark. The
anyon eigenbasis of the second one is, however, not transparent and its
successful construction provides a nontrivial corroboration of our method.
