  Gaussian fermionic matrix product states (GfMPS) form a class of ansatz
quantum states for 1d systems of noninteracting fermions. We show, for a simple
critical model of free hopping fermions, that: (i) any GfMPS approximation to
its ground state must have bond dimension scaling superpolynomially with the
system size, whereas (ii) there exists a non-Gaussian fermionic MPS
approximation to this state with polynomial bond dimension. This proves that,
in general, imposing Gaussianity at the level of the tensor network may
significantly alter its capability to efficiently approximate critical Gaussian
states. We also provide numerical evidence that the required bond dimension is
subexponential, and thus can still be simulated with moderate resources.
