  The representation of ground states of fermionic quantum impurity problems as
superpositions of Gaussian states has recently been given a rigorous
mathematical foundation. [S. Bravyi and D. Gosset, Comm. Math. Phys. 356, 451
(2017)]. It is natural to ask how many parameters are required for an efficient
variational scheme based on this representation. An upper bound is $\mathcal
O(N^2)$, where $N$ is the system size, which corresponds to the number
parameters needed to specify an arbitrary Gaussian state. We provide an
alternative representation, with more favorable scaling, only requiring
$\mathcal O(N)$ parameters, that we illustrate for the interacting resonant
level model. We achieve the reduction by associating mean-field-like parent
Hamiltonians with the individual terms in the superposition, using physical
insight to retain only the most relevant channels in each parent Hamiltonian.
We benchmark our variational ansatz against the Numerical Renormalization
Group, and compare our results to existing variational schemes of a similar
nature to ours. Apart from the ground state energy, we also study the spectrum
of the correlation matrix -- a very stringent measure of accuracy. Our approach
outperforms some existing schemes and remains quantitatively accurate in the
numerically challenging near-critical regime.
