  We consider the problem of approximating the ground state energy of a
fermionic Hamiltonian using a Gaussian state. In sharp contrast to the dense
case, we prove that strictly $q$-local $\rm {\textit {sparse}}$ fermionic
Hamiltonians have a constant Gaussian approximation ratio; the result holds for
any connectivity and interaction strengths. Sparsity means that each fermion
participates in a bounded number of interactions, and strictly $q$-local means
that each term involves exactly $q$ fermionic (Majorana) operators. We extend
our proof to give a constant Gaussian approximation ratio for sparse fermionic
Hamiltonians with both quartic and quadratic terms. With additional work, we
also prove a constant Gaussian approximation ratio for the so-called sparse SYK
model with strictly $4$-local interactions (sparse SYK-$4$ model). In each
setting we show that the Gaussian state can be efficiently determined. Finally,
we prove that the $O(n^{-1/2})$ Gaussian approximation ratio for the normal
(dense) SYK-$4$ model extends to SYK-$q$ for even $q>4$, with an approximation
ratio of $O(n^{1/2 - q/4})$. Our results identify non-sparseness as the prime
reason that the SYK-$4$ model can fail to have a constant approximation ratio.
