  We give a classical $1/(qk+1)$-approximation for the maximum eigenvalue of a
$k$-sparse fermionic Hamiltonian with strictly $q$-local terms, as well as a
$1/(4k+1)$-approximation when the Hamiltonian has both $2$-local and $4$-local
terms. More generally we obtain a $1/O(qk^2)$-approximation for $k$-sparse
fermionic Hamiltonians with terms of locality at most $q$. Our techniques also
yield analogous approximations for $k$-sparse, $q$-local qubit Hamiltonians
with small hidden constants and improved dependence on $q$.
