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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 $4$-localterms. More generally we obtain a $1/O(qk^2)$-approximation for $k$-sparsefermionic Hamiltonians with terms of locality at most $q$. Our techniques alsoyield analogous approximations for $k$-sparse, $q$-local qubit Hamiltonianswith small hidden constants and improved dependence on $q$.

Improved Approximations for Extremal Eigenvalues of Sparse Hamiltonians