  We discuss classical algorithms for approximating the largest eigenvalue of
quantum spin and fermionic Hamiltonians based on semidefinite programming
relaxation methods. First, we consider traceless $2$-local Hamiltonians $H$
describing a system of $n$ qubits. We give an efficient algorithm that outputs
a separable state whose energy is at least $\lambda_{\max}/O(\log n)$, where
$\lambda_{\max}$ is the maximum eigenvalue of $H$. We also give a simplified
proof of a theorem due to Lieb that establishes the existence of a separable
state with energy at least $\lambda_{\max}/9$. Secondly, we consider a system
of $n$ fermionic modes and traceless Hamiltonians composed of quadratic and
quartic fermionic operators. We give an efficient algorithm that outputs a
fermionic Gaussian state whose energy is at least $\lambda_{\max}/O(n\log n)$.
Finally, we show that Gaussian states can vastly outperform Slater determinant
states commonly used in the Hartree-Fock method. We give a simple family of
Hamiltonians for which Gaussian states and Slater determinants approximate
$\lambda_{\max}$ within a fraction $1-O(n^{-1})$ and $O(n^{-1})$ respectively.
