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 $lambda_{max}/O(nlog n)$. 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.
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