Improved classical optimization of fermionic Hamiltonians beyond Gaussian states

Abstract: We give a polynomial-time randomized classical algorithm that approximates the largest eigenvalue $λ<em>{\max}$ of a traceless quartic fermionic Hamiltonian $H$ on $n$ modes. The algorithm achieves energy at least $λ</em>{\max}/O(\sqrt n)$ and outputs either a fermionic Gaussian state or a polynomially sparse state in the Fock basis. This is achieved through a new upper bound on $λ_{\max}$ in terms of the maximum energy of a Gaussian state and $\sqrt{2<sup>{-n}\operatorname{Tr}(H<sup>2)}$.