Improved classical optimization of fermionic Hamiltonians beyond Gaussian states

We give a polynomial-time randomized classical algorithm that approximates the largest eigenvalue ${\lambda }_{\max }$ of a traceless quartic fermionic Hamiltonian $H$ on $n$ modes. The algorithm achieves energy at least ${\lambda }_{\max }\mathrm{/}O(\sqrt{n})$ 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 ${\lambda }_{\max }$ in terms of the maximum energy of a Gaussian state and $\sqrt{2^{-n}\mathrm{Tr}(H^2)}$.