From Heisenberg to Hubbard: An initial state for the shallow quantum simulation of correlated electrons

The widespread use of the noninteracting ground state as the initial state for the digital quantum simulation of the Fermi-Hubbard model is largely due to the scarcity of alternative easy-to-prepare approximations to the exact ground state in the literature. Exploiting the fact that the spin-$\frac{1}{2}$ Heisenberg model is the effective low-energy theory of the Fermi-Hubbard model at half-filling in the strongly interacting limit, here we propose a three-step deterministic quantum routine to prepare an educated guess of the ground state of the Fermi-Hubbard model through a shallow circuit suitable for near-term quantum hardware. First, the ground state of the Heisenberg model is initialized via a hybrid variational method using an ansatz that explores only the correct symmetry subspace. Second, a general method is devised to convert a multi-spin-$\frac{1}{2}$ wave function into its fermionic version. Third, taking inspiration from the Baeriswyl ansatz, a constant-depth single-parameter layer that adds doublon-holon pairs is applied to this fermionic state. Numerical simulations on chains and ladders with up to 12 sites confirm the improvement over the noninteracting ground state of the overlap with the exact ground state for the intermediate values of the interaction strength at which quantum simulation is found to be most relevant. More broadly, the general scheme to convert a multi-spin-$\frac{1}{2}$ state into a half-filled fermionic state may bridge the gap between quantum spin models and lattice models of correlated fermions in the realm of digital quantum simulation.