An efficient quantum algorithm for building effective Hamiltonians on fault-tolerant hardware*

The construction of accurate effective Hamiltonians for interacting quantum systems is a long-standing problem in many-body quantum mechanics. We present a novel quantum algorithm for constructing effective Hamiltonians built upon the methodology of density matrix downfolding (DMD). We provide formal theoretical developments in downfolding, including the novel concept of Hamiltonian compressibility, show that a quantum implementation of DMD addresses the main systematic errors present in the classical implementation --- sampling of low-energy states in the physical theory --- and provide circuits for implementing DMD on fault-tolerant quantum hardware. We also provide rigorous resource estimates at the logical and physical level for the case of a doped 2-D Fermi-Hubbard model and a large cuprate supercell.