Unitary Route to Correlated Many-Fermion States

Recent advances in variational approaches to correlated fermionic systems—such as neural quantum states and variational quantum eigensolvers (VQE)—have reinvigorated the quest to efficiently represent interacting many-body ground states. We explore a unifying perspective that views an interacting ground state as the result of a unitary transformation acting on a non-interacting reference state. This "unitary route" naturally organizes complexity by the number of fermionic operators involved in the unitary—one-body, two-body, and beyond. Using the Hubbard model as a testbed, we construct the exact unitary transformation that connects non-interacting and interacting eigenstates obtained via exact diagonalization. We then examine how faithfully this transformation can be approximated by one-body or two-body unitaries. Our results show that, in the absence of level crossings, a purely two-body unitary can reproduce the interacting ground state with high fidelity, while a one-body unitary cannot capture the essential correlations. We discuss the implications of our findings for the unitary coupled-cluster framework, neural quantum state architectures, and VQE implementations on fermionic quantum processors, where two-body unitaries correspond directly to native two-fermion gates.