Engineering two-body interaction for the Moore-Read State

Engineering interactions that stabilize non-Abelian fractional quantum Hall phases is a central challenge in strongly correlated topological matter and quantum simulation. We introduce a differentiable framework for inverse Hamiltonian design, in which Haldane pseudopotentials are optimized by gradient-based exact diagonalization to stabilize target fractional quantum Hall phases. In spherical geometry, the Haldane pseudopotentials are treated as variational parameters and optimized in a JAX-based exact-diagonalization framework. By directly maximizing the overlap between the many-body ground state and the Moore-Read state, we obtain a robust pseudopotential profile that has Pfaffian overlaps exceeding $99\%$ for systems up to $N_e=12$, substantially improving over conventional Coulomb interactions. Analyses of the neutral excitation spectrum and orbital entanglement spectrum further confirm that the optimized interaction stabilizes the Pfaffian topological phase. Our results demonstrate that essential features of the three-body Pfaffian parent Hamiltonian can be effectively encoded in a suitably designed two-body interaction. Furthermore, they identify a nearly universal exponentially decaying pseudopotential profile that stabilizes the Pfaffian phase and establishes a general framework toward engineering non-Abelian topological order in quantum simulation.