NQOF: Neural Operator Flow for Quantum Optimization

Ground energy estimation is central to quantum chemistry and many-body physics, yet its computational cost grows rapidly with system size. State-centered methods depend strongly on the wavefunction ansatz, while operator-flow methods identify energy-lowering directions within an exponentially large operator space. To address this challenge, we reformulate ground energy optimization as learning a Brockett-Wegner flow on the Hamiltonian’s operator manifold. Building on this formulation, we introduce Neural Quantum Operator Flow (NQOF), a learn-and-propagate framework that uses an autoregressive neural network to discover a compact set of Pauli operators without enumerating the full operator basis. A quadratic projection then determines their coefficients to approximate the energy-lowering direction of the Brockett-Wegner flow. The resulting unitary updates preserve the Hamiltonian spectrum while progressively driving the variational energy toward the ground energy. We conduct systematic experiments on molecular and quantum many-body systems with up to 30 qubits. The results show that NQOF achieves competitive accuracy while requiring fewer learnable parameters, and remains robust across different systems. Our approach establishes operator-flow learning as a new paradigm for quantum optimization and opens a promising avenue toward solving complex quantum problems.