We present a quantum algorithm for solving algebraic Riccati equations, with applications to quantum-chemical random-phase approximation (RPA) and higher-order RPA theories. Our method block-encodes stabilizing Riccati solutions via Riesz projectors onto invariant subspaces of an associated non-normal matrix. Applied to m-particle, m-hole RPA under localized-orbital sparsity assumptions, our algorithm’s end-to-end cost scales linearly with system size and polynomially with excitation rank m, suggesting an exponential advantage in m over plausible classical heuristics. More broadly, this work provides a framework for solving nonlinear matrix equations with quantum computers and motivates further developments in quantum algorithms for nonlinear quantum-chemistry methods.