Unleashed from Constrained Optimization: Quantum Computing for Quantum Chemistry Employing Generator Coordinate Method

Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they also introduce challenges. These challenges include addressing the barren plateau and ensuring the accuracy of the ans\"{a}tze, which often manifest as constrained optimization problems. In this work, we explore the interconnection between constrained optimization and generalized eigenvalue problems through \textcolor{black}{the Unitary Coupled Cluster (UCC) excitation generators. These generators often serve as building blocks constituting the ans\"{a}tze in variational quantum eigensolver (VQE) and adaptive derivative-assembled pseudo-Trotter VQE (ADAPT-VQE) simulations. Here, inspired by the generator coordinate method, we employ these UCC excitation generators to construct non-orthogonal, overcomplete many-body generating functions, projecting the system Hamiltonian into a practical working subspace. This approach results in a generalized eigenvalue problem that provides rigorous lower bounds to VQE/ADAPT-VQE energies, effectively bypassing issues related to barren plateaus and heuristic numerical minimizers typical in standard VQE methods. Diverging from conventional quantum subspace expansion methods, we introduce an adaptive scheme that robustly constructs many-body basis sets from a pool of the UCC excitation generators. This scheme supports the development of a hierarchical ADAPT quantum-classical strategy, enabling a balanced interplay between subspace expansion and ansatz optimization to address complex, strongly correlated quantum chemical systems efficiently and cost-effectively. The effective Hamiltonian generated by our approach also supports the computation of excited states and dynamic properties, setting the stage for more advanced quantum simulations in chemistry.