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Hybrid quantum-classical approaches offer potential solutions for quantumchemistry problems, but they also introduce challenges such as the barrenplateau and the exactness of the ansatze. These challenges often manifest asconstrained optimization problems without a guarantee of identifying globalminima. In this work, we highlight the interconnection between constrainedoptimization and generalized eigenvalue problems, using a unique class ofnon-orthogonal and overcomplete basis sets generated by Givens rotation-typecanonical transformations on a reference state. Employing the generatorcoordinate approach, we represent the wave function in terms of these basissets. The ensuing generalized eigenvalue problem yields rigorous lower boundson energy, outperforming the conventional variational quantum eigensolver (VQE)that employs the same canonical transformations in its ansatze. Our approacheffectively tackles the barren plateau issue and the heuristic nature ofnumerical minimizers in the standard VQE, making it ideal for intricate quantumchemical challenges. For real-world applications, we propose an adaptive schemefor selecting these transformations, emphasizing the linear expansion of thenon-orthogonal basis sets. This ensures a harmonious balance between accuracyand efficiency in hybrid quantum-classical simulations. Our analysis andsuggested methodology further broaden the applications of quantum computing inquantum chemistry. Notably, they pave the way for alternative strategies inexcited state computation and Hamiltonian downfolding, laying the groundworkfor sophisticated quantum simulations in chemistry.

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