  Hybrid quantum-classical approaches offer potential solutions for quantum
chemistry problems, but they also introduce challenges such as the barren
plateau and the exactness of the ansatze. These challenges often manifest as
constrained optimization problems without a guarantee of identifying global
minima. In this work, we highlight the interconnection between constrained
optimization and generalized eigenvalue problems, using a unique class of
non-orthogonal and overcomplete basis sets generated by Givens rotation-type
canonical transformations on a reference state. Employing the generator
coordinate approach, we represent the wave function in terms of these basis
sets. The ensuing generalized eigenvalue problem yields rigorous lower bounds
on energy, outperforming the conventional variational quantum eigensolver (VQE)
that employs the same canonical transformations in its ansatze. Our approach
effectively tackles the barren plateau issue and the heuristic nature of
numerical minimizers in the standard VQE, making it ideal for intricate quantum
chemical challenges. For real-world applications, we propose an adaptive scheme
for selecting these transformations, emphasizing the linear expansion of the
non-orthogonal basis sets. This ensures a harmonious balance between accuracy
and efficiency in hybrid quantum-classical simulations. Our analysis and
suggested methodology further broaden the applications of quantum computing in
quantum chemistry. Notably, they pave the way for alternative strategies in
excited state computation and Hamiltonian downfolding, laying the groundwork
for sophisticated quantum simulations in chemistry.
