Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they often manifest as constrained optimization problems. Here, we explore the interconnection between constrained optimization and generalized eigenvalue problems through the Unitary Coupled Cluster (UCC) excitation generators. Inspired by the generator coordinate method, we employ these UCC excitation generators to construct non-orthogonal, overcomplete many-body bases, projecting the system Hamiltonian into an effective Hamiltonian, which bypasses issues such as barren plateaus that heuristic numerical minimizers often encountered in standard variational quantum eigensolver (VQE). Diverging from conventional quantum subspace expansion methods, we introduce an adaptive scheme that robustly constructs the 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 cost-effectively, setting the stage for more advanced quantum simulations in chemistry.

混合量子经典方法为量子化学问题提供了潜在的解决方案，但它们通常表现为约束优化问题。在本文中，我们通过酉耦合簇 （UCC） 激励生成器探讨了约束优化和广义特征值问题之间的相互联系。受生成器坐标方法的启发，我们使用这些 UCC 激发生成器来构建非正交、超完全的多体基，将系统哈密顿量投影到有效的哈密顿量中，从而绕过启发式数值最小化器在标准变分量子特征求解器 （VQE） 中经常遇到的贫瘠高原等问题。与传统的量子子空间扩展方法不同，我们引入了一种自适应方案，该方案从 UCC 激发发生器池中稳健地构建多体基集。该方案支持分层 ADAPT 量子经典策略的开发，实现子空间扩展和拟设优化之间的平衡相互作用，以经济高效地解决复杂、强相关的量子化学系统，为更先进的化学量子模拟奠定基础。