An Optimal Framework for Constructing Lie-Algebra Generator Pools: Application to Variational Quantum Eigensolvers for Chemistry

Lie Algebras are powerful mathematical structures used in physics to describe sets of operators and associated combinations. A central task is to identify a minimal set of generators from which the algebra can be constructed. The classical search for such generators has so far relied on greedy construction steps applied to an exponentially growing number of candidate operators, making it rapidly computationally intractable. We propose a general, polynomial-scaling and optimal strategy, based on Lie-Algebraic basic properties, to overcome this bottleneck. It allows for the efficient construction of these generators, also known as Minimal Complete Pools (MCPs), for a target Lie Algebra. As an immediate application, efficiently constructing user-defined MCPs that respect fermionic algebra is crucial in the context of adaptive Variational Quantum Eigensolver for quantum chemistry. Thus, we introduce MB-ADAPT-VQE, which incorporates optimally constructed MCPs into batched ADAPT-VQE to reduce quantum resources and improve convergence under strong correlation. These MCPs also unlock fixed-ansatz methods based on a Lie-algebraic structure such as the gradient-free NI-DUCC-VQE, enabling simulations surpassing prior MCP limits. The presented mathematical framework is general and applicable well beyond chemistry in fields including quantum error correction, quantum control, quantum machine learning, and more universally wherever compact Pauli basis are required.