Theoretical Guarantees of Variational Quantum Algorithm with Guiding States

Abstract

Variational quantum algorithms (VQAs) are prominent candidates for near-term quantum advantage but lack rigorous guarantees of convergence and generalization. By contrast, quantum phase estimation (QPE) provides provable performance under the guiding state assumption, where access to a state with non-trivial overlap with the ground state enables efficient energy estimation. In this work, we ask whether similar guarantees can be obtained for VQAs. We introduce a variational quantum algorithm with guiding states aiming towards predicting ground-state properties of quantum many-body systems. We then develop a proof technique-the linearization trick-that maps the training dynamics of the algorithm to those of a kernel model. This connection yields the first theoretical guarantees on both convergence and generalization for the VQA under the guiding state assumption. Our analysis shows that guiding states accelerate convergence, suppress finite-size error terms, and ensure stability across system dimensions. Finally, we validate our findings with numerical experiments on 2D random Heisenberg models.