Crossing The Gap Using Variational Quantum Eigensolver: A Comparative Study

Within the evolving domain of quantum computational chemistry, the Variational Quantum Eigensolver (VQE) has been developed to explore not only the ground state but also the excited states of molecules. In this study, we compare the performance of Variational Quantum Deflation (VQD) and Subspace-Search Variational Quantum Eigensolver (SSVQE) methods in determining the low-lying excited states of $LiH$. Our investigation reveals that while VQD exhibits a slight advantage in accuracy, SSVQE stands out for its efficiency, allowing the determination of all low-lying excited states through a single parameter optimization procedure. We further evaluate the effectiveness of Gradient Descent (GD), Quantum Natural Gradient (QNG), and Adam optimizers in obtaining $LiH$'s first excited state, with the Adam optimizer demonstrating superior efficiency in requiring the fewest iterations. Moreover, we propose a novel approach combining Folded Spectrum VQE (FS-VQE) with either VQD or SSVQE, enabling the exploration of highly excited states. We test the new approaches for finding all three of $H_4$'s excited states. Folded Spectrum SSVQE (FS-SSVQE) can find all three highly excited states near $-1.0$ Ha with only one optimization procedure, but the procedure converges slowly. In contrast, although Folded spectrum VQD (FS-VQD) finds highly excited states with multiple optimization procedures, the optimization converges faster.