Quantum Davidson algorithm for excited states

Abstract

Excited state properties play a pivotal role in various chemical and physical phenomena, such as charge separation and light emission. However, the primary focus of most existing quantum algorithms has been the ground state, as seen in quantum phase estimation and the variational quantum eigensolver (VQE). Although VQE-type methods have been extended to explore excited states, these methods grapple with optimization challenges. In contrast, the quantum Krylov subspace (QKS) method has been introduced to address both ground and excited states, positioning itself as a cost-effective alternative to quantum phase estimation. However, conventional QKS methodologies depend on a pre-generated subspace through either real or imaginary-time evolutions. This subspace is inherently expansive and can be plagued with issues like slow convergence or numerical instabilities, often leading to relatively deep circuits. In our research, we present an economic QKS algorithm, which we term the quantum Davidson (QDavidson) algorithm. This innovation hinges on the iterative expansion of the Krylov subspace and the incorporation of a pre-conditioner within the Davidson framework. By using the residues of eigenstates to expand the Krylov subspace, we manage to formulate a compact subspace that aligns closely with the exact solutions. This iterative subspace expansion paves the way for a more rapid convergence in comparison to other QKS techniques, such as the quantum Lanczos. We employ the novel QDavidson algorithm to delve into the excited state properties of various systems, spanning from the Heisenberg spin model to real molecules, utilizing quantum simulators. Compared to the existing QKS methods, the QDavidson algorithm not only converges swiftly but also demands a significantly shallower circuits. This efficiency establishes the QDavidson method as a pragmatic tool for elucidating both ground and excited state properties on quantum computing platforms.