Quantum Inverse Algorithm via Adaptive Variational Quantum Linear Solver: Applications to General Eigenstates

Abstract

We propose a quantum inverse algorithm (QInverse) to directly determine general eigenstates by repeatedly applying the inverse power of a shifted Hamiltonian to an arbitrary initial state. To properly deal with the strongly entangled inverse power states and the resultant excited states, we solved the underlying linear equation, both variationally and adaptively, to obtain a faithful inverse power state with a shallow quantum circuit. QInverse is singularity-free and successfully obtains the target excited states with an energy closest to the shift ω, which is difficult to reach using variational methods. We also propose a subspace expansion approach to accelerate convergence and show that it is helpful to determine the two nearest eigenvalues when they are equally close to ω. These approaches were compared with the folded-spectrum method, which aims to generate excited states through variational optimization. It is shown that, whereas the folded-spectrum approach often fails to predict the target state by falling into a local minimum owing to its variational features, the success rate and accuracy of our algorithms are systematically improvable.

Many-Body-Expansion Based on Variational Quantum Eigensolver and Deflation for Dynamical Correlation

“…In the framework of variational quantum algorithms, there has been considerable effort in obtaining excited states. ,,− Among them, the most prominent approach is perhaps variational quantum deflation (VQD), which introduces the following modification to the Hamiltonian to determine the κth excited state Ĥ false( κ false) = Ĥ + ∑ λ = 0 κ − 1 α | Ψ λ false⟩ false⟨ Ψ λ | where α = prefix− ⟨ Ψ λ | Ĥ | Ψ λ ⟩ is typically chosen. This modified Hamiltonian ensures approximate orthogonality between |Ψ λ ⟩ (λ = 0, 1, 2, ..., κ), which are the ground state, and the first, second, ..., and κth excited states, obtained by the corresponding VQE calculations.…”

Physical Chemistry of Quantum Information Science

“…We investigate the topic of quantum algorithms tailored for quantum chemistry, molecular dynamics, and statistical mechanics. This includes a quest to enhance the accuracy of classical computations for difficult chemistry problems involving strongly correlated systems in the works by A. Tammaro et al., A. Khamoshi et al, and N. T. Le and L. N. Tran., and calculations of excited states in articles by Y. Kim and A. I. Krylov and by T. Yoshikura et al Outstanding problems of quantum state preparation were discussed by I. Magoulas and F. A. Evangelista, S. G. Mehendale et al, S. E. Ghasempouri et al, J. H. Zhang et al., and L. M. Sager-Smith et al, for near-term quantum algorithms. An equally important topic of quantum measurement is touched upon by Z. P. Bansingh et al and T. Kurita et al In addition, Hamiltonian learning from quantum dynamics is presented by R. Gupta et al, and an interesting and accessible approach to visualization of quantum algorithms is described by I. Ganti and S. S. Iyengar …”