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

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.