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

We propose a quantum inverse algorithm (QInverse) todirectly determinegeneral eigenstates by repeatedly applying the inverse power of ashifted Hamiltonian to an arbitrary initial state. To properly dealwith the strongly entangled inverse power states and the resultantexcited states, we solved the underlying linear equation, both variationallyand adaptively, to obtain a faithful inverse power state with a shallowquantum circuit. QInverse is singularity-free and successfully obtainsthe target excited states with an energy closest to the shift & omega;,which is difficult to reach using variational methods. We also proposea subspace expansion approach to accelerate convergence and show thatit is helpful to determine the two nearest eigenvalues when they areequally close to & omega;. These approaches were compared with thefolded-spectrum method, which aims to generate excited states throughvariational optimization. It is shown that, whereas the folded-spectrumapproach often fails to predict the target state by falling into alocal minimum owing to its variational features, the success rateand accuracy of our algorithms are systematically improvable.