Approximating Ground and Excited State Energies on a Quantum Computer

Approximating ground and a fixed number of excited state energies, or equivalently low order Hamiltonian eigenvalues, is an important but computationally hard problem. Typically, the cost of classical deterministic algorithms grows exponentially with the number of degrees of freedom. Under general conditions, and using a perturbation approach, we provide a quantum algorithm that produces estimates of a constant number $j$ of different low order eigenvalues. The algorithm relies on a set of trial eigenvectors, whose construction depends on the particular Hamiltonian properties. We illustrate our results by considering a special case of the time-independent Schr\"odinger equation with $d$ degrees of freedom. Our algorithm computes estimates of a constant number $j$ of different low order eigenvalues with error $O(\epsilon)$ and success probability at least $\frac34$, with cost polynomial in $\frac{1}{\epsilon}$ and $d$. This extends our earlier results on algorithms for estimating the ground state energy. The technique we present is sufficiently general to apply to problems beyond the application studied in this paper.