  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.

Quantum Physics