  We study the complexity of approximating the smallest eigenvalue of a
univariate Sturm-Liouville problem on a quantum computer. This general problem
includes the special case of solving a one-dimensional Schroedinger equation
with a given potential for the ground state energy.
  The Sturm-Liouville problem depends on a function q, which, in the case of
the Schroedinger equation, can be identified with the potential function V.
Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an
exponential reduction in the number of queries over the number needed in the
classical worst-case and randomized settings for smooth functions q. Their
method uses the (discretized) unitary propagator and arbitrary powers of it as
a query ("power queries"). They showed that the Sturm-Liouville equation can be
solved with O(log(1/e)) power queries, while the number of queries in the
worst-case and randomized settings on a classical computer is polynomial in
1/e. This proves that a quantum computer with power queries achieves an
exponential reduction in the number of queries compared to a classical
computer.
  In this paper we show that the number of queries in Papageorgiou's and
Wozniakowski's algorithm is asymptotically optimal. In particular we prove a
matching lower bound of log(1/e) power queries, therefore showing that log(1/e)
power queries are sufficient and necessary. Our proof is based on a frequency
analysis technique, which examines the probability distribution of the final
state of a quantum algorithm and the dependence of its Fourier transform on the
input.
