  Quantum computation by adiabatic evolution, as described in quant-ph/0001106,
will solve satisfiability problems if the running time is long enough. In
certain special cases (that are classically easy) we know that the quantum
algorithm requires a running time that grows as a polynomial in the number of
bits. In this paper we present numerical results on randomly generated
instances of an NP-complete problem and of a problem that can be solved
classically in polynomial time. We simulate a quantum computer (of up to 16
qubits) by integrating the Schrodinger equation on a conventional computer. For
both problems considered, for the set of instances studied, the required
running time appears to grow slowly as a function of the number of bits.
