Adiabatic quantum computing solution of the knapsack problem

We illustrate the adiabatic quantum computing solution of the knapsack problem with both integer profits and weights. For problems with $n$ objects (or items) and integer capacity $c$, we give specific examples using both an Ising class problem Hamiltonian requiring $n+c$ qubits and a much more efficient one using $n+[\log_2 c]+1$ qubits. The discussion includes a brief mention of classical algorithms for knapsack, applications of this commonly occurring problem, and the relevance of further studies both theoretically and numerically of the behavior of the energy gap. Included too is a demonstration and commentary on a version of quantum search using a certain Ising model. Furthermore, an Appendix presents analytic results concerning the boundary for the easy-versus-hard problem-instance phase transition for the special case subset sum problem.