  Quantum optimization algorithms hold the promise of solving classically hard,
discrete optimization problems in practice. The requirement of encoding such
problems in a Hamiltonian realized with a finite -- and currently small --
number of qubits, however, poses the risk of finding only the optimum within
the restricted space supported by this Hamiltonian. We describe an iterative
algorithm in which a solution obtained with such a restricted problem
Hamiltonian is used to define a new problem Hamiltonian that is better suited
than the previous one. In numerical examples of the shortest vector problem, we
show that the algorithm with a sequence of improved problem Hamiltonians
converges to the desired solution.
