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.