Multiobjective Optimization in a Quantum Adiabatic Computer

In this work we propose what we consider the first quantum algorithm for multiobjective combinatorial optimization, at least to the best of our knowledge. The proposed algorithm is based on the adiabatic algorithm of Farhi et al. and it is constructed by mapping a multiobjective combinatorial optimization problem into a Hamiltonian using a convex combination among objectives. We present mathematical properties of the eigenspectrum of the associated Hamiltonian and prove that the quantum adiabatic algorithm can find Pareto-optimal solutions provided certain convex combinations of objectives are used and the underlying multiobjective problem meets certain restrictions.