Challenges in Multiobjective Optimization using a Quantum Adiabatic Computer

Bin Cao, Shanshan Fan, Jianwei Zhao +3 more

Daniel Ruffinelli, Benjamı́n Barán

Benjamı́n Barán, Marcos Villagra

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.

V. Akshay, Mile Gu

Benjamı́n Barán, Marcos Villagra

In this work we show how to use a quantum adiabatic algorithm to solve multiobjective optimization problems. For the first time, we demonstrate a theorem proving that the quantum adiabatic algorithm can find Pareto-optimal solutions in finite-time, provided some restrictions to the problem are met. A numerical example illustrates an application of the theorem to a well-known problem in multiobjective optimization. This result opens the door to solve multiobjective optimization problems using current technology based on quantum annealing.

Xiaodong Yang, Ran Liu, Jun Li +1 more

Designing proper time-dependent control fields for slowly varying the system to the ground state that encodes the problem solution is crucial for adiabatic quantum computation. However, inevitable perturbations in real applications demand us to accelerate the evolution so that the adiabatic errors can be prevented from accumulation. Here, by treating this trade-off task as a multiobjective optimization problem, we propose a gradient-free learning algorithm with pulse smoothing technique to search optimal adiabatic quantum pathways and apply it to the Landau-Zener Hamiltonian and Grover search Hamiltonian. Numerical comparisons with a linear schedule, local adiabatic theorem induced schedule, and gradient-based algorithm searched schedule reveal that the proposed method can achieve significant performance improvements in terms of the adiabatic time and the instantaneous ground-state population maintenance. The proposed method can be used to solve more complex and real adiabatic quantum computation problems.