A Rigorous Quantum Framework for Inequality-Constrained and Multi-Objective Binary Optimization: Quadratic Cost Functions and Empirical Evaluations

Abstract: The prospect of quantum solutions for complicated optimization problems is contingent on mapping the original problem onto a tractable quantum energy landscape, e.g. an Ising-type Hamiltonian. Subsequently, techniques like adiabatic optimization, quantum annealing, and the Quantum Approximate Optimization Algorithm (QAOA) can be used to find the ground state of this Hamiltonian. Quadratic Unconstrained Binary Optimization (QUBO) is one prominent problem class for which this entire pipeline is well understood and has received considerable attention over the past years. In this work, we provide novel, tractable mappings for the maxima of multiple QUBO problems. Termed Multi-Objective Quantum Approximations, or MOQA for short, our framework allows us to recast new types of classical binary optimization problems as ground state problems of a tractable Ising-type Hamiltonian. This, in turn, opens the possibility of new quantum- and quantum-inspired solutions to a variety of problems that frequently occur in practical applications. In particular, MOQA can handle various types of routing and partitioning problems, as well as inequality-constrained binary optimization problems.