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Solving combinatorial optimization problems of the kind that can be codifiedby quadratic unconstrained binary optimization (QUBO) is a promisingapplication of quantum computation. Some problems of this class suitable forpractical applications such as the traveling salesman problem (TSP), the binpacking problem (BPP), or the knapsack problem (KP) have inequality constraintsthat require a particular cost function encoding. The common approach is theuse of slack variables to represent the inequality constraints in the costfunction. However, the use of slack variables considerably increases the numberof qubits and operations required to solve these problems using quantumdevices. In this work, we present an alternative method that does not requireextra slack variables and consists of using an unbalanced penalization functionto represent the inequality constraints in the QUBO. This function ischaracterized by larger penalization when the inequality constraint is notachieved than when it is. We evaluate our approach on the TSP, BPP, and KP,successfully encoding the optimal solution of the original optimization problemnear the ground state cost Hamiltonian. Additionally, we employ D-WaveAdvantage and D-Wave hybrid solvers to solve the BPP, surpassing theperformance of the slack variables approach by achieving solutions for up to 29items, whereas the slack variables approach only handles up to 11 items. Thisnew approach can be used to solve combinatorial problems with inequalityconstraints with a reduced number of resources compared to the slack variablesapproach using quantum annealing or variational quantum algorithms.

Unbalanced penalization: A new approach to encode inequality constraints of combinatorial problems for quantum optimization algorithms