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Combinatorial optimization problems are one of the target applications ofcurrent quantum technology, mainly because of their industrial relevance, thedifficulty of solving large instances of them classically, and theirequivalence to Ising Hamiltonians using the quadratic unconstrained binaryoptimization (QUBO) formulation. Many of these applications have inequalityconstraints, usually encoded as penalization terms in the QUBO formulationusing additional variables known as slack variables. The slack variables havetwo disadvantages: (i) these variables extend the search space of optimal andsuboptimal solutions, and (ii) the variables add extra qubits and connectionsto the quantum algorithm. Recently, a new method known as unbalancedpenalization has been presented to avoid using slack variables. This methodoffers a trade-off between additional slack variables to ensure that theoptimal solution is given by the ground state of the Ising Hamiltonian, andusing an unbalanced heuristic function to penalize the region where theinequality constraint is violated with the only certainty that the optimalsolution will be in the vicinity of the ground state. This work tests theunbalanced penalization method using real quantum hardware on D-Wave Advantagefor the traveling salesman problem (TSP). The results show that the unbalancedpenalization method outperforms the solutions found using slack variables andsets a new record for the largest TSP solved with quantum technology.

Improving Performance in Combinatorial Optimization Problems with  Inequality Constraints: An Evaluation of the Unbalanced Penalization Method  on D-Wave Advantage