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

TL;DR

The paper addresses the challenge of encoding inequality constraints in combinatorial optimization for quantum hardware by evaluating unbalanced penalization as an alternative to slack variables. It applies the method to the DFJ traveling salesman problem, mapping to QUBO and Ising formulations and testing on D-Wave Advantage, the D-Wave hybrid solver, and classical solvers. Results show that unbalanced penalization outperforms slack variables on quantum hardware and remains competitive on classical solvers, enabling larger instances (up to 45 cities) and reducing qubit-connectivity demands. This work strengthens the case for practical quantum optimization by providing a robust inequality-constraint encoding that improves scalability and accelerates benchmarking of quantum devices.

Abstract

Combinatorial optimization problems are one of the target applications of current quantum technology, mainly because of their industrial relevance, the difficulty of solving large instances of them classically, and their equivalence to Ising Hamiltonians using the quadratic unconstrained binary optimization (QUBO) formulation. Many of these applications have inequality constraints, usually encoded as penalization terms in the QUBO formulation using additional variables known as slack variables. The slack variables have two disadvantages: (i) these variables extend the search space of optimal and suboptimal solutions, and (ii) the variables add extra qubits and connections to the quantum algorithm. Recently, a new method known as unbalanced penalization has been presented to avoid using slack variables. This method offers a trade-off between additional slack variables to ensure that the optimal solution is given by the ground state of the Ising Hamiltonian, and using an unbalanced heuristic function to penalize the region where the inequality constraint is violated with the only certainty that the optimal solution will be in the vicinity of the ground state. This work tests the unbalanced penalization method using real quantum hardware on D-Wave Advantage for the traveling salesman problem (TSP). The results show that the unbalanced penalization method outperforms the solutions found using slack variables and sets a new record for the largest TSP solved with quantum technology.

Figures (7)

• Figure 1: Unbalanced penalization function $\xi(\mathrm{x})$ in Eq. (\ref{['eq:xi']}) as a function of $h(x) = W - \sum_i w_i x_i$. The region $h(x)\ge0$ where the inequality constraint Eq. (\ref{['Eq:ineq']}) is satisfied (violated) is shown in blue (orange). Markers represent special points at $h(x)\in\{-W,-W/2,0,W/2,W\}$.
• Figure 2: TSP problem with 11 cities. (a) the degree LP relaxation of the problem. (b) the optimal solution found after adding the sub-tour $Q = \{2, 5, 7\}$.
• Figure 3: Probability of finding a valid solution on the QPU using the different methods, the unbalanced penalization method (yellow stars), the degree LP relaxation (blue circles), and the slack variables approach (red diamonds). Lines are guides to the eye.
• Figure 4: Average tour distance of the valid solutions found using the different methods. The error bars indicate one standard deviation of the distances. The markers are the same as in Fig. \ref{['probability-qpu']}. For reference, the black points are the results of the CPLEX solver. Lines are guides to the eye.
• Figure 5: Minimum distance of the valid solutions found using the different methods. The markers are the same as in Fig. \ref{['mean-qpu']}.