Solving combinatorial optimization problems of the kind that can be codified by quadratic unconstrained binary optimization (QUBO) is a promising application of quantum computation. Some problems of this class suitable for practical applications such as the traveling salesman problem (TSP), the bin packing problem (BPP), or the knapsack problem (KP) have inequality constraints that require a particular cost function encoding. The common approach is the use of slack variables to represent the inequality constraints in the cost function. However, the use of slack variables considerably increases the number of qubits and operations required to solve these problems using quantum devices. In this work, we present an alternative method that does not require extra slack variables and consists of using an unbalanced penalization function to represent the inequality constraints in the QUBO. This function is characterized by larger penalization when the inequality constraint is not achieved than when it is. We evaluate our approach on the TSP, BPP, and KP, successfully encoding the optimal solution of the original optimization problem near the ground state cost Hamiltonian. Additionally, we employ D-Wave Advantage and D-Wave hybrid solvers to solve the BPP, surpassing the performance of the slack variables approach by achieving solutions for up to 29 items, whereas the slack variables approach only handles up to 11 items. This new approach can be used to solve combinatorial problems with inequality constraints with a reduced number of resources compared to the slack variables approach using quantum annealing or variational quantum algorithms.

翻译：解决可通过二次无约束二进制优化（QUBO）编码的组合优化问题是量子计算的一个前景广阔的应用领域。此类问题中一些适合实际应用的案例，如旅行商问题（TSP）、装箱问题（BPP）或背包问题（KP），包含需要特定成本函数编码的不等式约束。常见方法是使用松弛变量在成本函数中表示不等式约束。然而，使用松弛变量会显著增加利用量子设备解决这些问题所需的量子比特数量和操作量。在本工作中，我们提出了一种无需额外松弛变量的替代方法，该方法通过使用不平衡惩罚函数在QUBO中表示不等式约束。该函数的特点是，当不等式约束未满足时施加的惩罚大于约束满足时。我们在TSP、BPP和KP上评估了我们的方法，成功地将原始优化问题的最优解编码在基态成本哈密顿量附近。此外，我们利用D-Wave Advantage和D-Wave混合求解器解决了BPP，其性能超越了松弛变量方法，可处理多达29个物品的问题，而松弛变量方法最多只能处理11个物品。与使用松弛变量的方法相比，这种新方法能够以更少的资源用量子退火或变分量子算法解决具有不等式约束的组合问题。