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.

翻译：组合优化问题是当前量子技术的重要应用目标之一，这主要源于其工业相关性、经典计算求解大规模实例的困难性，以及通过二次无约束二元优化(QUBO)形式与伊辛哈密顿量的等价性。此类问题中常涉及不等式约束，通常采用松弛变量作为额外变量，以惩罚项形式编码至QUBO框架中。松弛变量存在两个缺陷：（i）该类变量会扩展最优解与次优解的搜索空间；（ii）变量为量子算法引入额外量子比特与连接。近期提出的非平衡惩罚方法可避免使用松弛变量。该方法在两种策略间取得权衡：一是通过引入额外松弛变量确保最优解对应伊辛哈密顿量基态，二是采用非平衡启发函数惩罚违反不等式约束的区域，其确定性仅在于最优解将位于基态邻近区域。本研究利用D-Wave Advantage真实量子硬件，以旅行商问题(TSP)为测试案例验证非平衡惩罚方法。结果表明，该方法的求解性能优于采用松弛变量的方案，并刷新了量子技术求解最大规模TSP的纪录。