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 tested our approach for the TSP, the BPP, and the KP. For all of them, we are able to encode the optimal solution of the original optimization problem in the vicinity of the ground state cost Hamiltonian. 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:解决那种可以由二次不受限制的二进制优化(QUBO)来编纂的组合优化问题,是一种有希望的量子计算应用。本类中适合实际应用的一些问题,如旅行推销员问题(TSP)、垃圾包装问题(BPP)或 knapsack 问题(KP)有不平等的制约,需要特定的成本函数编码。共同的方法是使用松软变量来代表成本功能中的不平等制约。然而,使用松软变量大大增加了使用量子装置解决这些问题所需的量子和操作的数量。在这项工作中,我们提出了一种不要求额外松软变量的替代方法,包括使用不平衡的处罚功能来代表QUBO的不平等制约。这一功能的特点是,在不平等制约没有达到时,惩罚力度就会更大。我们测试了我们针对TSP、BPP和KP的方法,以体现成本功能上的不平等制约。对于所有这些,我们都能将最初优化问题的最佳解决方案用量子体来编码。在地面州Crimeritonian附近地区解决这些问题。我们提出的一种新方法可以用来用软体化的量态变量来解决不平等性变制的量级资源,以降低量级变数。