The quantum approximate optimization algorithm (QAOA) is a method of approximately solving combinatorial optimization problems. While QAOA is developed to solve a broad class of combinatorial optimization problems, it is not clear which classes of problems are best suited for it. One factor in demonstrating quantum advantage is the relationship between a problem instance and the circuit depth required to implement the QAOA method. As errors in noisy intermediate-scale quantum (NISQ) devices increase exponentially with circuit depth, identifying lower bounds on circuit depth can provide insights into when quantum advantage could be feasible. Here, we identify how the structure of problem instances can be used to identify lower bounds for circuit depth for each iteration of QAOA and examine the relationship between problem structure and the circuit depth for a variety of combinatorial optimization problems including MaxCut and MaxIndSet. Specifically, we show how to derive a graph, G, that describes a general combinatorial optimization problem and show that the depth of circuit is at least the chromatic index of G. By looking at the scaling of circuit depth, we argue that MaxCut, MaxIndSet, and some instances of vertex covering and Boolean satisfiability problems are suitable for QAOA approaches while knapsack and traveling salesperson problems are not.

量子近似优化算法（QAOA）是一种近似解决组合优化问题的方法。虽然开发QAOA是为了解决广泛的组合优化问题，但尚不清楚哪种类型的问题最适合它。证明量子优势的一个因素是问题实例与实施QAOA方法所需的电路深度之间的关系。随着嘈杂的中级量子（NISQ）器件中的误差随着电路深度的增加而呈指数增加，识别电路深度的下限可以提供何时量子优势可行的见解。这里，我们确定问题实例的结构如何用于为QAOA的每次迭代确定电路深度的下限，并针对各种组合优化问题（包括MaxCut和MaxIndSet）检查问题结构与电路深度之间的关系。具体来说，我们展示了如何得出图，G，它描述了一般的组合优化问题，并表明电路的深度至少为G的色指数。通过查看电路深度的缩放比例，我们认为MaxCut，MaxIndSet以及一些顶点覆盖和布尔可满足性问题的实例适用于QAOA方法，而背包和旅行销售人员问题则不适合。