Lower Bounds on Circuit Depth of the Quantum Approximate Optimization Algorithm

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 NISQ devices increases 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 satisifiability problems are suitable for QAOA approaches while Knapsack and Traveling Sales Person problems are not.