A sharp interaction-degree threshold for simulating QAOA

A critical threshold has been discovered for simulating the Quantum Approximate Optimization Algorithm (QAOA) with 2-local cost functions, where the interaction degree sharply affects the feasibility of classical simulation. At an interaction degree of 3, simulating QAOA with minimal error would have significant implications for computational complexity, potentially collapsing the polynomial hierarchy to its third level¹. In contrast, at an interaction degree of 2, classical simulation of QAOA can be achieved efficiently, with a runtime of n^O(1) for logarithmic depth and n qubits. This distinction highlights the importance of interaction degree in determining the computational resources required for simulating QAOA. The discovery of this threshold has significant implications for the development of quantum algorithms and the study of computational complexity, as it underscores the limits of classical simulation and the potential advantages of quantum computing.