Connecting Stabilizer Rényi Entropy to Problem Hardness in the Quantum Approximate Optimization Algorithm

The quantum approximate optimization algorithm (QAOA) is a promising near-term approach to solving optimization problems on quantum hardware. However, its applicability to large-scale problems depends on the scaling of algorithmic performance and quantum resource requirements. Here, we explore the relationship between classical problem hardness and the stabilizer Rényi entropy—also known as magic—within QAOA. Using numerical simulations of the Maximum Satisfiability (MAX SAT) problem as a test case, we leverage clause density (the ratio of clauses to variables) to systematically tune problem hardness. Our results reveal that the dynamics of magic in QAOA depend strongly on clause density, demonstrating a clear link between magic growth and problem characteristics. We further identify a volume-law scaling "magic barrier" separating the initial and solution states, with a slope that increases with clause density. Analysis of the magic spectrum shows that its distribution expands and contracts depending on the problem's clause density. Together, these findings establish a direct connection between classical hardness and the quantum resources required by QAOA.