A Unified Complexity-Algorithm Account of Constant-Round QAOA Expectation Computation

The Quantum Approximate Optimization Algorithm (QAOA) is widely studied for combinatorial optimization and has achieved significant advances both in theoretical guarantees and practical performance, yet for general combinatorial optimization problems the expected performance and classical simulability of fixed-round QAOA remain unclear. Focusing on Max-Cut, we first show that for general graphs and any fixed round pge2, exactly evaluating the expectation of fixed-round QAOA at prescribed angles is NP-hard, and that approximating this expectation within additive error 2^-O(n) in the number n of vertices is already NP-hard. To evaluate the expected performance of QAOA, we propose a dynamic programming algorithm leveraging tree decomposition. As a byproduct, when the p-local treewidth grows at most logarithmically with the number of vertices, this yields a polynomial-time exact evaluation algorithm in the graph size n. Beyond Max-Cut, we extend the framework to general Binary Unconstrained Combinatorial Optimization (BUCO). Finally, we provide reproducible evaluations for rounds up to p=3 on representative structured families, including the generalized Petersen graph GP(15,2), double-layer triangular 2-lifts, and the truncated icosahedron graph C₆₀, and report cut ratios while benchmarking against locality-matched classical baselines.