A Unified Complexity-Algorithm Account of Constant-Round QAOA Expectation Computation
Jingheng Wang, Shengminjie Chen, Xiaoming Sun, Jialin Zhang
TL;DR
The paper addresses the fundamental question of how hard it is to compute the fixed-round QAOA expectation for Max-Cut and, more generally, BUCO. It proves NP-hardness for exact evaluation at any fixed p≥2 and shows additive-error hardness, framing a divide between intractability and tractable evaluation via structural parameters. It then introduces a bag-level dynamic-programming evaluator that runs in time exp(O(p · ltw_p(G))) times poly(n), enabling exact computation on graph families with local treewidth growing only logarithmically with n, and extends the framework to BUCO. The authors provide reproducible experiments on structured graphs (GP(15,2), double-layer triangular lattices, and C60), comparing QAOA against locality-matched classical baselines and observing instance-dependent advantages for QAOA at modest depths. Overall, the work supplies a unifying theory linking complexity and algorithmic structure to the practical evaluation of quantum-inspired optimization, with implications for benchmarking and problem-design in near-term quantum devices.
Abstract
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 $p\ge2$, exactly evaluating the expectation of fixed-round QAOA at prescribed angles is $\mathrm{NP}$-hard, and that approximating this expectation within additive error $2^{-O(n)}$ in the number $n$ of vertices is already $\mathrm{NP}$-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 \emph{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 $C_{60}$, and report cut ratios while benchmarking against locality-matched classical baselines.