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

TL;DR

This paper investigates the computational complexity of evaluating the expectation of fixed-round QAOA for Max-Cut, showing NP-hardness results, proposing a dynamic programming evaluation method, and providing empirical benchmarks for small p.

Contribution

It establishes NP-hardness of expectation evaluation for fixed-round QAOA, introduces a dynamic programming approach leveraging tree decomposition, and extends the framework to general BUCO problems.

Findings

Expectation evaluation is NP-hard for fixed p≥2.

A dynamic programming algorithm enables exact evaluation under certain graph conditions.

Empirical benchmarks for p=3 on structured graphs show QAOA performance relative to classical baselines.

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.