A sharp interaction-degree threshold for simulating QAOA

2026-05-21

AI Summary

This work investigates the classical simulability of the Quantum Approximate Optimization Algorithm (QAOA) under varying degrees of interaction graphs to delineate the boundary of its quantum advantage. Focusing on QAOA with two-local cost functions, the study integrates computational complexity theory, quantum sampling analysis, and graph-theoretic techniques to establish a sharp threshold between graph degrees 2 and 3. It demonstrates that for degree-2 graphs and circuit depth O(log n), an n-qubit QAOA instance admits efficient exact classical sampling. In contrast, for degree-3 graphs—even when the cost function is trivially optimizable—approximate sampling would imply a collapse of the polynomial hierarchy to its third level, indicating computational hardness for classical simulation. This result precisely characterizes the complexity boundary for classically simulating QAOA.

Abstract

We identify a sharp interaction-degree threshold for the classical simulation of QAOA with $2$-local cost functions. At degree $3$, classical sampling from depth-$1$ QAOA with small multiplicative error would collapse the polynomial hierarchy to its third level. At degree $2$, exact classical sampling from depth-$p$ QAOA on $n$ qubits runs in time $n^{O(1)}$ whenever $p = O(\log n)$. The hard degree-$3$ instances have trivially optimizable cost functions, so sampling hardness does not by itself imply a quantum optimization advantage.

Problem

• QAOA
• classical simulation
• interaction degree
• sampling hardness
• polynomial hierarchy

Innovation

• QAOA
• classical simulation
• interaction-degree threshold
• sampling hardness
• polynomial hierarchy