A sharp interaction-degree threshold for simulating QAOA

TL;DR

This paper establishes a precise threshold in interaction degree that determines when classical simulation of QAOA becomes computationally hard, highlighting a boundary at degree 3.

Contribution

It identifies a sharp interaction-degree threshold for classical simulation of QAOA, revealing complexity distinctions at degrees 2 and 3.

Findings

Classical sampling from degree-2 QAOA at depth O(log n) is efficient.

Degree-3 QAOA at depth 1 would cause a collapse in the polynomial hierarchy.

Hard instances at degree 3 are trivially optimizable, so sampling hardness does not imply quantum advantage.

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.