Quantum Supremacy through the Quantum Approximate Optimization Algorithm

TL;DR

This paper argues that the Quantum Approximate Optimization Algorithm (QAOA) can demonstrate quantum supremacy by producing output distributions that are hard to simulate classically, even at low depths, making it promising for near-term quantum computers.

Contribution

It establishes that QAOA's output distribution cannot be efficiently simulated classically under reasonable assumptions, highlighting its potential for quantum supremacy beyond optimization.

Findings

QAOA's output distribution is hard to simulate classically.

Classical simulation of QAOA would collapse the Polynomial Hierarchy.

QAOA is a promising candidate for demonstrating quantum supremacy.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is designed to run on a gate model quantum computer and has shallow depth. It takes as input a combinatorial optimization problem and outputs a string that satisfies a high fraction of the maximum number of clauses that can be satisfied. For certain problems the lowest depth version of the QAOA has provable performance guarantees although there exist classical algorithms that have better guarantees. Here we argue that beyond its possible computational value the QAOA can exhibit a form of Quantum Supremacy in that, based on reasonable complexity theoretic assumptions, the output distribution of even the lowest depth version cannot be efficiently simulated on any classical device. We contrast this with the case of sampling from the output of a quantum computer running the Quantum Adiabatic Algorithm (QADI) with the restriction that the Hamiltonian that governs the evolution is gapped and stoquastic. Here we show that there is an oracle that would allow sampling from the QADI but even with this oracle, if one could efficiently classically sample from the output of the QAOA, the Polynomial Hierarchy would collapse. This suggests that the QAOA is an excellent candidate to run on near term quantum computers not only because it may be of use for optimization but also because of its potential as a route to establishing quantum supremacy.