[Paper Review] Quantum Supremacy through the Quantum Approximate Optimization Algorithm

Edward Farhi, Aram W. Harrow|arXiv (Cornell University)|Feb 24, 2016

Quantum Computing Algorithms and ArchitectureComputer Science35 references189 citations

TL;DR

The paper argues that the Quantum Approximate Optimization Algorithm (QAOA) can exhibit quantum supremacy: even its shallowest depth makes its output distribution hard to classically simulate, under standard complexity-theoretic assumptions, and may offer near-term computational advantages for optimization.

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.

Motivation & Objective

• Motivate and formalize how QAOA operates on combinatorial optimization problems.
• Explain why sampling from QAOA outputs is computationally hard for classical devices under standard complexity assumptions.
• Compare QAOA-based supremacy arguments with those for quantum adiabatic computation under stoquastic constraints.
• Bridge practical near-term quantum computing prospects with foundational complexity-theoretic implications.

Proposed method

• Define the QAOA circuit family and its p-depth generalization.
• Relate QAOA to CSPs and MAX-CUT via the cost operator C and mixing operator B.
• Argue hardness of classically computing or sampling from QAOA outputs using complexity theory and postselection.
• Use PostBQP and PostBPP to connect sampling from quantum circuits to the collapse of the Polynomial Hierarchy (PH).
• Contrast QAOA with Quantum Adiabatic Algorithm in the stoquastic and gapped Hamiltonian setting to delineate where similar hardness arguments apply or fail.

Experimental results

Research questions

• RQ1 Can the output distribution of the lowest-depth QAOA be efficiently simulated or sampled by a classical computer?
• RQ2 Do assumptions about the PH collapse imply that efficient classical sampling of QAOA outputs is unlikely?
• RQ3 How does QAOA's hardness compare to that of the Quantum Adiabatic Algorithm under stoquastic constraints?
• RQ4 Under what conditions might QAOA provide not only optimization benefits but a route to quantum supremacy?
• RQ5 What is the role of postselection and related complexity tools in establishing hardness of simulating quantum circuits like QAOA?

Key findings

• Even the shallowest depth version of QAOA is argued to be hard to simulate classically, under reasonable complexity-theoretic assumptions.
• Efficiently sampling from the output distribution of an arbitrary quantum circuit implies PH collapse, and analogous arguments extend to QAOA.
• Post-selected quantum computing (PostBQP) can solve counting problems, suggesting a distinct power gap between quantum and classical sampling models.
• The QAOA may achieve approximation advantages for certain CSPs, but its supremacy argument rests on hardness of classical simulation rather than purely on approximation guarantees.
• In contrast, sampling from a stoquastic QADI (Quantum Adiabatic Algorithm) under certain conditions does not yield the same supremacy proof, highlighting a boundary between QAOA and QADI-based arguments.