[Paper Review] Performance of the Quantum Approximate Optimization Algorithm on the Maximum Cut Problem

Gavin E. Crooks|arXiv (Cornell University)|Nov 20, 2018

Quantum Computing Algorithms and ArchitectureComputer Science103 citations

TL;DR

The paper simulates QAOA on MaxCut using automatic differentiation and SGD, shows that QAOA can outperform Goemans-Williamson at modest depths, and that depth-8 or more sustains advantage as graph size grows.

ABSTRACT

The Quantum Approximate Optimization Algorithm (QAOA) is a promising approach for programming a near-term gate-based hybrid quantum computer to find good approximate solutions of hard combinatorial problems. However, little is currently know about the capabilities of QAOA, or of the difficulty of the requisite parameters optimization. Here, we study the performance of QAOA on the MaxCut combinatorial optimization problem, optimizing the quantum circuits on a classical computer using automatic differentiation and stochastic gradient descent, using QuantumFlow, a quantum circuit simulator implemented with TensorFlow. We find that we can amortize the training cost by optimizing on batches of problems instances; that QAOA can exceed the performance of the classical polynomial time Goemans-Williamson algorithm with modest circuit depth, and that performance with fixed circuit depth is insensitive to problem size. Moreover, MaxCut QAOA can be efficiently implemented on a gate-based quantum computer with limited qubit connectivity, using a qubit swap network. These observations support the prospects that QAOA will be an effective method for solving interesting problems on near-term quantum computers.

Motivation & Objective

• Motivate the use of QAOA for near-term quantum optimization on combinatorial problems.
• Investigate the performance of QAOA on MaxCut through classical simulation and automatic differentiation.
• Assess whether batch training over ensembles can amortize training costs.
• Compare QAOA performance against the classical Goemans-Williamson algorithm across graph sizes.

Proposed method

• Encode MaxCut into a diagonal cost Hamiltonian for QAOA.
• Use a two-Hamiltonian alternation with cost and driver unitaries parameterized by angles (gamma_p, beta_p).
• Train QAOA parameters via automatic differentiation and stochastic gradient descent on batches of instances.
• Employ a quantum virtual machine built on TensorFlow to enable back-propagation through quantum circuits.
• Draw initial parameters from a controlled normal distribution to avoid chaotic training.
• Evaluate performance by averaging over instances from Erdős–Rényi graphs and compare to Goemans-Williamson.

Experimental results

Research questions

• RQ1 Can QAOA achieve higher approximation ratios than classical baselines on MaxCut with modest circuit depth?
• RQ2 How does QAOA performance scale with graph size and circuit depth?
• RQ3 Is batch-based training over problem ensembles effective for finding useful QAOA parameters?
• RQ4 What is the resource cost, in terms of two-qubit gates, for implementing QAOA on near-term hardware?
• RQ5 How do optimized QAOA protocols vary with the number of steps P?

Key findings

• QAOA with five steps matches Goemans-Williamson on 10-node graphs in their training/test setup.
• QAOA performance improves with circuit depth and surpasses Goemans-Williamson by P=8 on their test sets.
• For fixed depth, QAOA performance degrades with larger graphs, but P≥8 maintains a relative advantage over the classical algorithm.
• QAOA can be efficiently implemented on hardware with limited qubit connectivity using a qubit swap network.
• Training cost can be amortized by optimizing on batches of problem instances from the same ensemble.