Calibrating the Classical Hardness of the Quantum Approximate Optimization Algorithm

TL;DR

This paper investigates how the fidelity of the Quantum Approximate Optimization Algorithm scales with entanglement and classical simulation parameters, providing benchmarks and insights into the classical-quantum computational boundary.

Contribution

It introduces a scaling law for fidelity based on entanglement per qubit, calibrates classical simulation resources, and benchmarks quantum hardware performance in optimization tasks.

Findings

Fidelity follows a specific scaling law with entanglement per qubit.

Classical simulation resources can be calibrated to match quantum hardware performance.

Results quantify the classical hardness of outperforming noisy quantum processors.

Abstract

Trading fidelity for scale enables approximate classical simulators such as matrix product states (MPS) to run quantum circuits beyond exact methods. A control parameter, the so-called bond dimension $\chi$ for MPS, governs the allocated computational resources and the output fidelity. Here, we characterize the fidelity for the quantum approximate optimization algorithm by the expectation value of the cost function it seeks to minimize and find that it follows a scaling law $F\bigl(\ln\chi\bigr/N\bigr)$ with $N$ the number of qubits. With $\ln\chi$ amounting to the entanglement that an MPS can encode, we show that the relevant variable for investigating the fidelity is the entanglement per qubit. Importantly, our results calibrate the classical computational power required to achieve the desired fidelity and benchmark the performance of quantum hardware in a realistic setup. For instance, we quantify the hardness of performing better classically than a noisy superconducting quantum processor by readily matching its output to the scaling function. Moreover, we relate the global fidelity to that of individual operations and establish its relationship with $\chi$ and $N$. We sharpen the requirements for noisy quantum computers to outperform classical techniques at running a quantum optimization algorithm in speed, size, and fidelity.