The Quantum Approximate Optimization Algorithm (QAOA) is suggested as a promising appli- cation on early quantum computers. Here, a quantum-inspired classical algorithm, the mean-field Approximate Optimization Algorithm (mean-field AOA), is developed by replacing the quantum evolution of the QAOA with classical spin dynamics through the mean-field approximation. Due to the alternating structure of the QAOA, this classical dynamics can be found exactly for any number of QAOA layers. We benchmark its performance against the QAOA on the Sherrington-Kirkpatrick (SK) model and the partition problem, and find that the mean-field AOA outperforms the QAOA in both cases for most instances. Our algorithm can thus serve as a tool to delineate optimization problems that can be solved classically from those that cannot, i.e. we believe that it will help to identify instances where a true quantum advantage can be expected from the QAOA. To quantify quantum fluctuations around the mean-field trajectories, we solve an effective scattering problem in time, which is characterized by a spectrum of time-dependent Lyapunov exponents. These provide an indicator for the hardness of a given optimization problem relative to the mean-field AOA.
For an introduction to using the mean-field AOA code in this repository, please look into our introductory notebook.
The remaining notebooks can be used to reproduce the plots of our paper from the data. Note that some of the notebooks are written in Julia.
If you are using code from this repository, please cite our work:
@article{PRXQuantum.4.030335,
title = {Mean-Field Approximate Optimization Algorithm},
author = {Misra-Spieldenner, Aditi and Bode, Tim and Schuhmacher, Peter K. and Stollenwerk, Tobias and Bagrets, Dmitry and Wilhelm, Frank K.},
journal = {PRX Quantum},
volume = {4},
issue = {3},
pages = {030335},
numpages = {19},
year = {2023},
month = {Sep},
publisher = {American Physical Society},
doi = {10.1103/PRXQuantum.4.030335},
url = {https://link.aps.org/doi/10.1103/PRXQuantum.4.030335}
}