Warm-Started QAOA with Custom Mixers Provably Converges and Computationally Beats Goemans-Williamson’s Max-Cut at Low Circuit Depths
1CCS-3 Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87544, USA
2Georgia Institute of Technology, Atlanta, GA 30332, USA
3Georgia Tech Research Institute, Atlanta, GA 30332, USA
4Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02142, USA
Published: | 2023-09-26, volume 7, page 1121 |
Eprint: | arXiv:2112.11354v4 |
Doi: | https://doi.org/10.22331/q-2023-09-26-1121 |
Citation: | Quantum 7, 1121 (2023). |
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Abstract
We generalize the Quantum Approximate Optimization Algorithm (QAOA) of Farhi et al. (2014) to allow for arbitrary separable initial states with corresponding mixers such that the starting state is the most excited state of the mixing Hamiltonian. We demonstrate this version of QAOA, which we call $QAOA-warmest$, by simulating Max-Cut on weighted graphs. We initialize the starting state as a $warm-start$ using $2$ and $3$-dimensional approximations obtained using randomized projections of solutions to Max-Cut's semi-definite program, and define a warm-start dependent $custom mixer$. We show that these warm-starts initialize the QAOA circuit with constant-factor approximations of $0.658$ for $2$-dimensional and $0.585$ for $3$-dimensional warm-starts for graphs with non-negative edge weights, improving upon previously known trivial (i.e., $0.5$ for standard initialization) worst-case bounds at $p=0$. These factors in fact lower bound the approximation achieved for Max-Cut at higher circuit depths, since we also show that QAOA-warmest with any separable initial state converges to Max-Cut under the adiabatic limit as $p\rightarrow \infty$. However, the choice of warm-starts significantly impacts the rate of convergence to Max-Cut, and we show empirically that our warm-starts achieve a faster convergence compared to existing approaches. Additionally, our numerical simulations show higher quality cuts compared to standard QAOA, the classical Goemans-Williamson algorithm, and a warm-started QAOA without custom mixers for an instance library of $1148$ graphs (upto $11$ nodes) and depth $p=8$. We further show that QAOA-warmest outperforms the standard QAOA of Farhi et al. in experiments on current IBM-Q and Quantinuum hardware.
![](https://quantum-journal.org/wp-content/uploads/2023/09/featured-image.png)
Featured image: Performance of QAOA-warmest (with warm-starts) and standard QAOA as a function of QAOA depth for an ideal (dashed) and noisy simulation (dotted). For the chosen 20-node graph, GW achieves an approximation ratio of 0.912, while in the ideal case, QAOA-warmest outperforms GW for $p \geq 2$ while standard QAOA requires $p > 4$. The noise simulation is based on calibration data from IBM-Q’s Guadalupe device.
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