Warm-Started QAOA with Custom Mixers Provably Converges and Computationally Beats Goemans-Williamson’s Max-Cut at Low Circuit Depths

Reuben Tate1, Jai Moondra2, Bryan Gard3, Greg Mohler3, and Swati Gupta4

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

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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.

Quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical technique for combinatorial optimization that promises to be more powerful than any classical optimizer. In this work, we exemplify its potential on a fundamental combinatorial optimization problem, known as Max-Cut, where the best possible classical algorithm is that by Goemans and Williamson (GW). We achieve this by introducing classically obtained warm-starts to the QAOA, with modified mixing operators, and show computationally that this outperforms GW. We modify the quantum algorithm appropriately to maintain the connection to quantum adiabatic computing; we discuss theory and provide numerical and experimental evidence showing the promise of our approach.

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