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We generalize the Quantum Approximate Optimization Algorithm (QAOA) of Farhiet al. (2014) to allow for arbitrary separable initial states withcorresponding mixers such that the starting state is the most excited state ofthe mixing Hamiltonian. We demonstrate this version of QAOA, which we callQAOA-warmest, by simulating Max-Cut on weighted graphs. We initialize thestarting state as a warm-start using $2$ and $3$-dimensional approximationsobtained using randomized projections of solutions to Max-Cut's semi-definiteprogram, and define a warm-start dependent custom mixer. We show that thesewarm-starts initialize the QAOA circuit with constant-factor approximations of$0.658$ for $2$-dimensional and $0.585$ for $3$-dimensional warm-starts forgraphs with non-negative edge weights, improving upon previously known trivial(i.e., $0.5$ for standard initialization) worst-case bounds at $p=0$. Thesefactors in fact lower bound the approximation achieved for Max-Cut at highercircuit depths, since we also show that QAOA-warmest with any separable initialstate converges to Max-Cut under the adiabatic limit as $p\rightarrow \infty$.However, the choice of warm-starts significantly impacts the rate ofconvergence to Max-Cut, and we show empirically that our warm-starts achieve afaster convergence compared to existing approaches. Additionally, our numericalsimulations show higher quality cuts compared to standard QAOA, the classicalGoemans-Williamson algorithm, and a warm-started QAOA without custom mixers foran instance library of $1148$ graphs (upto $11$ nodes) and depth $p=8$. Wefurther show that QAOA-warmest outperforms the standard QAOA of Farhi et al. inexperiments on current IBM-Q and Quantinuum hardware.

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