We generalize the Quantum Approximate Optimization Algorithm (QAOA) of Farhi et al. (2014) to allow for arbitrary separable initial states and corresponding mixers such that the starting state is the most excited state of the mixing Hamiltonian. We demonstrate this version of QAOA by simulating Max-Cut on weighted graphs. We initialize the starting state as a warm-start inspired by classical rank-2 and rank-3 approximations obtained using Burer-Monteiro's heuristics, and choose a corresponding custom mixer. Our numerical simulations with this generalization, which we call QAOA-Warmest, yield 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 hardware.

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