Warm-starting quantum optimization

There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers such employ relaxations, which replace binary variables with continuous ones, instance the form higher-dimensional matrix-valued (semidefinite programming). Under Unique Games Conjecture, these relaxations often provide best performance ratios available classically polynomial time. Here, we discuss how to warm-start optimization initial state corresponding solution a relaxation problem analyze properties associated algorithms. In particular, this allows algorithm inherit guarantees classical algorithm. We illustrate context portfolio optimization, where our results indicate that warm-starting Quantum Approximate Optimization Algorithm (QAOA) particularly beneficial at low depth. Likewise, Recursive QAOA MAXCUT shows systematic increase size obtained cut fully connected graphs random weights, when Goemans-Williamson randomized rounding utilized warm start. It straightforward apply same ideas other randomized-rounding schemes problems.