  We generalize Farhi et al.'s 0.6924-approximation result technique of the
Max-Cut Quantum Approximate Optimization Algorithm (QAOA) on 3-regular graphs
to obtain provable lower bounds on the approximation ratio for warm-started
QAOA. Given an initialization angle $\theta$, we consider warm-starts where the
initial state is a product state where each qubit position is angle $\theta$
away from either the north or south pole of the Bloch sphere; of the two
possible qubit positions the position of each qubit is decided by some
classically obtained cut encoded as a bitstring $b$. We illustrate through
plots how the properties of $b$ and the initialization angle $\theta$ influence
the bound on the approximation ratios of warm-started QAOA. We consider various
classical algorithms (and the cuts they produce which we use to generate the
warm-start). Our results strongly suggest that there does not exist any choice
of initialization angle that yields a (worst-case) approximation ratio that
simultaneously beats standard QAOA and the classical algorithm used to create
the warm-start.
  Additionally, we show that at $\theta=60^\circ$, warm-started QAOA is able to
(effectively) recover the cut used to generate the warm-start, thus suggesting
that in practice, this value could be a promising starting angle to explore
alternate solutions in a heuristic fashion. Finally, for any combinatorial
optimization problem with integer-valued objective values, we provide bounds on
the required circuit depth needed for warm-started QAOA to achieve some change
in approximation ratio; more specifically, we show that for small $\theta$, the
bound is roughly proportional to $1/\theta$.
