Warm-Started QAOA with Aligned Mixers Converges Slowly Near the Poles of the Bloch Sphere

In order to boost the performance of the Quantum Approximate Optimization Algorithm (QAOA) to solve problems in combinatorial optimization, researchers have leveraged the solutions returned from classical algorithms in order to create a warm-started quantum initial state for QAOA that is biased towards "good" solutions. Cain et al. showed that if the classically-obtained solutions are mapped to the poles of the Bloch sphere, then vanilla QAOA with the standard mixer "gets stuck". If the classically-obtained solution is instead mapped to within some angle $\theta$ from the poles of the Bloch sphere, creating an initial product state, then QAOA with optimal variational parameters is known to converge to the optimal solution with increased circuit depth if the mixer is modified to be "aligned" with the warm-start initial state. Leveraging recent work of Benchasattabuse et al., we provide theoretical lower bounds on the circuit depth necessary for this form of warm-started QAOA to achieve a desired change $\Delta \lambda$ in approximation ratio; in particular, we show that for small $\theta$, the lower bound on the circuit depth roughly scales proportionally with $\Delta \lambda/\theta$.