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 $1/\theta$.

翻译：我们推广了Farhi等人关于三正则图Max-Cut量子近似优化算法（QAOA）的0.6924近似比结果技术，为热启动QAOA的近似比建立了可证明的下界。给定初始化角度$\theta$，考虑初始态为乘积态的热启动方案，其中每个量子比特位置在布洛赫球面上偏离北极或南极的角度为$\theta$；两个可能的量子比特位置中，每个量子比特的位置由经典算法获得的切割结果（编码为比特串$b$）决定。我们通过图表展示了$b$的性质和初始化角度$\theta$如何影响热启动QAOA近似比的下界。我们考虑了多种经典算法（及其产生的用于生成热启动的切割结果）。研究结果强烈表明：不存在任何初始化角度能使（最坏情况下的）近似比同时超越标准QAOA和用于生成热启动的经典算法。此外，我们证明在$\theta=60^\circ$时，热启动QAOA能够（有效地）恢复用于生成热启动的切割结果，这表明在实践中该角度可作为启发式探索替代方案的有前景的起始值。最后，对于任意具有整数值目标函数的组合优化问题，我们给出了热启动QAOA在实现近似比变化时所需电路深度的界限；特别地，我们证明当$\theta$较小时，该界限大致与$1/\theta$成正比。