The quantum alternating operator ansatz (QAOA) is a heuristic hybrid quantum-classical algorithm for finding high-quality approximate solutions to combinatorial optimization problems, such as Maximum Satisfiability. While QAOA is well-studied, theoretical results as to its runtime or approximation ratio guarantees are still relatively sparse. We provide some of the first lower bounds for the number of rounds (the dominant component of QAOA runtimes) required for QAOA. For our main result, (i) we leverage a connection between quantum annealing times and the angles of QAOA to derive a lower bound on the number of rounds of QAOA with respect to the guaranteed approximation ratio. We apply and calculate this bound with Grover-style mixing unitaries and (ii) show that this type of QAOA requires at least a polynomial number of rounds to guarantee any constant approximation ratios for most problems. We also (iii) show that the bound depends only on the statistical values of the objective functions, and when the problem can be modeled as a $k$-local Hamiltonian, can be easily estimated from the coefficients of the Hamiltonians. For the conventional transverse field mixer, (iv) our framework gives a trivial lower bound to all bounded occurrence local cost problems and all strictly $k$-local cost Hamiltonians matching known results that constant approximation ratio is obtainable with constant round QAOA for a few optimization problems from these classes. Using our novel proof framework, (v) we recover the Grover lower bound for unstructured search and -- with small modification -- show that our bound applies to any QAOA-style search protocol that starts in the ground state of the mixing unitaries.

翻译：量子交替算子拟设（QAOA）是一种启发式混合量子-经典算法，用于寻找组合优化问题（如最大可满足性问题）的高质量近似解。尽管QAOA已被广泛研究，但关于其运行时间或近似比保证的理论结果仍相对稀少。我们首次为QAOA所需轮数（其运行时间的主要组成部分）提供了若干下界。关于主要结果：（i）利用量子退火时间与QAOA角度之间的关联，推导了关于保证近似比的QAOA轮数下界。我们使用Grover式混合酉算子应用并计算了这一下界，并（ii）表明：对于大多数问题，此类QAOA至少需要多项式轮数才能保证任何常数近似比。我们还（iii）证明该下界仅取决于目标函数的统计值，且当问题可建模为$k$-局域哈密顿量时，可由哈密顿量的系数轻松估算。对于传统横向场混合器，（iv）我们的框架为所有有界出现局域代价问题及所有严格$k$-局域代价哈密顿量给出了平凡下界，这与已知结果一致：对于这些类别中的少数优化问题，常数轮QAOA即可获得常数近似比。利用这一新型证明框架，（v）我们恢复了无结构搜索的Grover下界，并通过小幅修改表明：该下界适用于任何从混合酉算子的基态出发的QAOA式搜索协议。