The Quantum Alternating Operator Ansatz (QAOA) is a prominent variational quantum algorithm for solving combinatorial optimization problems. Its effectiveness depends on identifying input parameters that yield high-quality solutions. However, understanding the complexity of training QAOA remains an under-explored area. Previous results have given analytical performance guarantees for a small, fixed number of parameters. At the opposite end of the spectrum, barren plateaus are likely to emerge at $\Omega(n)$ parameters for $n$ qubits. In this work, we study the difficulty of training in the intermediate regime, which is the focus of most current numerical studies and near-term hardware implementations. Through extensive numerical analysis of the quality and quantity of local minima, we argue that QAOA landscapes can exhibit a superpolynomial growth in the number of low-quality local minima even when the number of parameters scales logarithmically with $n$. This means that the common technique of gradient descent from randomly initialized parameters is doomed to fail beyond small $n$, and emphasizes the need for good initial guesses of the optimal parameters.

翻译：量子交替算子拟设（QAOA）是一种用于解决组合优化问题的著名变分量子算法。其有效性取决于能否识别出产生高质量解的输入参数。然而，理解训练QAOA的复杂性仍是一个探索不足的领域。以往的研究成果为少量固定参数提供了分析性能保证。而在另一个极端，对于$n$量子比特，当参数数量达到$\Omega(n)$时，荒芜高原现象很可能出现。在本工作中，我们研究了中间参数规模下的训练难度，这恰是当前大多数数值研究和近期硬件实现关注的焦点。通过对局部极小值的质量和数量进行广泛数值分析，我们论证了即使参数数量随$n$对数增长，QAOA景观中低质量局部极小值的数量也可能呈现超多项式增长。这意味着从随机初始化参数出发的梯度下降法在超出小规模$n$后将必然失败，从而凸显了对最优参数进行良好初始估计的必要性。