Quantum optimization allows for up to exponential quantum speedups for specific, possibly industrially relevant problems. As the key algorithm in this field, we motivate and discuss the Quantum Approximate Optimization Algorithm (QAOA), which can be understood as a slightly generalized version of Quantum Annealing for gate-based quantum computers. We delve into the quantum circuit implementation of the QAOA, including Hamiltonian simulation techniques for higher-order Ising models, and discuss parameter training using the parameter shift rule. An example implementation with Pennylane source code demonstrates practical application for the Maximum Cut problem. Further, we show how constraints can be incorporated into the QAOA using Grover mixers, allowing to restrict the search space to strictly valid solutions for specific problems. Finally, we outline the Variational Quantum Eigensolver (VQE) as a generalization of the QAOA, highlighting its potential in the NISQ era and addressing challenges such as barren plateaus and ansatz design.

翻译：量子优化算法能够针对特定（可能具有工业应用价值）的问题实现高达指数级的量子加速。作为该领域的核心算法，本文阐释并探讨了量子近似优化算法（QAOA），该算法可视为基于门电路的量子计算机上量子退火的一种略微广义化版本。我们深入分析了QAOA的量子电路实现，包括高阶伊辛模型的哈密顿量模拟技术，并讨论了利用参数平移规则进行参数训练的方法。通过结合Pennylane源代码的示例实现，展示了该算法在最大割问题中的实际应用。进一步，我们展示了如何通过Grover混合器将约束条件整合到QAOA中，从而将搜索空间严格限制为特定问题的有效解。最后，我们将变分量子本征求解器（VQE）概述为QAOA的广义化形式，强调其在含噪声中等规模量子（NISQ）时代的潜力，并探讨了诸如贫瘠高原和拟设设计等挑战。