Quantum Annealing (QA) relies on mixing two Hamiltonian terms, a simple driver and a complex problem Hamiltonian, in a linear combination. The time-dependent schedule for this mixing is often taken to be linear in time: improving on this linear choice is known to be essential and has proven to be difficult. Here, we present different techniques for improving on the linear-schedule QA along two directions, conceptually distinct but leading to similar outcomes: 1) the first approach consists of constructing a Trotter-digitized QA (dQA) with schedules parameterized in terms of Fourier modes or Chebyshev polynomials, inspired by the Chopped Random Basis algorithm for optimal control in continuous time; 2) the second approach is technically a Quantum Approximate Optimization Algorithm (QAOA), whose solutions are found iteratively using linear interpolation or expansion in Fourier modes. Both approaches emphasize finding smooth optimal schedule parameters, ultimately leading to hybrid quantum–classical variational algorithms of the alternating Hamiltonian Ansatz type. We apply these techniques to MaxCut problems on weighted 3-regular graphs with N = 14 sites, focusing on hard instances that exhibit a small spectral gap, for which a standard linear-schedule QA performs poorly. We characterize the physics behind the optimal protocols for both the dQA and QAOA approaches, discovering shortcuts to adiabaticity-like dynamics. Furthermore, we study the transferability of such smooth solutions among hard instances of MaxCut at different circuit depths. Finally, we show that the smoothness pattern of these protocols obtained in a digital setting enables us to adapt them to continuous-time evolution, contrarily to generic non-smooth solutions. This procedure results in an optimized QA schedule that is implementable on analog devices.

中文翻译：

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超越量子退火：maxcut 问题的最优控制解决方案


量子退火 (QA) 依赖于以线性组合形式混合两个哈密顿量、一个简单驱动器和一个复杂问题哈密顿量。这种混合的时间相关时间表通常被认为是时间上线性的：众所周知，改进这种线性选择是必要的，但事实证明是困难的。在这里，我们提出了沿两个方向改进线性时间表 QA 的不同技术，概念上不同，但会产生相似的结果：1）第一种方法包括构建 Trotter 数字化 QA (dQA)，其时间表根据傅立叶模式进行参数化或切比雪夫多项式，受到连续时间内最优控制的切碎随机基础算法的启发； 2) 第二种方法从技术上讲是量子近似优化算法（QAOA），其解决方案是使用线性插值或傅里叶模式展开迭代地找到的。两种方法都强调寻找平滑的最优调度参数，最终导致交替哈密顿 Ansatz 类型的混合量子经典变分算法。我们将这些技术应用于加权 3-正则图上的 MaxCut 问题氮= 14 个站点，重点关注表现出较小光谱间隙的困难实例，对于这些实例，标准线性时间表 QA 表现不佳。我们描述了 dQA 和 QAOA 方法的最佳协议背后的物理原理，发现绝热的捷径类似的动力学。此外，我们研究了不同电路深度的 MaxCut 硬实例之间这种平滑解的可转移性。 最后，我们表明，在数字环境中获得的这些协议的平滑模式使我们能够使它们适应连续时间演化，而不是通用的非平滑解决方案。此过程会产生可在模拟设备上实施的优化 QA 计划。