Quantum annealing (QA) and the quantum approximate optimization algorithm (QAOA) are two special cases of the following control problem: apply a combination of two Hamiltonians to minimize the energy of a quantum state. Which is more effective has remained unclear. Here we analytically apply the framework of optimal control theory to show that generically, given a fixed amount of time, the optimal procedure has the pulsed (or “bang-bang”) structure of QAOA at the beginning and end but can have a smooth annealing structure in between. This is in contrast to previous works which have suggested that bang-bang (i.e., QAOA) protocols are ideal. To support this theoretical work, we carry out simulations of various transverse field Ising models, demonstrating that bang-anneal-bang protocols are more common. The general features identified here provide guideposts for the nascent experimental implementations of quantum optimization algorithms.

中文翻译：

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量子退火中的最佳协议和量子近似优化算法问题

量子退火（QA）和量子近似优化算法（QAOA）是以下控制问题的两种特殊情况：应用两个哈密顿量的组合以最小化量子态的能量。哪个更有效尚不清楚。在这里，我们分析性地应用了最佳控制理论的框架，以表明一般而言，在固定的时间量下，最佳过程在开始和结束时具有QAOA的脉冲（或“砰砰”）结构，但可以平滑退火之间的结构。这与以前的研究相反，之前的研究表明，bang-bang（即QAOA）协议是理想的。为了支持这项理论工作，我们对各种横向场Ising模型进行了仿真，证明了Bang-Annual-Bang协议更为常见。