量子近似优化算法中最优参数的对称性可迁移性
Symmetry-informed transferability of optimal parameters in the Quantum Approximate Optimization Algorithm
AI总结：
本文研究量子近似优化算法中MaxCut问题最优参数的对称性，证明其存在多组最优参数，并发现仅部分可迁移，通过对称性转换实现有效迁移，并推广至伊辛模型及递归和多角度变体。
摘要：
变分量子算法的主要局限性之一在于对高维非凸变分参数景观进行经典优化。为简化此优化，我们可以利用问题对称性来缩减搜索空间，并在典型最优参数集中时将其作为初始点。本文考虑了d-正则树子图上的MaxCut问题的量子近似优化算法最优参数的典型值，并将其复用于不同的图实例。我们证明了若干类加权与未加权图优化景观中的对称性，这解释了多组最优参数存在的原因。然而，我们观察到并非所有最优参数集都能在问题实例间成功迁移。我们在搜索空间中找到了特定的可迁移域，并展示了如何利用所研究的对称性将任意一组最优参数转换到合适的域中。最后，我们将这些结果推广到由伊辛哈密顿量描述的通用经典优化问题、相关物理模型的哈密顿量变分拟设，以及递归和多角度量子近似优化算法。
英文摘要：
One of the main limitations of variational quantum algorithms is the classical optimization of the highly dimensional non-convex variational parameter landscape. To simplify this optimization, we can reduce the search space using problem symmetries and typical optimal parameters as initial points if they concentrate. In this article, we consider typical values of optimal parameters of the quantum approximate optimization algorithm for the MaxCut problem with d-regular tree subgraphs and reuse them in different graph instances. We prove symmetries in the optimization landscape of several kinds of weighted and unweighted graphs, which explains the existence of multiple sets of optimal parameters. However, we observe that not all optimal sets can be successfully transferred between problem instances. We find specific transferable domains in the search space and show how to translate an arbitrary set of optimal parameters into the adequate domain using the studied symmetries. Finally, we extend these results to general classical optimization problems described by Ising Hamiltonians, the Hamiltonian variational ansatz for relevant physical models, and the recursive and multi-angle quantum approximate optimization algorithms.
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