Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups. In contrast, in the adjacency list model for bounded-degree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).

中文翻译：

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对称性、图属性和量子加速

Aaronson 和 Ambainis (2009) 以及 Chailloux (2018) 表明，完全对称（部分）函数不允许指数量子查询加速。这提出了一个自然的问题：一个函数必须有多对称才能表现出大的量子加速？在这项工作中，我们证明了邻接矩阵模型中的超图对称性最多允许随机和量子查询复杂性之间的多项式分离。我们还表明，值得注意的是，由这些对称性构成的置换群本质上是唯一阻止超多项式量子加速的置换群。我们通过完全表征允许超多项式量子加速的原始置换群来证明这一点。相比之下，在有界度图的邻接表模型中（图对称性表现不同），我们展示了一个属性测试问题，该问题显示了指数量子加速。这些结果解决了 Ambainis、Childs 和 Liu（2010）以及 Montanaro 和 de Wolf（2013）提出的开放性问题。