置换群、极小度与量子计算
Permutation groups, minimal degrees and quantum computing
摘要：
研究了给定最小次数的置换群，在不需要经典的置换性假设的情况下。我们提供了清晰的上界的顺序置换群的最小程度的m和它的元素的数量的任何给定的支持。这些结果有助于非交换编码理论的基础。我们的结果的一个主要应用涉及量子计算中对称群的隐子群问题。我们完全刻画了对称群的隐子群，这些隐子群可以用弱量子傅立叶采样从恒等式中区分出来，证明了它们是具有有界极小度的子群。这意味着对称群的弱标准方法与经典的穷举搜索相比没有任何优势。
英文摘要：
We study permutation groups of given minimal degree without the classical primitivity assumption. We provide sharp upper bounds on the order of a permutation group of minimal degree m and on the number of its elements of any given support. These results contribute to the foundations of a non-commutative coding theory. A main application of our results concerns the Hidden Subgroup Problem for the symmetric group in Quantum Computing. We completely characterize the hidden subgroups of the symmetric group that can be distinguished from identity with weak Quantum Fourier Sampling, showing these are exactly the subgroups with bounded minimal degree. This implies that the weak standard method for the symmetric group has no advantage whatsoever over classical exhaustive search.
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