关于图同构的量子筛算法的不可能性：无条件结果
On the impossibility of a quantum sieve algorithm for graph isomorphism: unconditional results
摘要：
众所周知，任何在隐子群问题（HSP）框架内工作的图同构的量子算法都必须在Ω（n \log n）陪集态上执行高度纠缠的测量。如何有效地进行这种测量的唯一已知模型之一是用于二面体群中HSP的Kuperberg算法，其中量子态根据张量积分解为不可约表示而自适应地组合和测量。这种“量子筛”从陪集态开始，一路向下，直到概率不同的表示，例如，隐藏的子群是平凡的还是非平凡的。在本文中，我们表明，没有这样的方法可以产生一个多项式时间的量子算法图同构。特别地，我们考虑图同构到圈积S_n \wr Z_2上的HSP的自然约化。利用最近证明的S_n的不可约特征标上的一个界，我们证明了这类算法中没有一个能在e^{Ω（\sqrt{n}）}时间内求解图同构，无论它使用什么自适应规则来选择和组合量子态.特别是，这种类型的算法可以提供基本上没有改进的最知名的经典算法，运行时间为e^{O（\sqrt{n \log n}）}。
英文摘要：
It is known that any quantum algorithm for Graph Isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across Ω(n \log n) coset states. One of the only known models for how such a measurement could be carried out efficiently is Kuperberg's algorithm for the HSP in the dihedral group, in which quantum states are adaptively combined and measured according to the decomposition of tensor products into irreducible representations. This ``quantum sieve'' starts with coset states, and works its way down towards representations whose probabilities differ depending on, for example, whether the hidden subgroup is trivial or nontrivial. In this paper we show that no such approach can produce a polynomial-time quantum algorithm for Graph Isomorphism. Specifically, we consider the natural reduction of Graph Isomorphism to the HSP over the the wreath product S_n\wr Z_2. Using a recently proved bound on the irreducible characters of S_n, we show that no algorithm in this family can solve Graph Isomorphism in less than e^{Ω(\sqrt{n})} time, no matter what adaptive rule it uses to select and combine quantum states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time e^{O(\sqrt{n \log n})}.
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