同构问题作为非交换隐移位问题的量子困难性
On the quantum hardness of solving isomorphism problems as nonabelian hidden shift problems
摘要：
我们考虑一种方法来决定刚性n-顶点图的同构（和相关的同构问题），通过解决一个非交换的隐藏移位问题的量子计算机上使用的标准方法。这样的方法可以说比将问题视为隐藏的子群问题更自然。我们证明了刚性图同构的隐移位方法在两个意义上是困难的。首先，我们证明了Omega（n）个隐藏移位状态的副本是解决问题所必需的（而O（n log n）个副本就足够了）。其次，我们证明，如果一个被限制为单寄存器测量，指数数量的隐藏移位状态是必需的。
英文摘要：
We consider an approach to deciding isomorphism of rigid n-vertex graphs (and related isomorphism problems) by solving a nonabelian hidden shift problem on a quantum computer using the standard method. Such an approach is arguably more natural than viewing the problem as a hidden subgroup problem. We prove that the hidden shift approach to rigid graph isomorphism is hard in two senses. First, we prove that Omega(n) copies of the hidden shift states are necessary to solve the problem (whereas O(n log n) copies are sufficient). Second, we prove that if one is restricted to single-register measurements, an exponential number of hidden shift states are required.
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