量子陪集状态在图同构问题上的局限性
Limitations of Quantum Coset States for Graph Isomorphism
AI总结：
本文研究了量子陪集状态在图同构问题上的局限性，证明需要对至少Ω(n log n)个陪集状态进行纠缠量子测量才能获得有用信息，且该结论可推广至其他群。
摘要：
人们早就知道图同构可以归约到隐藏子群问题（HSP）。此外，量子计算中的大多数指数级加速都是通过求解HSP的实例获得的。这些算法的一个共同特征是使用量子陪集状态，这些状态编码了隐藏子群。一个悬而未决的问题是，使用这些状态求解图同构有多困难。Moore、Russell和Schulman最近证明，从单个或一对陪集状态中只能获得指数级少量的信息。利用其能力的潜在来源是同时对多个状态进行联合操作的纠缠量子测量。我们证明，对于图同构的情况，要对至少Ω(n log n)个陪集状态进行纠缠量子测量才能获得有用信息，这与信息论上界相匹配。这可以被视为一个负面结果，因为高度纠缠的测量通常似乎难以实现。我们的主定理非常通用，并且也排除了在其他一些群（例如GL(n, F_{p^m})和G^n，其中G是有限的且满足适当属性）上对少量陪集状态使用联合测量。
英文摘要：
It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because highly entangled measurements seem hard to implement in general. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n, F_{p^m}) and G^n where G is finite and satisfies a suitable property.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
