We present a negative result regarding the hidden subgroup problem on the powers $G^n$ of a fixed group $G$. Under a condition on the base group $G$, we prove that strong Fourier sampling cannot distinguish some subgroups of $G^n$. Since strong sampling is in fact the optimal measurement on a coset state, this shows that we have no hope of efficiently solving the hidden subgroup problem over these groups with separable measurements on coset states (that is, using any polynomial number of single-register coset state experiments). Base groups satisfying our condition include all nonabelian simple groups. We apply our results to show that there exist uniform families of nilpotent groups whose normal series factors have constant size and yet are immune to strong Fourier sampling.
The Hidden Subgroup Problem in Affine Groups: Basis Selection in Fourier Sampling
许多量子算法，包括Shor著名的因式分解和离散对数算法，都是通过归约到隐子群问题来进行的，其中群G的子群H必须从H的左陪集上一致支撑的量子态y确定。这些隐藏的子群问题，然后解决傅里叶采样：量子傅里叶变换y的计算和测量。当基础群是非阿贝尔群时，傅立叶抽样范式的两个重要变体已经被确定：弱标准方法，其中仅测量表示名称，以及强标准方法，其中发生完全测量。强标准方法是否真的更强，也就是说，是否存在可以通过强方法重建而不能通过弱方法或任何其他已知方法重建的隐藏子群，这一点仍然是开放的。在本文中，我们肯定地解决了这个问题。证明了强标准方法可以有效地确定Zp与Zq的半直积的隐子群，其中q整除（p-1）且q = p / polylog（p）.此外，弱标准方法和"遗忘“阿贝尔方法对这些群体是不够的。我们将其推广到Z_p的半直积上的隐子群问题的信息论解，通过\Z_q，其中q整除（p-1），特别是仿射群A_p.最后，我们证明了一类群的闭包性质，在这类群上隐子群问题可以有效地得到解决.
Part I of this paper showed that the hidden subgroup problem over the symmetric group--including the special case relevant to Graph Isomorphism--cannot be efficiently solved by strong Fourier sampling, even if one may perform an arbitrary POVM on the coset state. In this paper, we extend these results to entangled measurements. Specifically, we show that the hidden subgroup problem on the symmetric group cannot be solved by any POVM applied to pairs of coset states. In particular, these hidden subgroups cannot be determined by any polynomial number of one- or two-register experiments on coset …
University of Oxford（牛津大学）；University College London（伦敦大学学院）；Shanghai Jiao Tong University（上海交通大学）
We show that DNF formulae can be quantum PAC-learned in polynomial time under product distributions using a quantum example oracle. The best classical algorithm (without access to membership queries) runs in superpolynomial time. Our result extends the work by Bshouty and Jackson (1998) that proved that DNF formulae are efficiently learnable under the uniform distribution using a quantum example oracle. Our proof is based on a new quantum algorithm that efficiently samples the coefficients of a μ-biased Fourier transform.
