发表机构
Joint Center for Quantum Information and Computer Science (QuICS), University of Maryland, USA; Diraq, Sydney, New South Wales, Australia（马里兰大学量子信息与计算科学联合中心; 迪拉克（悉尼，新南威尔士州））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对两类含非阿贝尔群的族，即特定半直积群与满足温和输入假设的有限拟哈密顿群，给出了求解隐子群问题的多项式时间量子算法。
AI 中文摘要
若干早期量子算法，包括Simon算法与Shor周期发现算法，均是有限阿贝尔群上隐子群问题（HSP）的实例。目前尚未发现针对任意有限非阿贝尔群HSP的多项式时间量子算法。非阿贝尔情形备受关注，因为部分实例（如二面体群HSP、对称群HSP）分别与格问题、图同构问题相关。本研究针对两类含非阿贝尔群的族给出多项式时间量子算法：第一类为形如$G=A\rtimes_{\varphi} \mathbb{Z}_{p^k}$的群，其中$A$为有限阿贝尔群，$p$为素数，$k\in\mathbb{N}$，且$\mathbb{Z}_{p^k}$的作用由标量自同构$a\mapsto\mu a$生成，$\mu\in\mathbb{Z}_{\operatorname{Exp}(A)}^\times$，$\operatorname{Exp}(A)$为$A$的指数。当$A$具有有界生成元秩且满足$\operatorname{Exp}(A)/p=\operatorname{polylog}(|G|)$时，该算法高效，涵盖Bacon、Childs与van Dam（FOCS 2005）研究的$A=\mathbb{Z}_N$（$N\in\mathbb{N}$）且$k=1$的情形，以及van Dam与Dey（TQC 2014）研究的$A=\mathbb{Z}_{q^r}$（$q$为素数，$r\in\mathbb{N}$）的情形；第二类在输入结构的温和假设下，针对有限拟哈密顿群给出多项式时间量子算法，拟哈密顿群是具有模子群格的有限幂零群，等价于每个子群均可置换的有限群。据作者所知，这是首个利用子群格的模性求解HSP的量子算法，在上述结构化输入假设下，扩展了Hallgren、Russell与Ta-Shma（SIAM J. Comput. 32, 2003）给出的戴德金群量子算法。
英文摘要
Several early quantum algorithms, including Simon's algorithm and Shor's period-finding are instances of the hidden subgroup problem (HSP) over finite abelian groups. No polynomial-time quantum algorithm is known for the HSP over arbitrary non-abelian finite groups. The non-Abelian case is of particular interest because some instances, such as the dihedral and symmetric group HSPs, are connected to lattice problems and graph isomorphism, respectively. In this work, we give polynomial-time quantum algorithms for two further families containing non-Abelian groups. First, we consider groups of the form $G=A\rtimes_φ \mathbb{Z}_{p^k}$, with $A$ finite Abelian, $p$ prime, $k\in \mathbb{N}$ and the action of $\mathbb Z_{p^k}$ is generated by the scalar automorphism $a\mapstoμa$, for some $μ\in\mathbb Z_{\operatorname{Exp}(A)}^\times$, where $\mathrm{Exp}(A)$ is the exponent of $A$. Our algorithm is efficient when $A$ has bounded generator rank and $\mathrm{Exp}(A)/p=\mathrm{polylog}(|G|)$. This includes the case $A=\mathbb{Z}_N$ for $N\in\mathbb{N}$ and $k=1$, studied by Bacon, Childs and van Dam (FOCS 2005), and $A=\mathbb{Z}_{q^r}$ with $q$ prime and $r\in \mathbb{N}$ studied by van Dam and Dey (TQC 2014). Second, we give a polynomial-time quantum algorithm for finite quasi-Hamiltonian groups under a mild assumption on the input structure. Quasi-Hamiltonian groups are finite nilpotent groups with modular subgroup lattice, or equivalently the finite groups in which every subgroup is permutable. As far as we know, this is the first quantum algorithm to exploit the modularity of the subgroup lattice for solving the HSP. This extends, under the aforementioned structured input assumption, the quantum algorithm for Dedekind groups given by Hallgren, Russell, and Ta-Shma (SIAM J. Comput. 32, 2003).
Comments30+9 pages, 1 figure
