The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism. We fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined. We show, however, that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism.
The Hidden Subgroup Problem is the foundation of many quantum algorithms.An efficient solution is known for the problem over Abelian groups and this was used in Simon's algorithm and Shor's Factoring and Discrete Log algorithms.The non-Abelian case is open; an efficient solution would give rise to an efficient quantum algorithm for Graph Isomorphism.We fully analyze a natural generalization of the Abelian case solution to the non-Abelian case, and give an efficient solution to the problem for normal subgroups.We show, however, that this immediate generalization of the Abelian algorithm does not efficiently solve Graph Isomorphism.
The Hidden Subgroup Problem is used in many quantum algorithms such as Simon's algorithm and Shor's factoring and discrete log algorithms. A polynomial time solution is known in case of abelian groups, and normal subgroups of arbitrary finite groups. The general case is still open. An efficient solution of the problem for symmetric group
Sn would give rise to an efficient quantum algorithm for Graph Isomorphism Problem. We formulate a hidden sub-hypergroup problem for finite hypergroups and solve it for finite commutative hypergroups. The given algorithm is efficient if the corresponding QFT could be calculated efficiently.
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⋊φZpk, with
A finite Abelian,
p prime,
k∈N and the action of
Zpk is generated by the scalar automorphism
a↦μa, for some
μ∈ZExp(A)×, where
Exp(A) is the exponent of
A. Our algorithm is efficient when
A has bounded generator rank and
Exp(A)/p=polylog(∣G∣). This includes the case
A=ZN for
N∈N and
k=1, studied by Bacon, Childs and van Dam (FOCS 2005), and
A=Zqr with
q prime and
r∈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).
After Shor's discovery of an efficient quantum algorithm for integer factoring, hidden subgroup problems play a central role in developing efficient quantum algorithms. In spite of many intensive studies, no efficient quantum algorithms are known for hidden subgroup problems for many non-Abelian groups. Of particular interest are the hidden subgroup problems for the symmetric group and for the dihedral group, because an efficient algorithm for the former implies an efficient solution to the graph isomorphism problem, and that for the latter essentially solves a certain lattice-related problem whose hardness is assumed in cryptography. This paper focuses on the latter case and gives a comprehensive survey of known facts related to the dihedral hidden subgroup problem.
We study quantum algorithms for the hidden subgroup problem (HSP). The HSP provides a unified framework for many important problems, including factoring integers, graph isomorphism problem, etc. It has been known that the problem for abelian groups can be efficiently solved. However, for non-abelian groups, it remains widely open. In this dissertation we study the problem for some specific non-abelian groups, called q-hedral groups.
We start with a quantum algorithm, proposed by Moore et al. (2003), for finding some hidden conjugate subgroups of affine groups, a special case of q-hedral groups for large q. We then adapt their algorithm to construct an algorithm for solving the hidden conjugate problem for q-hedral groups for large q and point out a difficulty in complexity analysis that involves certain exponential sums. We discuss certain bounds on these sums and present some computational data on their values to make a conjecture.
For small q, Regev (2003) gave an efficient algorithm for solving the HSP for dihedral groups (q = 2) using an oracle that solves a certain subset sum problem (SSP). A crucial step of his algorithm relies on the probability that elements in a cyclic group ZN can be expressed as sums of random elements in the group. We generalize this problem to any finite group and prove the following theorem: for any finite group G and r ≥ 32 log2 |G| + 5, if r elements from G are chosen randomly, then the 2r possible sums (or products) of the r elements will cover all the elements in G with high probability. Also, by making a further generalization of the SSP, we show that Regev's algorithm can be generalized for q-hedral groups for any fixed q. We also study this generalized subset sum problem.
In addition to the algorithm developing for small q, we give an efficient quantum algorithm for solving the coset problem and an efficient quantum algorithm for solving the hidden conjugate problem. Furthermore, we also give a survey of an algorithm for finding the largest normal subgroup in a hidden subgroup of any finite group, introduced by Hallgren et al. (2000) and provide a simplified and unified proof.
The Hidden Subgroup Problem (HSP) provides the fundamental framework for most quantum algorithms. Until very recently, all known problems where quantum computation provides a super-polynomial speedup over classical algorithms have been variants of the HSP for particular abelian groups. Examples include factoring integers and computing the discrete log, for which Shor found efficient quantum algorithms. The algorithm for these problems may be viewed as consisting of the main HSP solution plus an ad hoc method of dealing with the particular variant. In this dissertation, we give a systematic way of dealing with all these variants. The key component of our solution is a new theorem about the robustness of Fourier sampling. The purpose of this theorem is to better understand the structure underlying existing algorithms as well as to provide an algorithmic tool for the construction of future quantum algorithms. In addition, we also derive a new algorithm for computing the quantum Fourier transform which is asymptotically faster than any previously known algorithm. By contrast, the nonabelian HSP is wide open. It includes as a special case the longstanding open question of graph isomorphism, where the group is the symmetric group, S n . It is natural to carry over the abelian HSP algorithm to the nonabelian case. We show that in the case that the hidden subgroup is normal, this algorithm succeeds in reconstructing the subgroup in polynomial time. On the other hand, we give evidence that this algorithm is inadequate to even distinguish a trivial subgroup from an involution in t...