Difference sets are basic combinatorial structures that have applications in signal processing, coding theory, and cryptography. We consider the problem of identifying a shifted version of the characteristic function of a (known) difference set. We present a generic quantum algorithm that can be used to tackle any hidden shift problem for any difference set in any abelian group. We discuss special cases of this framework where the resulting quantum algorithm is efficient. This includes: a) Paley difference sets based on quadratic residues in finite fields, which allows to recover the shifted Legendre function quantum algorithm, b) Hadamard difference sets, which allows to recover the shifted bent function quantum algorithm, and c) Singer difference sets based on finite geometries. The latter class allows us to define instances of the dihedral hidden subgroup problem that can be efficiently solved on a quantum computer.
We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over $\mathbb{D}_{2^n}$. The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm using this approach has time complexity in $2^{\mathcal{O}(\sqrt{n})}$. By focusing on structure encoded in the codomain of the problem, we develop a polynomial-time algorithm which uses this structure to direct a "walk" down the subgroup lattice of $\mathbb{D}_{2^n}$ terminating at the hidden subgroup.
Quantum Fourier transformation is important in many quantum algorithms. In this paper, we generalize quantum Fourier transformation over the Abelian group $\mathbb{Z}_N$ from two different points to get more efficient unitary transformations. The obtained unitary transformations are given in concise and explicit formula which can be used directly. A relationship between the generalized quantum Fourier transformation and the dihedral hidden subgroup problem are discussed in this paper. This may lead a way to solve the dihedral hidden subgroup problem. The second goal of this paper is to give an…
We study the hardness of the dihedral hidden subgroup problem. It is known that lattice problems reduce to it, and that it reduces to random subset sum with density $> 1$ and also to quantum sampling subset sum solutions. We examine a decision version of the problem where the question asks whether the hidden subgroup is trivial or order two. The decision problem essentially asks if a given vector is in the span of all coset states. We approach this by first computing an explicit basis for the coset space and the perpendicular space. We then look at the consequences of having efficient unitarie…
In the theory of algebraic groups, parabolic subgroups form a crucial building block in the structural studies. In the case of general linear groups over a finite field $F_q$, given a sequence of positive integers $n_1, ..., n_k$, where $n=n_1+...+n_k$, a parabolic subgroup of parameter $(n_1, ..., n_k)$ in $GL_n(F_q)$ is a conjugate of the subgroup consisting of block lower triangular matrices where the $i$th block is of size $n_i$. Our main result is a quantum algorithm of time polynomial in $\log q$ and $n$ for solving the hidden subgroup problem in $GL_n(F_q)$, when the hidden subgroup is …
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.
The study of entanglement properties of multi-qubit states that are invariant under permutations of qubits is motivated by potential applications in quantum computing, quantum communication, and quantum metrology. In this work, we generalize the notions of symmetrization, Dicke states, and the Majorana representation to the alternating, cyclic, and dihedral subgroups of the full group of permutations. We use these tools to characterize states that are invariant under these subgroups and analyze their entanglement properties.
In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum group-based cryptography. We review the relationship between HSP and other computational problems discuss an optimal solution method, and review the known results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in such infinite group platforms are not yet known.
