We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we present an exact quantum algorithm for the finite AHSP, our algorithm is more concise than the previous exact algorithm and applies to any finite Abelian group. Second, utilizing the Chinese Remainder Theorem, we propose a distributed exact quantum algorithm for finite AHSP, which requires fewer qudits, lower quantum query complexity, and no quantum communication. We further show that our distributed approach can be extended to certain classes of non-Abelian groups. Finally, we develop a parallel exact classical algorithm for finite AHSP with reduced query complexity; even without parallel execution, the total number of queries across all nodes does not exceed that of the original centralized algorithm under mild conditions.
We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we present an exact quantum algorithm for the finite AHSP, our algorithm is more concise than the previous exact algorithm and applies to any finite Abelian group. Second, utilizing the Chinese Remainder Theorem, we propose a distributed exact quantum algorithm for finite AHSP, which requires fewer qudits, lower quantum query complexity, and no quantum communication. We further show that our distributed approach can be extended to certain classes of non-Abelian groups. Finally, we develop a parallel exact classical algorithm for finite AHSP with reduced query complexity; even without parallel execution, the total number of queries across all nodes does not exceed that of the original centralized algorithm under mild conditions.
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.
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).
Abstract Simon’s problem is one of the most important problems demonstrating the power of quantum algorithms, as it greatly inspired the proposal of Shor’s algorithm. The generalized Simon’s problem is a natural extension of Simon’s problem, and also a special hidden subgroup problem: Given a function f : {0, 1} n → {0, 1} m , with the property that for any x, y ∈ {0, 1} n , there is some unknown hidden subgroup S ≤ Z n 2 such that f (x) = f (y) iff x ⊕ y ∈ S, where |S| = 2 k for some 0 ≤ k ≤ n (m ≥ n − k). The goal of generalized Simon’s problem is to find the hidden subgroup S. In this paper, we present two key contributions. Firstly, we characterize the structure of the generalized Simon’s problem in distributed scenario and introduce a corresponding distributed quantum algorithm. Secondly, we refine the algorithm to ensure exactness due to the application of quantum amplitude amplification technique. Our algorithm offers exponential acceleration compared to the distributed classical algorithm. When contrasted with the centralized quantum algorithm for the generalized Simon’s problem, our algorithm’s oracle requires fewer qubits, thus making it easier to be physically implemented. Particulary, the exact distributed quantum algorithm we develop for the generalized Simon’s problem outperforms the best previously proposed distributed quantum algorithm for Simon’s problem in terms of generalizability and exactness.
The ultimate objective of this paper is to create a stepping stone to the development of new quantum algorithms. The strategy chosen is to begin by focusing on the class of abelian quantum hidden subgroup algorithms, i.e., the class of abelian algorithms of the Shor/Simon genre. Our strategy is to make this class of algorithms as mathematically transparent as possible. By the phrase "mathematically transparent" we mean to expose, to bring to the surface, and to make explicit the concealed mathematical structures that are inherently and fundamentally a part of such algorithms. In so doing, we create symbolic abelian quantum hidden subgroup algorithms that are analogous to the those symbolic algorithms found within such software packages as Axiom, Cayley, Maple, Mathematica, and Magma. As a spin-off of this effort, we create three different generalizations of Shor's quantum factoring algorithm to free abelian groups of finite rank. We refer to these algorithms as wandering (or vintage Z_Q) Shor algorithms. They are essentially quantum algorithms on free abelian groups A of finite rank n which, with each iteration, first select a random cyclic direct summand Z of the group A and then apply one iteration of the standard Shor algorithm to produce a random character of the "approximating" finite group A=Z_Q, called the group probe. These characters are then in turn used to find either the order P of a maximal cyclic subgroup Z_P of the hidden quotient group H_phi, or the entire hidden quotient group H_phi. An integral part of these wandering quantum algorithms is the selection of a very special random transversal, which we refer to as a Shor transversal. The algorithmic time complexity of the first of these wandering Shor algorithms is found to be O(n^2(lgQ)^3(lglgQ)^(n+1)).
We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely~$ \Order(\sqrt{ n}\, ),wheren$ is the order of the group. The analogous algorithm for non-abelian groups comes within a~logn factor of the optimal randomized query complexity. The best known randomized algorithm for the Hidden Subgroup Problem has \emph{expected\/} query complexity that is sensitive to the input, namely~$ \Order(\sqrt{ n/m}\, ),wherem$ is the order of the hidden subgroup. In the first version of this article~\cite[Sec.~5]{Nayak21-hsp-classical}, we asked if there is a deterministic algorithm whose query complexity has a similar dependence on the order of the hidden subgroup. Prompted by this question, Ye and Li~\cite{YL21-hsp-classical} present deterministic algorithms for \emph{abelian\/} groups which solve the problem with~$ \Order(\sqrt{ n/m }\, )$ queries, and find the hidden subgroup with~$ \Order( \sqrt{ n (\log m) / m} + \log m ) $ queries. Moreover, they exhibit instances which show that in general, the deterministic query complexity of the problem may be~$\order(\sqrt{ n/m } \,)$, and that of \emph{finding\/} the entire subgroup may also be~$\order(\sqrt{ n/m } \,)$ or even~$\upomega(\sqrt{ n/m } \,) $.}We present a different deterministic algorithm for the Hidden Subgroup Problem that also has query complexity~$ \Order(\sqrt{ n/m }\, )$ for abelian groups. The algorithm is arguably simpler. Moreover, it works for non-abelian groups, and has query complexity~$ \Order(\sqrt{ (n/m) \log (n/m) }\,) $ for a large class of instances, such as those over supersolvable groups. We build on this to design deterministic algorithms to find the hidden subgroup for all abelian and some non-abelian instances, at the cost of a~logm multiplicative factor increase in the query complexity.
When classifying a collection of finite algebras (for instance, in the computational classification of finite semifields), an important task is the determination of substructures such as the right, middle, and left nuclei, the nucleus, and the center. Finding these structures may become computationally expensive when there is no additional information about the algebra properties. In this paper, we introduce quantum algorithms than solve this task efficiently, by formulating it as an instance of the Hidden Subgroup Problem (HSP) {over Abelian groups}. We give detailed constructions of the quantum circuits involved in the process and prove that the overall (quantum) complexity of our algorithm is polynomial in the dimension of the algebra, while a similar approach with classical computers would require an exponential number of queries to the HSP function