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Quantum Information and Computation

Efficient quantum algorithms to find substructures on finite algebras

J. M. Hernández Cáceres, I. F. Rúa, Elías F. Combarro

Published Dec 1, 2023• 0 citations
Publisher

Abstract

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

DOI
10.26421/qic23.15-16-2

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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
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⋊φZpkG=A\rtimes_{\varphi} \mathbb{Z}_{p^k}G=A⋊φ​Zpk​, with AAA finite Abelian, ppp prime, k∈Nk\in \mathbb{N}k∈N and the action of Zpk\mathbb Z_{p^k}Zpk​ is generated by the scalar automorphism a↦μaa\mapstoμaa↦μa, for some μ∈ZExp⁡(A)×μ\in\mathbb Z_{\operatorname{Exp}(A)}^\timesμ∈ZExp(A)×​, where Exp(A)\mathrm{Exp}(A)Exp(A) is the exponent of AAA. Our algorithm is efficient when AAA has bounded generator rank and Exp(A)/p=polylog(∣G∣)\mathrm{Exp}(A)/p=\mathrm{polylog}(|G|)Exp(A)/p=polylog(∣G∣). This includes the case A=ZNA=\mathbb{Z}_NA=ZN​ for N∈NN\in\mathbb{N}N∈N and k=1k=1k=1, studied by Bacon, Childs and van Dam (FOCS 2005), and A=ZqrA=\mathbb{Z}_{q^r}A=Zqr​ with qqq prime and r∈Nr\in \mathbb{N}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).
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) is one of the most important problems in quantum computation. Many problems for which quantum algorithm achieves exponential speedup over its classical counterparts can be reduced to the Abelian HSP. However, there is no efficient quantum algorithm for solving the non-Abelian HSP. We find that the HSP can be reduced to a nested structured search problem that is solved efficiently by using a quantum algorithm via multistep quantum computation. Then we solve the HSP and problems that can be reduced to both the Abelian and the non-Abelian HSP in polynomial time by using this algorithm.
The Hidden Subgroup Problem (HSP) is a computational problem which includes as special cases integer factorization, the discrete logarithm problem, graph isomorphism, and the shortest vector problem. The celebrated polynomial-time quantum algorithms for factorization and the discrete logarithm are restricted versions of a generic polynomial-time quantum solution to the HSP for abelian groups, but despite focused research no polynomial-time solution for general groups has yet been found. We propose a generalization of the HSP to include arbitrary algebraic structures and analyze this new problem on powers of 2-element algebras. We prove a complete classification of every such power as quantum tractable (i.e. polynomial-time), classically tractable, quantum intractable, or classically intractable. In particular, we identify a class of algebras for which the generalized HSP exhibits super-polynomial speedup on a quantum computer compared to a classical one.
The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms are special cases of the HSP. In this paper we show that there exist a new efficient quantum algorithm for the HSP on groups ZN⋊Zqs\Z_{N}\rtimes\Z_{q^s}ZN​⋊Zqs​ where NNN is an integer with a special prime factorization, qqq prime number and sss any positive integer.
The Hidden Subgroup Problem (HSP) unifies several landmark quantum algorithms, yet systematic exploration of its variants and modern applications has slowed. This paper revives HSP-based algorithm design by examining new group structures with direct relevance to post-quantum cryptography, lattice problems, and structured optimization. We classify a family of finite and continuous groups-extensions and semi-direct products of abelian groups, matrix groups, and certain non-abelian groups-according to their Fourier-theoretic properties and the feasibility of efficient quantum Fourier transforms (QFTs). For each class, we determine whether the HSP admits an efficient quantum algorithm, a subexponential-time quantum algorithm, or appears intractable even quantumly. We construct explicit algorithms for eight new problem families, including structured lattice enumeration, automorphism groups of certain graphs, and algebraic attack primitives appearing in code-based and multivariate cryptosystems. For these, we show superpolynomial quantum speedups over all known classical approaches under standard complexity assumptions. We also discuss how approximate and block-encoded QFTs can be combined with variational post-processing to make some of these HSP algorithms more NISQ-friendly. Our results demonstrate that algebraic-structure-based quantum algorithms remain a rich and underexploited direction with both theoretical depth and potential cryptanalytic and optimization applications.