Daniel Simon's 1994 discovery of an efficient quantum algorithm for solving the hidden subgroup problem (HSP) over Z_2^n provided one of the first algebraic problems for which quantum computers are exponentially faster than their classical counterparts. In this paper, we study the generalization of Simon's problem to arbitrary groups. Fixing a finite group G, this is the problem of recovering an involution m = (m_1, ..., m_n) in G^n from an oracle f with the property that f(x) = f(xy) iff y equals m or the identity. In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form G^n, where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order two. Although groups of the form G^n have a simple product structure, they share important representation-theoretic properties with the symmetric groups S_n, where a solution to the HSP would yield a quantum algorithm for Graph Isomorphism. In particular, solving their HSP with the so-called ``standard method'' requires highly entangled measurements on the tensor product of many coset states. Here we give quantum algorithms with time complexity 2^O(sqrt(n log n)) that recover hidden involutions m = (m_1, ..., m_n) in G^n where, as in Simon's problem, each m_i is either the identity or the conjugate of a known element k, and there is a character X of G for which X(k) = -X(1)$. Our approach combines the general idea behind Kuperberg's sieve for dihedral groups with the ``missing harmonic'' approach of Moore and Russell. These are the first nontrivial hidden subgroup algorithms for group families that require highly entangled multiregister Fourier sampling.
Daniel Simon's 1994 discovery of an efficient quantum algorithm for solving the hidden subgroup problem (HSP) over Z_2^n provided one of the first algebraic problems for which quantum computers are exponentially faster than their classical counterparts. In this paper, we study the generalization of Simon's problem to arbitrary groups. Fixing a finite group G, this is the problem of recovering an involution m = (m_1, ..., m_n) in G^n from an oracle f with the property that f(x) = f(xy) iff y equals m or the identity. In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form G^n, where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order two. Although groups of the form G^n have a simple product structure, they share important representation-theoretic properties with the symmetric groups S_n, where a solution to the HSP would yield a quantum algorithm for Graph Isomorphism. In particular, solving their HSP with the so-called standard method'' requires highly entangled measurements on the tensor product of many coset states. Here we give quantum algorithms with time complexity 2^O(sqrt(n log n)) that recover hidden involutions m = (m_1, ..., m_n) in G^n where, as in Simon's problem, each m_i is either the identity or the conjugate of a known element k, and there is a character X of G for which X(k) = -X(1)$. Our approach combines the general idea behind Kuperberg's sieve for dihedral groups with the missing harmonic'' approach of Moore and Russell. These are the first nontrivial hidden subgroup algorithms for group families that require highly entangled multiregister Fourier sampling.
Daniel Simon's 1994 discovery of an efficient quantum algorithm for finding “hidden shifts” of Z 2 n provided the first algebraic problem for which quantum computers are exponentially faster than their classical counterparts. In this article, we study the generalization of Simon's problem to arbitrary groups. Fixing a finite group G , this is the problem of recovering an involution m = ( m 1 ,…, m n ) ∈ G n from an oracle f with the property that f ( x ⋅ y ) = f ( x ) ⇔ y ∈ {1, m }. In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form G n , where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order two. Although groups of the form G n have a simple product structure, they share important representation--theoretic properties with the symmetric groups S n , where a solution to the HSP would yield a quantum algorithm for Graph Isomorphism. In particular, solving their HSP with the so-called “standard method” requires highly entangled measurements on the tensor product of many coset states. In this article, we provide quantum algorithms with time complexity 2 O (√ n ) that recover hidden involutions m = ( m 1 ,… m n ) ∈ G n where, as in Simon's problem, each m i is either the identity or the conjugate of a known element m which satisfies κ( m ) = −κ(1) for some κ ∈ Ĝ . Our approach combines the general idea behind Kuperberg's sieve for dihedral groups with the “missing harmonic” approach of Moore and Russell. These are the first nontrivial HSP algorithms for group families that require highly entangled multiregister Fourier sampling.
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.
It is known that any quantum algorithm for graph isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across Ω(nlogn) coset states. One of the only known models for how such a measurement could be carried out efficiently is Kuperberg's algorithm for the HSP in the dihedral group, in which quantum states are adaptively combined and measured according to the decomposition of tensor products into irreducible representations. This “quantum sieve” starts with coset states and works its way down toward representations whose probabilities differ depending on, for example, whether the hidden subgroup is trivial or nontrivial. In this paper we show that no such approach can produce a polynomial-time quantum algorithm for graph isomorphism. Specifically, we consider the natural reduction of graph isomorphism to the HSP over the wreath product Sn≀Z2. Using a recently proved bound on the irreducible characters of Sn, we show that no algorithm in this family can solve graph isomorphism in less than eΩ(n) time, no matter what adaptive rule it uses to select and combine quantum states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time eO(nlogn).
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).
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...
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.