Hidden translation and orbit coset in quantum computing

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called $${\mathcal{PT}}$$ -groups. In particular, it is shown that the finite solvable $${\mathcal{PT}}$$ -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not $${\mathcal{PT}}$$ -groups but all subnormal subgroups of defect two are permutable.

In this brief chapter we present a quantum algorithm, which can be seen as a generalization of Shor’s algorithm. We can here explain, in a more structural manner, why quantum computers can speed up some computational problems. The so-called Simon’s hidden subgroup problem can be stated in a general form as follows [18]: Input: A finite abelian group G and function ρ: G → R, where R is a finite set. Promise: There exists a nontrivial subgroup H ≤ G such that ρ is constant and distinct on each coset of H. Output: A generating set for H.

Let A be a finite group and let n be the number of primes, including multiplicities, which divide the order of A. Suppose further that G is a finite solvable group which admits A as a fixed-point-free operator group and such that (i A I, G|I ) = 1. Under these assumptions it has been conjectured that l(G), the nilpotent length of G, is at most n, while lp(G), the p-length of G, is at most [(n+1)/2]. These inequalities have been verified and shown to be best-possible in certain special cases ( [3 ][5], [1]). It is the purpose of this note to show that these conjectured upper bounds are actually obtained provided A is solvable. Specifically we will prove

In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum cryptography. We review the relationship between HSP and other computational problems, discuss an optimal solution method, and review results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in the proposed infinite group platforms are not yet known.

Regular and p-regular orbits of solvable linear groups

It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was...

Several authors have recently studied the problem of finding circumstances under which the amenability of a semigroup is inherited by a subsemigroup. It is known ([2] and [3]) that subgroups of an amenable group are amenable, but that in general subsemigroups of an amenable semigroup are not. We show here that subsemigroups of an amenbable group need not be amenable. Frey proved in [4] that subsemigroups of a group inherit amenability and the group contains no copy of the free semigroup S on two generators (S is not amenable). The problem is discussed in detail in [51. It is known [11 that a group generated by a copy of S contained in it need not be free; the question is whether it can be far enough from free to be amenable. Since solvable groups are amenable [6], our claim that subsemigroups of amenable groups need not be amenable will be established we can embed S in a solvable group. We proceed to do this. Granirer conjectured in his review [MR 35 #3423] of Wilde and Witz [7] that a group is amenable and it contains no copy of S. It will follow that the only if part of this conjecture is quite false. Let B be the free additive abelian group on the family of generators {b(m, n) }, m, n integers. Any permutation of these generators induces an automorphism of B. Let x and y be the automorphisms induced by b(m, n)F->b(m+1, n) and b(m, n)'->b(m, n+1), respectively. It is easy to see that the subgroup C of Aut B generated by x and y is the free abelian group on the generators x, y. We write C additively, and we indicate the action of an element c of C on an element b of B by be. Let be the normal product of B by C. consists of pairs [c, b] and the multiplication is given by -[c, b1i [c, b] [cl+c, b1+b]. c1->[c, 0], bF-*[O, b] are embeddings of C, B respectively into G, and B is thus a normal abelian subgroup with quotient G/B-C. Hence, is solvable and in fact the second derived group G of is trivial. The same holds for every subgroup of G. Now, let s= [x, b(O, 0)] and t= [y, b(0, 0)].

Let \(G\) be a finite solvable group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). A fundamental result by P.P. Palfy asserts that the complement $\bar{\Delta}(G)$ of the graph \(\Delta(G)\) does not contain any cycle of length \(3\). In this paper we generalize Palfy's result, showing that $\bar{\Delta}(G)$ does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of \(\Delta(G)\) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if \(n\) is the clique number of \(\Delta(G)\), then \(\Delta(G)\) has at most \(2n\) vertices. This confirms a conjecture by Z. Akhlaghi and H.P. Tong-Viet, and provides some evidence for the famous \emph{\(\rho\)-\(\sigma\) conjecture} by B. Huppert.

Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.

Minimal sumsets in finite solvable groups

Solvable and nonsolvable finite groups of the same order type Pawe PiwekWe construct two groups of size 2 365 3 105 7 104 : a solvable group G and a nonsolvable group H such that for every integer n the groups have the same number of elements of order n.This answers a question posed in 1987 by John G. Thompson.

Let G be a finite group. I. M. Isaacs and G. Seitz have conjectured that if G is a solvable group, then Taketa’s inequality holds, where is the set of irreducible character degrees of G and is the derived length of G. In this paper, we show that this inequality holds if G is a solvable Frobenius group. Also, we prove that if for some normal Sylow p-subgroup P of G, then . Moreover, we investigate this conjecture for some sets of the real-valued irreducible character degrees.

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we generalize a result of J. G. Thompson and characterize finite solvable groups as those finite groups which do not contain nontrivial elements xi , i = 1,2,3, with x 1 x 2 x 3 = 1 and xi a pi -element for distinct primes pi . We note some connections with coverings of curves.

Finite Solvable Groups in Which Only Two Nonlinear Irreducible Characters Have Equal Degrees