相关论文: The Hidden Subgroup Problem - Review and Open Prob…

The word problem for discrete groups is well-known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real…

In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of…

In this research notebook in the four-part, quantum computation and applications, quantum computation and algorithms, quantum communication protocol, and universal quantum computation for quantum engineers, researchers, and scientists, we…

A coherent mathematical overview of computation and its generalisations is described. This conceptual framework is sufficient to comfortably host a wide range of contemporary thinking on embodied computation and its models.

While efficient algorithms are known for solving many important problems related to groups, no efficient algorithm is known for determining whether two arbitrary groups are isomorphic. The particular case of 2-nilpotent groups, a special…

Simon's problem is to find a hidden period (a bitstring) encoded into an unknown 2-to-1 function. It is one of the earliest problems for which an exponential quantum speedup was proven for ideal, noiseless quantum computers, albeit in the…

We present efficient quantum algorithms for the hidden subgroup problem (HSP) on the semidirect product of cyclic groups $\Z_{p^r}\rtimes_{\phi}\Z_{p^2}$, where $p$ is any odd prime number and $r$ is any integer such that $r>4$. We also…

This chapter summarizes quantum computation, including the motivation for introducing quantum resources into computation and how quantum computation is done. Finally, this chapter articulates advantages and limitations of quantum…

Using an algebraic framework we solve a problem posed in [5] and [7] about the axiomatizability of a quantum computational type logic related to fuzzy logic. A Hilbert-style calculus is developed obtaining an algebraic strong completeness…

Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then…

We study the behaviors of quantum groups under an edge contraction. We show that there exists an explicit embedding induced by an edge contraction operation. We further conjecture that this explicit embedding is a section of an explicit…

We survey group-theoretic algorithms for finding (some or all) subgroups of a finite group and discuss the implementation of these algorithms in the computer algebra system GAP

A central goal in quantum computing is the development of quantum hardware and quantum algorithms in order to analyse challenging scientific and engineering problems. Research in quantum computation involves contributions from both physics…

The measurement process for hidden-configuration formulations of quantum mechanics is analysed. It is shown how a satisfactory description of quantum measurement can be given in this framework. The unified treatment of hidden-configuration…

Computations in the cohomology of finite groups.

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier…

This is a survey on the state-of-the-art of the classification of finite-dimensional complex Hopf algebras. This general question is addressed through the consideration of different classes of such Hopf algebras. Pointed Hopf algebras…

Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its…

We give an algorithm to solve the quantum hidden subgroup problem for maximal cyclic non-normal subgroups of the affine group of a finite field (if the field has order $q$ then the group has order $q(q-1)$) with probability $1-\varepsilon$…

We consider the hidden subgroup problem on the semi-direct product of cyclic groups $\Z_{N}\rtimes\Z_{p}$ with some restriction on $N$ and $p$. By using the homomorphic properties, we present a class of semi-direct product groups in which…