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 $1-varepsilon$ with (polylog) complexity $O(log(q)^{R}log(varepsilon)^{2})$ where $R<infty.$
Let $pi_1(C)$ be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic $p>0$ of countable cardinality. Let $N$ be a normal (resp. characteristic) subgroup of $pi_1(C)$. Under the hypothesis that the quotient $pi_1(C)/N$ admits an infinitely generated Sylow $p$-subgroup, we prove that $N$ is indeed isomorphic to a normal (resp. characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of $N$ is a free profinite group of countable cardinality.
We characterize the algebraic structure of semi-direct product of cyclic groups, $Z_{N}rtimesZ_{p}$, where $p$ is an odd prime number which does not divide $q-1$ for any prime factor $q$ of $N$, and provide a polynomial-time quantum computational algorithm solving hidden symmetry subgroup problem of the groups.
We report on the existence of the phenomenon of sudden birth of maximal hidden quantum correlations in open quantum systems. Specifically, we consider the CHSH-inequality for Bell-nonlocality, the $rm F_3$-inequality for EPR-steering, and usefulness for teleportation as quantum correlations in a bipartite qubit system under collective decoherence due to an environment. We show that, even though the system may undergo a sudden birth of entanglement, neither of the aforementioned correlations are present and yet, these can still be revealed by means of local filtering operations, thus evidencing the presence of hidden quantum correlations. Furthermore, there are extrem
The orbifold group of the Borromean rings with singular angle 90 degrees, $U$, is a universal group, because every closed oriented 3--manifold $M^{3}$ occurs as a quotient space $M^{3} = H^{3}/G$, where $G$ is a finite index subgroup of $U$. Therefore, an interesting, but quite difficult problem, is to classify the finite index subgroups of the universal group $U$. One of the purposes of this paper is to begin this classification. In particular we analyze the classification of the finite index subgroups of $U$ that are generated by rotations.
Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results from the bottom level up by determining completely the non-cyclic finite p-groups whose number of subgroups among p-groups of a given order is minimal.
Nolan Wallach
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(2013)
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"A quantum polylog algorithm for non-normal maximal cyclic hidden subgroups in the affine group of a finite field"
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Nolan Wallach
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