scholarly journals Quantum algorithms for one-dimensional infrastructures

2014 ◽  
Vol 14 (1&2) ◽  
pp. 56-90
Author(s):  
Pradeep Sarvepalli ◽  
Pawel M. Wocjan
Keyword(s):  
Quantum Algorithms ◽  
Function Fields ◽  
Success Probability ◽  
Number Fields ◽  
Fourth Power ◽  
Quadratic Number ◽  
Problem Size ◽  
One Dimensional ◽  
Quadratic Number Fields ◽  
And Function

Infrastructures are group-like objects that make their appearance in arithmetic geometry in the study of computational problems related to number fields and function fields over finite fields. The most prominent computational tasks of infrastructures are the computation of the circumference of the infrastructure and the generalized discrete logarithms. Both these problems are not known to have efficient classical algorithms for an arbitrary infrastructure. Our main contributions are polynomial time quantum algorithms for one-dimensional infrastructures that satisfy certain conditions. For instance, these conditions are always fulfilled for infrastructures obtained from number fields and function fields, both of unit rank one. Since quadratic number fields give rise to such infrastructures, this algorithm can be used to solve Pell's equation and the principal ideal problem. In this sense we generalize Hallgren's quantum algorithms for quadratic number fields, while also providing a polynomial speedup over them. Our more general approach shows that these quantum algorithms can also be applied to infrastructures obtained from complex cubic and totally complex quartic number fields. Our improved way of analyzing the performance makes it possible to show that these algorithms succeed with constant probability independent of the problem size. In contrast, the lower bound on the success probability due to Hallgren decreases as the fourth power of the logarithm of the circumference. Our analysis also shows that fewer qubits are required. We also contribute to the study of infrastructures, and show how to compute efficiently within infrastructures.

scholarly journals Non-abelian Cohen–Lenstra heuristics over function fields

2017 ◽  
Vol 153 (7) ◽  
pp. 1372-1390 ◽  
Author(s):  
Nigel Boston ◽  
Melanie Matchett Wood
Keyword(s):  
Function Fields ◽  
Quadratic Function ◽  
Number Fields ◽  
Galois Groups ◽  
Quadratic Number ◽  
Quadratic Extensions ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Unique Distribution ◽  
Quadratic Function Fields

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-$p$extensions of imaginary quadratic number fields for$p$an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of$\mathbb{F}_{q}(t)$, the Galois groups of the maximal unramified pro-$p$extensions, as$q\rightarrow \infty$, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

scholarly journals On quaternion algebras over the composite of quadratic number fields

2021 ◽  
Vol 56 (1) ◽  
pp. 63-78
Author(s):  
Vincenzo Acciaro ◽  
◽  
Diana Savin ◽  
Mohammed Taous ◽  
Abdelkader Zekhnini ◽  
...  
Keyword(s):  
Number Fields ◽  
Complete Characterization ◽  
Quadratic Number ◽  
Quaternion Algebras ◽  
Quadratic Number Fields

Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.

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