Title

Quantum Algorithms For One-Dimensional Infrastructures

Keywords

Circumference; Discrete log problem; Infrastructures; Period finding; Quantum algorithms; Quantum fourier transform

Abstract

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. © Rinton Press.

Publication Date

1-1-2014

Publication Title

Quantum Information and Computation

Volume

14

Issue

1-2

Number of Pages

56-90

Document Type

Article

Personal Identifier

scopus

Socpus ID

84890405918 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84890405918

This document is currently not available here.

Share

COinS