QUANTUM ALGORITHMS FOR ONE-DIMENSIONAL INFRASTRUCTURES

    Authors

    P. Sarvepalli;P. Wocjan

    Comments

    Authors: contact us about adding a copy of your work at STARS@ucf.edu

    Abbreviated Journal Title

    Quantum Inform. Comput.

    Keywords

    quantum algorithms; infrastructures; circumference; period finding; quantum Fourier transform; discrete log problem; DISCRETE LOGARITHMS; PELLS EQUATION; NUMBER-FIELD; Computer Science, Theory & Methods; Physics, Particles & Fields; Physics, Mathematical

    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.

    Journal Title

    Quantum Information & Computation

    Volume

    14

    Issue/Number

    1-2

    Publication Date

    1-1-2014

    Document Type

    Article

    Language

    English

    First Page

    56

    Last Page

    90

    WOS Identifier

    WOS:000333067900004

    ISSN

    1533-7146

    Share

    COinS