Quantum Algorithm for Computing the Period Lattice of an Infrastructure

TL;DR

This paper introduces a quantum algorithm that efficiently computes the period lattice of infrastructures, especially from global fields, with improved sampling success probability and polynomial runtime in key parameters.

Contribution

It presents a novel quantum sampling method, provides a rigorous runtime analysis, and offers explicit success probability bounds for infrastructures from global fields.

Findings

Improved sampling method increases success probability exponentially.

Algorithm runs in polynomial time relative to the logarithm of the lattice determinant.

Efficient reduction to the abelian hidden subgroup problem for function field infrastructures.

Abstract

We present a quantum algorithm for computing the period lattice of infrastructures of fixed dimension. The algorithm applies to infrastructures that satisfy certain conditions. The latter are always fulfilled for infrastructures obtained from global fields, i.e., algebraic number fields and function fields with finite constant fields. The first of our main contributions is an exponentially better method for sampling approximations of vectors of the dual lattice of the period lattice than the methods outlined in the works of Hallgren and Schmidt and Vollmer. This new method improves the success probability by a factor of at least 2^{n^2-1} where n is the dimension. The second main contribution is a rigorous and complete proof that the running time of the algorithm is polynomial in the logarithm of the determinant of the period lattice and exponential in n. The third contribution is the determination of an explicit lower bound on the success probability of our algorithm which greatly improves on the bounds given in the above works. The exponential scaling seems inevitable because the best currently known methods for carrying out fundamental arithmetic operations in infrastructures obtained from algebraic number fields take exponential time. In contrast, the problem of computing the period lattice of infrastructures arising from function fields can be solved without the exponential dependence on the dimension n since this problem reduces efficiently to the abelian hidden subgroup problem. This is also true for other important computational problems in algebraic geometry. The running time of the best classical algorithms for infrastructures arising from global fields increases subexponentially with the determinant of the period lattice.