Quantum algorithms for number fields

Abstract

This is a survey of recent results on quantum algorithms for the computation of invariants of number fields, namely the class number and the regulator. Most known classical algorithms for the computation of these values are of subexponential complexity and depend on the truth of a still unproven hypothesis of analytic number theory. We use an important number theoretic concept, Minkowski's Geometry of Numbers, to visualize these invariants, and describe the quantum algorithms developed by Hallgren, Schmidt and Vollmer which compute these invariants using a polynomial number of steps. Computational techniques in number fields, which are necessary to justify the classical part of these quantum algorithms, are the focus of the research of our project group, and are explained in detail.

Degradation of Grover’s search under collective phase flips in queries to the oracle

“…In the standard formulation, the initial distribution of amplitudes is taken in the form (3). Then the evolution of amplitudes can be described within the two-dimensional picture.…”

“…In last decades, quantum effects have found a new field of applications for information processing [1]. Famous Shor's discovery [2] had lead to numerous quantum algorithms for algebraic problems [3][4][5]. Grover's search algorithm [6][7][8] is another fundamental result.…”

On the role of dealing with quantum coherence in amplitude amplification

“…The Grover search algorithm [1][2][3] is one of fundamental discoveries that motivate quantum computations. Celebrated Shor's results [4] have led to numerous quantum algorithms for algebraic problems [5][6][7]. The authors of [8] gave arguments that Grover's and Shor's algorithms are more closely related than one might expect at first.…”