Two Problems on Quantum Computing in Finite Abelian Groups

TL;DR

This paper addresses two quantum computing problems in finite Abelian groups, including the well-known Hidden Subgroup Problem and a new Fully Balanced Image Problem, using Boolean conversion and phase-kick back techniques.

Contribution

It introduces solutions to the Fully Balanced Image Problem and applies quantum techniques to both problems, expanding quantum algorithms in finite Abelian group contexts.

Findings

Solved the Hidden Subgroup Problem using quantum algorithms.

Introduced the Fully Balanced Image Problem and provided a quantum solution.

Demonstrated the effectiveness of Boolean conversion and phase-kick back methods.

Abstract

In the context of finite Abelian groups two problems are presented and solved using quantum computing techniques. The first is the well--known Hidden Subgroup Problem, originally solved by Simon in a landmark work. The second is the Fully Balanced Image Problem, originally introduced by the authors (joint with J. Ossorio--Castillo), which is related to a certain class of mappings (which contains strictly, for instance, the family of group morphisms). Both problems are tackled using a combination of two techniques: first, a conversion into Boolean objects, better suited for quantum computing arguments, and subsequently a custom--tailored algorithm which takes advantage of the Generalised Phase--Kick Back technique.