An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups

TL;DR

This paper presents a quantum algorithm that efficiently solves the hidden subgroup problem over Weyl-Heisenberg groups by leveraging non-commutative Fourier analysis and innovative representation techniques, reducing resource requirements.

Contribution

The authors introduce a novel quantum algorithm that improves efficiency by operating on two coset states simultaneously and employs new methods for handling irreducible representations.

Findings

Efficient quantum solution for HSP over Weyl-Heisenberg groups.

Reduction in required coset states from four to two per iteration.

Application of new label-changing technique for low-dimensional irreducible representations.

Abstract

Many exponential speedups that have been achieved in quantum computing are obtained via hidden subgroup problems (HSPs). We show that the HSP over Weyl-Heisenberg groups can be solved efficiently on a quantum computer. These groups are well-known in physics and play an important role in the theory of quantum error-correcting codes. Our algorithm is based on non-commutative Fourier analysis of coset states which are quantum states that arise from a given black-box function. We use Clebsch-Gordan decompositions to combine and reduce tensor products of irreducible representations. Furthermore, we use a new technique of changing labels of irreducible representations to obtain low-dimensional irreducible representations in the decomposition process. A feature of the presented algorithm is that in each iteration of the algorithm the quantum computer operates on two coset states simultaneously. This is an improvement over the previously best known quantum algorithm for these groups which required four coset states.