Quantum Algorithm, Gaussian Sums, and Topological Invariants

TL;DR

This paper demonstrates that quantum computers can efficiently approximate certain topological invariants of three-manifolds, expressed as Gaussian sums, with potential applications in NMR techniques.

Contribution

It introduces quantum algorithms for approximating topological invariants related to gauge theories, extending applicability to various gauge groups.

Findings

Quantum algorithms efficiently approximate topological invariants.

Invariants can be estimated using NMR techniques.

Applicable to Abelian, finite, and some non-Abelian gauge groups.

Abstract

Certain quantum topological invariants of three manifolds can be written in the form of the Gaussian sum. It is shown that such topological invariants can be approximated efficiently by a quantum computer. The invariants discussed here are obtained as a partition function of the gauge theory on three manifolds with various gauge groups. Our algorithms are applicable to Abelian and finite gauge groups and to some classes of non-Abelian gauge groups. These invariants can be directly estimated by the nuclear magnetic resonance (NMR) technique used for evaluating the Gaussian sum.