Constant-time Quantum Algorithm for Homology Detection in Closed Curves

TL;DR

This paper introduces a quantum algorithm that detects homology of closed curves on surfaces in constant time, significantly outperforming classical methods that require linear or logarithmic time.

Contribution

The paper presents the first quantum algorithm for homology detection on surfaces that operates in constant time with respect to the loop size, using only a single oracle query.

Findings

Quantum homology detection runs in constant time.

Classical algorithms require linear or logarithmic time.

The quantum method extends to homology class comparison and homotopy detection.

Abstract

Given a loop or more generally 1-cycle $r$ of size L on a closed two-dimensional manifold or surface, represented by a triangulated mesh, a question in computational topology asks whether or not it is homologous to zero. We frame and tackle this problem in the quantum setting. Given an oracle that one can use to query the inclusion of edges on a closed curve, we design a quantum algorithm for such a homology detection with a constant running time, with respect to the size or the number of edges on the loop $r$, requiring only a single usage of oracle. In contrast, classical algorithm requires $\Omega(L)$ oracle usage, followed by a linear time processing and can be improved to logarithmic by using a parallel algorithm. Our quantum algorithm can be extended to check whether two closed loops belong to the same homology class. Furthermore, it can be applied to a specific problem in the homotopy detection, namely, checking whether two curves are \textit{not} homotopically equivalent on a closed two-dimensional manifold.