Constant-time Quantum Algorithm for Homology Detection in Closed Curves

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.

References in corpus (9)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Non-Abelian Anyons and Topological Quantum Computation
• Quantum algorithm for solving linear systems of equations
• Quantum random access memory
• The quest for a Quantum Neural Network
• Quantum Data Fitting
• Quantum-state preparation with universal gate decompositions
• A new quantum ripple-carry addition circuit
• Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits