TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.15.2.049
TI  - Constant-time quantum algorithm for homology detection in closed curves
PY  - 2023/08/04
UR  - https://scipost.org/SciPostPhys.15.2.049
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 15
IS  - 2
SP  - 049
A1  - Nghiem Vu, Nhat Anh
AU  - Gu, Xianfeng David
AU  - Wei, Tzu-Chieh
AB  - 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 not homotopically equivalent on a closed two-dimensional manifold.
ER  -

@Article{10.21468/SciPostPhys.15.2.049,
	title={{Constant-time quantum algorithm for homology detection in closed curves}},
	author={Nhat A. Nghiem Vu and Xianfeng David Gu and Tzu-Chieh Wei},
	journal={SciPost Phys.},
	volume={15},
	pages={049},
	year={2023},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.15.2.049},
	url={https://scipost.org/10.21468/SciPostPhys.15.2.049},
}