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Constant-time Quantum Algorithm for Homology Detection in Closed Curves
by Nhat A. Nghiem Vu, Xianfeng David Gu, Tzu-Chieh Wei
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Submission summary
| Authors (as registered SciPost users): | Nhat A. Nghiem Vu |
| Submission information | |
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| Preprint Link: | scipost_202302_00042v1 (pdf) |
| Date submitted: | 2023-02-27 19:51 |
| Submitted by: | Nghiem Vu, Nhat A. |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Given a loop or more generally 1-cycle $r$ 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 $r$. In contrast, classical algorithms take a linear time. 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.
List of changes
- We have corrected grammar mistakes that were pointed out by first reviewer
- We add further detail to explain why exact chains are closed
- We have additionally emphasized the significant improvement of our quantum algorithm with respect to query/ oracle usage