A new quantum computational set-up for algebraic topology via simplicial sets

  • Roberto Zucchini Department of Physics and Astronomy, University di Bologna, and INFN, Bologna Division, 40126 Bologna, Italy
Article ID: 3011
DOI: https://doi.org/10.59400/jam3011
Keywords: algebraic topology; simplicial sets; homology; computational topology; quantum information; quantum computation

Abstract

In this paper, a quantum computational framework for algebraic topology based on simplicial set theory is presented. This extends previous work, which was limited to simplicial complexes and aimed mostly at topological data analysis. The proposed set-up applies to any parafinite simplicial set and proceeds by associating with it a finite dimensional simplicial Hilbert space, whose simplicial operator structure is studied in some depth. It is shown in particular how the problem of determining the simplicial set’s homology can be solved within the simplicial Hilbert framework. Further, the conditions under which simplicial set theoretic algorithms can be implemented in a quantum computational setting with finite resources are examined. Finally a quantum algorithmic scheme capable of computing the simplicial homology spaces and Betti numbers of a simplicial set combining a number of basic quantum algorithms is outlined.

Published
2025-07-01
How to Cite
Zucchini, R. (2025). A new quantum computational set-up for algebraic topology via simplicial sets. Journal of AppliedMath, 3(4), 3011. https://doi.org/10.59400/jam3011
Issue
Vol. 3 No. 4 (2025)
Section
Review

Copyright (c) 2025 Author(s)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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