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

TL;DR

This paper introduces a quantum computational framework for algebraic topology using simplicial sets, enabling efficient homology computation and Betti number determination within quantum resource constraints.

Contribution

It extends previous simplicial complex methods to simplicial sets and develops a quantum algorithmic scheme for computing their homology and Betti numbers.

Findings

Quantum framework for simplicial sets developed

Homology computation implemented within quantum setting

Algorithmic scheme for Betti numbers outlined

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 to 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 to compute the simplicial homology spaces and Betti numbers of a simplicial set combining a number of basic quantum algorithms is outlined.