Quantum algorithm for persistent Betti numbers and topological data analysis
Abstract
Topological data analysis (TDA) is an emergent field of data analysis. The critical step of TDA is computing the persistent Betti numbers. Existing classical algorithms for TDA are limited if we want to learn from highdimensional topological features because the number of highdimensional simplices grows exponentially in the size of the data. In the context of quantum computation, it has been previously shown that there exists an efficient quantum algorithm for estimating the Betti numbers even in high dimensions. However, the Betti numbers are less general than the persistent Betti numbers, and there have been no quantum algorithms that can estimate the persistent Betti numbers of arbitrary dimensions. This paper shows the first quantum algorithm that can estimate the (normalized) persistent Betti numbers of arbitrary dimensions. Our algorithm is efficient for simplicial complexes such as the VietorisRips complex and demonstrates exponential speedup over the known classical algorithms.
 Publication:

Quantum
 Pub Date:
 December 2022
 DOI:
 10.22331/q20221207873
 arXiv:
 arXiv:2111.00433
 Bibcode:
 2022Quant...6..873H
 Keywords:

 Quantum Physics
 EPrint:
 27 pages