# Quantum Distance Approximation for Persistence Diagrams

arXiv: https://arxiv.org/abs/2402.17295 · HTML: https://mlsanity.com/paper/2402.17295/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 62 references that are in the map.
- First posted on arXiv 27 Feb 2024.

## Builds on (most cited first, 11 of 11)

- [q-means: A quantum algorithm for unsupervised machine learning](https://mlsanity.com/paper/1812.03584/) (2018)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [Persistent Laplacians: properties, algorithms and implications](https://mlsanity.com/paper/2012.02808/) (2020)
- [Persistent Homology of Attractors For Action Recognition](https://mlsanity.com/paper/1603.05310/) (2016)
- [Computing Wasserstein Distance for Persistence Diagrams on a Quantum Computer](https://mlsanity.com/paper/1809.06433/) (2018)
- [Bayesian Topological Learning for Classifying the Structure of Biological Networks](https://mlsanity.com/paper/2009.11974/) (2020)
- [A Random Persistence Diagram Generator](https://mlsanity.com/paper/2104.07737/) (2021)
- [Quantum Topological Data Analysis with Linear Depth and Exponential Speedup](https://mlsanity.com/paper/2108.02811/) (2021)
- [Quantum Persistent Homology for Time Series](https://mlsanity.com/paper/2211.04465/) (2022)
- [Persistent Homology in Sparse Regression and its Application to Brain Morphometry](https://mlsanity.com/paper/1409.0177/) (2014)
- [Applying the Quantum Alternating Operator Ansatz to the Graph Matching Problem](https://mlsanity.com/paper/2011.11918/) (2020)

## Built on by (most cited first, 1 of 1)

- [Lipschitz Bounds for Persistent Laplacian Eigenvalues under One-Simplex Insertions](https://mlsanity.com/paper/2506.21352/) (2025)

## Abstract (arXiv)

Topological Data Analysis methods can be useful for classification and clustering tasks in many different fields as they can provide two dimensional persistence diagrams that summarize important information about the shape of potentially complex and high dimensional data sets. The space of persistence diagrams can be endowed with various metrics such as the Wasserstein distance which admit a statistical structure and allow to use these summaries for machine learning algorithms. However, computing the distance between two persistence diagrams involves finding an optimal way to match the points of the two diagrams and may not always be an easy task for classical computers. In this work we explore the potential of quantum computers to estimate the distance between persistence diagrams, in particular we propose variational quantum algorithms for the Wasserstein distance as well as the dcp distance. Our implementation is a weighted version of the Quantum Approximate Optimization Algorithm that relies on control clauses to encode the constraints of the optimization problem.