# Higher-order topological kernels via quantum computation

arXiv: https://arxiv.org/abs/2307.07383 · HTML: https://mlsanity.com/paper/2307.07383/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 36 references that are in the map.
- First posted on arXiv 14 Jul 2023.

## Builds on (most cited first, 15 of 15)

- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Quantum anomaly detection in the latent space of proton collision events at the LHC](https://mlsanity.com/paper/2301.10780/) (2023)
- [Topological Machine Learning with Persistence Indicator Functions](https://mlsanity.com/paper/1907.13496/) (2019)
- [Towards quantum advantage via topological data analysis](https://mlsanity.com/paper/2005.02607/) (2020)
- [Topological data analysis on noisy quantum computers](https://mlsanity.com/paper/2209.09371/) (2022)
- [Quantum Persistent Homology for Time Series](https://mlsanity.com/paper/2211.04465/) (2022)
- [Review of a Quantum Algorithm for Betti Numbers](https://mlsanity.com/paper/1906.07673/) (2019)

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

- [Quantum Topological Data Encoding](https://mlsanity.com/paper/2607.13847/) (2026)

## Abstract (arXiv)

Topological data analysis (TDA) has emerged as a powerful tool for extracting meaningful insights from complex data. TDA enhances the analysis of objects by embedding them into a simplicial complex and extracting useful global properties such as the Betti numbers, i.e. the number of multidimensional holes, which can be used to define kernel methods that are easily integrated with existing machine-learning algorithms. These kernel methods have found broad applications, as they rely on powerful mathematical frameworks which provide theoretical guarantees on their performance. However, the computation of higher-dimensional Betti numbers can be prohibitively expensive on classical hardware, while quantum algorithms can approximate them in polynomial time in the instance size. In this work, we propose a quantum approach to defining topological kernels, which is based on constructing Betti curves, i.e. topological fingerprint of filtrations with increasing order. We exhibit a working prototype of our approach implemented on a noiseless simulator and show its robustness by means of some empirical results suggesting that topological approaches may offer an advantage in quantum machine learning.