Towards quantum advantage via topological data analysis

Even after decades of quantum computing development, examples generally useful algorithms with exponential speedups over classical counterparts are scarce. Recent progress in for linear-algebra positioned machine learning (QML) as a potential source such improvements. Yet, an unexpected recent series "dequantization" results has equally rapidly removed the promise several QML algorithms. This raises critical question whether other linear-algebraic persist. In this paper, we study quantum-algorithmic methods behind algorithm topological data analysis Lloyd, Garnerone and Zanardi through lens. We provide evidence that problem solved by is classically intractable showing its natural generalization hard simulating one clean qubit model -- which widely believed to require superpolynomial time on computer thus very likely immune dequantizations. Based result, number new problems rank estimation complex network analysis, along complexity-theoretic their intractability. Furthermore, analyze suitability proposed near-term implementations. Our applications full-blown, restricted computers guaranteed speedup methods, recovering some become computing's killer applications.