A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits

TL;DR

This paper introduces an improved quantum algorithm for calculating persistent Betti numbers in topological data analysis, offering significant polynomial and exponential resource savings over previous methods.

Contribution

The paper presents a new quantum algorithm with polynomial speedup and exponential space savings, along with a classical power method with comparable scaling to existing heuristics.

Findings

Quantum algorithm achieves almost quintic speedup over classical methods.

Exponential space savings compared to previous quantum algorithms.

Classical power method has quadratic scaling, comparable to heuristics.

Abstract

Topological invariants of a dataset, such as the number of holes that survive from one length scale to another (persistent Betti numbers) can be used to analyze and classify data in machine learning applications. We present an improved quantum algorithm for computing persistent Betti numbers, and provide an end-to-end complexity analysis. Our approach provides large polynomial time improvements, and an exponential space saving, over existing quantum algorithms. Subject to gap dependencies, our algorithm obtains an almost quintic speedup in the number of datapoints over previously known rigorous classical algorithms for computing the persistent Betti numbers to constant additive error - the salient task for applications. However, we also introduce a quantum-inspired classical power method with provable scaling only quadratically worse than the quantum algorithm. This gives a provable classical algorithm with scaling comparable to existing classical heuristics. We discuss whether quantum algorithms can achieve an exponential speedup for tasks of practical interest, as claimed previously. We conclude that there is currently no evidence for this being the case.