Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis

TL;DR

This paper provides complexity-theoretic evidence that quantum algorithms for topological data analysis, specifically for estimating Betti numbers, are unlikely to surpass polynomial speedups due to inherent computational hardness.

Contribution

It proves that computing Betti numbers exactly is #P-hard and approximating them is NP-hard, limiting quantum advantage in topological data analysis.

Findings

Quantum algorithms achieve only polynomial speedup for Betti number estimation.

Computing Betti numbers exactly is #P-hard; approximating them is NP-hard.

Quantum advantage may be exponential if data is given as simplices rather than vertex lists.

Abstract

Quantum algorithms for topological data analysis (TDA) seem to provide an exponential advantage over the best classical approach while remaining immune to dequantization procedures and the data-loading problem. In this paper, we give complexity-theoretic evidence that the central task of TDA -- estimating Betti numbers -- is intractable even for quantum computers. Specifically, we prove that the problem of computing Betti numbers exactly is #P-hard, while the problem of approximating Betti numbers up to multiplicative error is NP-hard. Moreover, both problems retain their hardness if restricted to the regime where quantum algorithms for TDA perform best. Because quantum computers are not expected to solve #P-hard or NP-hard problems in subexponential time, our results imply that quantum algorithms for TDA offer only a polynomial advantage in the worst case. We support our claim by showing that the seminal quantum algorithm for TDA developed by Lloyd, Garnerone and Zanardi achieves a quadratic speedup over the best known classical approach in asymptotically almost all cases. Finally, we argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices rather than as a list of vertices and edges.