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.
- Received 12 September 2023
- Accepted 6 December 2023
DOI:https://doi.org/10.1103/PRXQuantum.4.040349
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Quantum computing has the potential of revolutionizing computation, but the number of known problems exhibiting an exponential quantum speedup remains limited. Consequently, a critical open challenge in quantum information science is identifying new tasks that can harness exponential quantum advantage. Recently, a quantum algorithm for topological data analysis (TDA) has experienced a surge of attention because of its ability to estimate the Betti numbers of a simplicial complex seemingly exponentially faster than the best classical algorithm. We prove multiple theorems that show that the quantum advantage provided by current quantum algorithms for TDA is, in fact, not exponential.
Our arguments are based on complexity theory. We prove that the central task of TDA—computing Betti numbers—is at least as difficult as a notoriously hard class of problems called #P. Since it is widely believed that quantum computers cannot efficiently solve #P-hard problems, our results limit the regime in which quantum algorithms for TDA can achieve exponential quantum advantage. We support our complexity-theoretic arguments by showing that current quantum algorithms for TDA achieve a quadratic speedup over the best-known classical approach in asymptotically almost all cases. Finally, we discuss different input models that could recover exponential quantum advantage.
Our analysis highlights and isolates the specific bottlenecks of quantum algorithms for TDA. The study of quantum algorithms for related problems in algebraic topology is an exciting new field, and we hope our analysis provides guidance as to which problems are good candidates for exponential quantum speedups.
