Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis
Name
PRXQuantum.4.040349.pdf
Size
1.06 MB
Format
Adobe PDF
Checksum (MD5)
c459c902981e445b054c19ad4e6520df
Author(s)
Schmidhuber, Alexander
Lloyd, Seth
Date Issued
December 28, 2023
Journal
PRX Quantum
Publisher
American Physical Society
Citation
Schmidhuber, Alexander and Lloyd, Seth. 2023. "Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis." PRX Quantum, 4 (4).
Version
Final published version
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.
Subjects
General Physics and Astronomy
Mathematical Physics
Applied Mathematics
Electronic, Optical and Magnetic Materials
Electrical and Electronic Engineering
General Computer Science
MIT Department
Massachusetts Institute of Technology. Center for Theoretical Physics
Massachusetts Institute of Technology. Department of Mechanical Engineering
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1103/prxquantum.4.040349
