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Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis
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DOI：10.1103/PRXQuantum.4.040349.png)
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Abstract
En 中文
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 expo-nential 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.
Keywords:
INDEPENDENCE COMPLEXES
ALEXANDER DUALITY
COMPLETENESS
Journal
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11
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