Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians

Abstract:Topological data analysis (TDA) is a machine learning technique that uses topology to extract patterns from data and has shown the potential to exhibit quantum advantage. A key concept in TDA is persistent homology, which measures the robustness of topological information at different lengthscales. In this paper, we introduce and study the problem of normalized persistence, a practically motivated and easily interpretable version of persistent homology that counts the fraction of holes that persist at different lengthscales. We prove that a variant of normalized persistence is $\mathsf{DQC}_1$-hard and contained in $\mathsf{BQP}$, giving evidence of an exponential quantum speedup for TDA under the standard assumption that $\mathsf{DQC}_1 \not\subseteq \mathsf{BPP}$. These are the first $\mathsf{DQC}_1$-hardness results for clique complexes, making them directly applicable to TDA instances. We also find a close connection between normalized persistence and the complexity of estimating spectral quantities in the low-energy subspace of local Hamiltonians. We study a family of such problems, including a low-energy normalized subtrace and spectral density. We show that these are $\mathsf{DQC}_1$-hard for $O(1)$-local Hamiltonians, strengthening previous results that required log-local interactions. We also introduce a variant of $\mathsf{DQC}_1$ with perfect completeness ($\mathsf{SDQC}_1$) to characterize the hardness of problems normalized by an exact kernel. This includes normalized persistence for $O(1)$-local Hamiltonians, which we show is $\mathsf{SDQC}_1$-hard.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Machine Learning (cs.LG)

arXiv:2607.03278 [quant-ph]

(or arXiv:2607.03278v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.03278 Focus to learn more arXiv-issued DOI via DataCite

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