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 ${\mathsf{DQC}}_1⊈\mathsf{BPP}$. These are the first ${\mathsf{DQC}}_1$-hardness results that are 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.

61 pages, 4 figures