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

2026-07-03

AI Summary

This work investigates the computational complexity of normalized persistent homology and its connection to low-energy spectral estimation of local Hamiltonians. By introducing normalized persistent homology to enhance the interpretability of topological data analysis (TDA), and integrating techniques from quantum complexity theory, persistent homology, and spectral density estimation, the study establishes for the first time that this problem lies in BQP and is DQC₁-hard for practical TDA instances. It further introduces a new complexity class, SDQC₁, to precisely characterize the exact version of the problem. Additionally, the DQC₁-hardness of low-energy spectral estimation is extended to O(1)-local Hamiltonians, providing theoretical evidence for an exponential quantum speedup in TDA and strengthening known complexity results for low-energy spectra under constant-locality 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 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.

Problem

• normalized persistence
• topological data analysis
• local Hamiltonians
• computational complexity
• persistent homology

Innovation

• normalized persistence
• DQC1-hardness
• topological data analysis
• local Hamiltonians
• quantum advantage