Average-case hardness of Betti number estimation

2026-09-11

AI Summary

本文通过从植团问题的归约，证明了随机团复形上贝蒂数估计的平均情况难度，并讨论了其对拓扑数据分析中其他问题的影响。

Abstract

We establish the average-case hardness of Betti number estimation on random clique complexes via a reduction from the planted clique problem. We further show that our reduction implies a series of hardness results for many problems in both classical and quantum Topological Data Analysis (qTDA). Under the classical planted clique conjecture, no randomized polynomial-time Betti number estimator achieves additive error below $\tfrac12$ with constant advantage. Under a new quantum planted clique conjecture that we introduce, the same conclusion holds for quantum polynomial-time algorithms. We also obtain related conditional hardness results for homology vanishing, additive approximations with larger error tolerances, preparation of simplex and harmonic states, cycle recovery, and counting eigenvalues at low energy. Our reduction clarifies the structural requirements for quantum advantage in TDA and provides a new lens to investigate the classical and quantum complexity of related problems.

Problem

• Betti number estimation
• random clique complexes
• planted clique problem
• Topological Data Analysis
• quantum complexity

Innovation

• average-case hardness
• Betti number estimation
• planted clique problem
• quantum Topological Data Analysis (qTDA)
• quantum advantage