Betti数估计的平均情况困难性
Average-case hardness of Betti number estimation
- Department of Computer Science, University of Oxford（牛津大学计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
通过从种植团问题的归约，证明了随机团复形上Betti数估计的平均情况困难性，并推导出经典和量子TDA中多个问题的条件困难性结果，阐明了量子优势的结构性要求。
AI中文摘要：
我们通过从种植团问题的归约，建立了随机团复形上Betti数估计的平均情况困难性。我们进一步表明，我们的归约蕴含了经典和量子拓扑数据分析（qTDA）中许多问题的一系列困难性结果。在经典种植团猜想下，没有随机多项式时间的Betti数估计器能够以常数优势实现低于1/2的加性误差。在我们引入的一个新的量子种植团猜想下，同样的结论对量子多项式时间算法也成立。我们还获得了同调消失、更大误差容限下的加性近似、单纯形和谐波态的制备、循环恢复以及低能量本征值计数等相关条件困难性结果。我们的归约阐明了TDA中量子优势的结构性要求，并为研究相关问题的经典和量子复杂性提供了新的视角。
英文摘要：
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.
