In machine learning and particularly in topological data analysis, $\epsilon$-graphs are important tools but are generally hard to compute as the distance calculation between n points takes time O(n^2) classically. Recently, quantum approaches for calculating distances between n quantum states have been proposed, taking advantage of quantum superposition and entanglement. We investigate the potential for quantum advantage in the case of quantum distance calculation for computing $\epsilon$-graphs. We show that, relying on existing quantum multi-state SWAP test based algorithms, the query complexity for correctly identifying (with a given probability) that two points are not $\epsilon$-neighbours is at least O(n^3 / ln n), showing that this approach, if used directly for $\epsilon$-graph construction, does not bring a computational advantage when compared to a classical approach.

翻译：在机器学习，特别是拓扑数据分析中，$\varepsilon$-图是重要工具，但通常难以计算，因为经典方法在 $n$ 个点之间进行距离计算需要 $O(n^2)$ 时间。近年来，利用量子叠加与纠缠，已提出计算 $n$ 个量子态之间距离的量子方法。我们研究了在计算 $\varepsilon$-图时，量子距离计算领域实现量子优势的潜力。研究表明，基于现有的量子多态 SWAP 测试算法，以给定概率正确识别两个点并非 $\varepsilon$-邻接的查询复杂度至少为 $O(n^3 / \ln n)$，这表明直接将该方法用于 $\varepsilon$-图构建时，与经典方法相比并不具备计算优势。