Quantum Distance Calculation for $ε$-Graph Construction

In machine learning and particularly in topological data analysis, $ε$-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 $ε$-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 $ε$-neighbours is at least O(n^3 / ln n), showing that this approach, if used directly for $ε$-graph construction, does not bring a computational advantage when compared to a classical approach.