Quantum Distance Calculation for ε -Graph Construction

Abstract

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(n2) 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(n3/lnn) , showing that this approach, if used directly for ε -graph construction, does not bring a computational advantage when compared to a classical approach.

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