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We report on experimental measurement of the Hilbert-Schmidt distance betweentwo two-qubit states by many-particle interference. We demonstrate that ourthree-step method for measuring distances in Hilbert space is far less complexthan reconstructing density matrices and that it can be applied inquantum-enhanced machine learning to reduce the complexity of calculatingEuclidean distances between multidimensional points, which can be especiallyinteresting for near term quantum technologies and quantum artificialintelligence research. Our results are also a novel example of applying mixedstates in quantum information processing. Usually working with mixed states isundesired, but here it gives the possibility of encoding extra information ascoherence between given two dimensions of the density matrix.

Experimental measurement of Hilbert-Schmidt distance between two-qubit states as means for speeding-up machine learning