We report on the experimental measurement of the Hilbert-Schmidt distance between two two-qubit states by many-particle interference. We demonstrate that our three-step method for measuring distances in the Hilbert space is far less complex than reconstructing density matrices and that it can be applied in quantum-enhanced machine learning to reduce the complexity of calculating Euclidean distances between multidimensional points, which can be especially interesting for near term quantum technologies and quantum artificial intelligence research. Our results are also a novel example of applying mixed states in quantum information processing. Usually working with mixed states is undesired, but here it gives the possibility of encoding extra information as the degree of coherence between the given two dimensions of the density matrix.

Faculty of Physics Adam Mickiewicz University PL 61 614 Poznań Poland

Institute of Physics of the Czech Academy of Sciences Joint Laboratory of Optics of PU and IP AS CR 17 listopadu 50A 772 07 Olomouc Czech Republic

RCPTM Joint Laboratory of Optics of Palacký University and Institute of Physics of Czech Academy of Sciences 17 listopadu 12 771 46 Olomouc Czech Republic

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