  We report on 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 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
coherence between given two dimensions of the density matrix.
