The problem of an optimal mapping between Hilbert spaces IN and OUT, based on a series of density matrix mapping measurements ρ^{(l)}→ϱ^{(l)}, l=1⋯M, is formulated as an optimization problem maximizing the total fidelity F=∑_{l=1}^{M}ω^{(l)}F(ϱ^{(l)},∑_{s}B_{s}ρ^{(l)}B_{s}^{†}) subject to probability preservation constraints on Kraus operators B_{s}. For F(ϱ,σ) in the form that total fidelity can be represented as a quadratic form with superoperator F=∑_{s}〈B_{s}|S|B_{s}〉 (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: (1) IN/OUT states are represented as density matrices; (2) the mapping itself is formulated as a mixed unitary quantum channel A^{OUT}=∑_{s}|w_{s}|^{2}U_{s}A^{IN}U_{s}^{†} (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states ϕ_{l}=Uψ_{l} to a quantum channel, which allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping ϱ^{(l)}=Uρ^{(l)}U^{†}, in this case a quadratic on U fidelity can be constructed by considering sqrt[ρ^{(l)}]→sqrt[ϱ^{(l)}] mapping, and on a quantum channel, where quadratic on B_{s} fidelity is an approximation-a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more. A software product implementing the algorithm is available from the authors.
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