Partially unitary learning

The problem of an optimal mapping between Hilbert spaces IN of |ψ〉 and OUT of |ϕ〉 based on a set of wavefunction measurements (within a phase) ψ_{l}→ϕ_{l}, l=1,⋯,M, is formulated as an optimization problem maximizing the total fidelity ∑_{l=1}^{M}ω^{(l)}|〈ϕ_{l}|U|ψ_{l}〉|^{2} subject to probability preservation constraints on U (partial unitarity). The constructed operator U can be considered as an IN to OUT quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension dim(OUT)×dim(IN) transforming operators as A^{OUT}=UA^{IN}U^{†}. An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.