Partially Unitary Learning

The problem of an optimal mapping between Hilbert spaces $IN$ of $\mid \psi ⟩$ and $OUT$ of $\mid \phi ⟩$ based on a set of wavefunction measurements (within a phase) ${\psi }_l\to {\phi }_l$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity ${\sum }_{l=1}^M{\omega }^{(l)}{\mid ⟨{\phi }_l\mathrm{\mid }\mathcal{U}\mathrm{\mid }{\psi }_l⟩\mid }^2$ subject to probability preservation constraints on $\mathcal{U}$ (partial unitarity). The constructed operator $\mathcal{U}$ can be considered as an $IN$ to $OUT$ quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension $\dim (OUT)\times \dim (IN)$ transforming operators as $A^{OUT}=\mathcal{U}{A}^{IN}{\mathcal{U}}^{†}$. 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.