Partially unitary learning
Mikhail Gennadievich Belov, Vladislav Gennadievich Malyshkin · Physical review. E · 2024
The problem of an optimal mapping between Hilbert spaces IN of $|\ensuremath{\psi}\ensuremath{\rangle}$ and OUT of $|\ensuremath{\phi}\ensuremath{\rangle}$ based on a set of wavefunction measurements (within a phase) ${\ensuremath{\psi}}_{l}\ensuremath{\rightarrow}{\ensuremath{\phi}}_{l}, l=1,\ensuremath{\cdots},M$, is formulated as an optimization problem maximizing the total fidelity ${\ensuremath{\sum}}_{l=1}^{M}{\ensuremath{\omega}}^{(l)}|\ensuremath{\langle}{\ensuremath{\phi}}_{l}|\mathcal{U}|{\ensuremath{\psi}}_{l}{\ensuremath{\rangle}|}^{2}$ 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 $dim(\mathrm{OUT})\ifmmode\times\else\texttimes\fi{}dim(\mathrm{IN})$ transforming operators as ${A}^{\mathrm{OUT}}=\mathcal{U}{A}^{\mathrm{IN}}{\mathcal{U}}^{\ifmmode\dagger\else\textdagger\fi{}}$. 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.