On Partially Unitary Learning
<|reference_start|>On Partially Unitary Learning: The problem of an optimal mapping between Hilbert spaces $IN$ of $\left|\psi\right\rangle$ and $OUT$ of $\left|\phi\right\rangle$ 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 $\sum_{l=1}^{M} \omega^{(l)} \left|\langle\phi_l|\mathcal{U}|\psi_l\rangle\right|^2$ subject to probability preservation constraints on $\mathcal{U}$ (partial unitarity). Constructed operator $\mathcal{U}$ can be considered as a $IN$ to $OUT$ quantum channel; it is a partially unitary rectangular matrix of the dimension $\dim(OUT) \times \dim(IN)$ transforming operators as $A^{OUT}=\mathcal{U} A^{IN} \mathcal{U}^{\dagger}$. An iteration algorithm finding the global maximum of this optimization problem is developed and it's application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.<|reference_end|>
@article{belov2024on,
    title={On Partially Unitary Learning},
    author={Mikhail Gennadievich Belov, Vladislav Gennadievich Malyshkin},
    journal={arXiv preprint arXiv:2405.10263},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.10263},
    primaryClass={cs.LG cs.NA math.NA quant-ph stat.ML}
}