Partially Unitary Learning

Abstract

The problem of an optimal mapping between Hilbert spaces $IN$ of $|\ddot{I}ˆ⟩$ and $OUT$ of $|\ddot{I}•⟩$ based on a set of wavefunction measurements (within a phase) $\ddot{I}{ˆ}_l\to \ddot{I}{•}_l$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity ${\sum }_{l=1}^M\ddot{I}{‰}^{(l)}{\left|⟨\ddot{I}{•}_l|\mathcal{U}|\ddot{I}{ˆ}_l⟩\right|}^2$ 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.

A working algorithm implementing Partially Unitary Learning arXiv:2212.14810 has been developed and generalized. See arXiv:2407.04406 for further generalization to density matrix mappings