Partially Unitary Learning
This addresses the challenge of optimal quantum channel design for researchers in quantum information, but appears incremental as it builds on existing partial unitarity concepts.
The paper tackles the problem of finding an optimal mapping between Hilbert spaces using wavefunction measurements, by maximizing total fidelity under partial unitarity constraints, and develops an iterative algorithm for global optimization with a software implementation.
The problem of an optimal mapping between Hilbert spaces $IN$ of $\left|ψ\right\rangle$ and $OUT$ of $\left|φ\right\rangle$ based on a set of wavefunction measurements (within a phase) $ψ_l \to φ_l$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\sum_{l=1}^{M} ω^{(l)} \left|\langleφ_l|\mathcal{U}|ψ_l\rangle\right|^2$ 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 $A^{OUT}=\mathcal{U} A^{IN} \mathcal{U}^{\dagger}$. 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.