Related papers: On Partially Unitary Learning

On Partially Unitary Learning

URL: http://arxiv.org/abs/2405.10263v1
Date: Thu, 16 May 2024 17:13:55 GMT
Title: On Partially Unitary Learning
Authors: Mikhail Gennadievich Belov, Vladislav Gennadievich Malyshkin,
Abstract summary: An optimal mapping between Hilbert spaces $IN$ of $left|psirightrangle$ and $OUT$ of $left|phirightrangle$ is presented. An algorithm finding the global maximum of this optimization problem is developed and it's application to a number of problems is demonstrated.
Score: 0.0
License: http://creativecommons.org/licenses/by/4.0/
Abstract: 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.

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