Partially Unitary Learning

Abstract

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

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Cited by in corpus (3)

• Quantum Channel Learning
• On The Radon-Nikodym Spectral Approach With Optimal Clustering
• Nonlinear unitary circuits for photonic neural networks