Partially Unitary Learning
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915025696849920 |
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| author | Belov, Mikhail Gennadievich Malyshkin, Vladislav Gennadievich |
| author_facet | Belov, Mikhail Gennadievich Malyshkin, Vladislav Gennadievich |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10263 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partially Unitary Learning Belov, Mikhail Gennadievich Malyshkin, Vladislav Gennadievich Machine Learning Numerical Analysis Quantum Physics 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. |
| title | Partially Unitary Learning |
| topic | Machine Learning Numerical Analysis Quantum Physics |
| url | https://arxiv.org/abs/2405.10263 |