Quantum Channel Learning
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912175788916736 |
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| author | Belov, Mikhail Gennadievich Dubov, Victor Victorovich Filimonov, Alexey Vladimirovich Malyshkin, Vladislav Gennadievich |
| author_facet | Belov, Mikhail Gennadievich Dubov, Victor Victorovich Filimonov, Alexey Vladimirovich Malyshkin, Vladislav Gennadievich |
| contents | The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $ρ^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} ω^{(l)} F\left(\varrho^{(l)},\sum_s B_s ρ^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,σ)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a mixed unitary quantum channel $A^{OUT}=\sum_s |w_s|^2 \mathcal{U}_s A^{IN} \mathcal{U}_s^{\dagger}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states $ϕ_l=\mathcal{U} ψ_l$ to a quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping $\varrho^{(l)}=\mathcal{U} ρ^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{ρ^{(l)}} \to \sqrt{\varrho^{(l)}}$ mapping, and on a quantum channel, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04406 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum Channel Learning Belov, Mikhail Gennadievich Dubov, Victor Victorovich Filimonov, Alexey Vladimirovich Malyshkin, Vladislav Gennadievich Machine Learning Numerical Analysis Quantum Physics The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $ρ^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} ω^{(l)} F\left(\varrho^{(l)},\sum_s B_s ρ^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,σ)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a mixed unitary quantum channel $A^{OUT}=\sum_s |w_s|^2 \mathcal{U}_s A^{IN} \mathcal{U}_s^{\dagger}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states $ϕ_l=\mathcal{U} ψ_l$ to a quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping $\varrho^{(l)}=\mathcal{U} ρ^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{ρ^{(l)}} \to \sqrt{\varrho^{(l)}}$ mapping, and on a quantum channel, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more. |
| title | Quantum Channel Learning |
| topic | Machine Learning Numerical Analysis Quantum Physics |
| url | https://arxiv.org/abs/2407.04406 |