Query-optimal estimation of unitary channels in diamond distance
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911971234807808 |
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| author | Haah, Jeongwan Kothari, Robin O'Donnell, Ryan Tang, Ewin |
| author_facet | Haah, Jeongwan Kothari, Robin O'Donnell, Ryan Tang, Ewin |
| contents | We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a $\textsf{d}$-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O(\textsf{d}^2/\varepsilon)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O(\textsf{d}^3/\varepsilon^2)$ [via standard process tomography] or $O(\textsf{d}^{2.5}/\varepsilon)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to "bootstrap" an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $Ω(\textsf{d}^2/\varepsilon)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_14066 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Query-optimal estimation of unitary channels in diamond distance Haah, Jeongwan Kothari, Robin O'Donnell, Ryan Tang, Ewin Quantum Physics Data Structures and Algorithms We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a $\textsf{d}$-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O(\textsf{d}^2/\varepsilon)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O(\textsf{d}^3/\varepsilon^2)$ [via standard process tomography] or $O(\textsf{d}^{2.5}/\varepsilon)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to "bootstrap" an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $Ω(\textsf{d}^2/\varepsilon)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity. |
| title | Query-optimal estimation of unitary channels in diamond distance |
| topic | Quantum Physics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2302.14066 |