Query-optimal estimation of unitary channels in diamond distance

• URL: http://arxiv.org/abs/2302.14066v2
• Date: Mon, 29 Jul 2024 20:16:13 GMT
• Title: Query-optimal estimation of unitary channels in diamond distance
• Authors: Jeongwan Haah, Robin Kothari, Ryan O'Donnell, Ewin Tang,
• Abstract summary: We consider process tomography for unitary quantum channels. We output a classical description of a unitary that is $varepsilon$-close to the unknown unitary in diamond norm.
• Score: 3.087385668501741
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\Omega(\textsf{d}^2/\varepsilon)$ 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.

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