• DOI:10.48550/arXiv.2302.14066
• Corpus ID: 257232399

Query-optimal estimation of unitary channels in diamond distance

@article{Haah2023QueryoptimalEO,
  title={Query-optimal estimation of unitary channels in diamond distance},
  author={Jeongwan Haah and Robin Kothari and Ryan O'Donnell and Ewin Tang},
  journal={ArXiv},
  year={2023},
  volume={abs/2302.14066}
}

• Jeongwan Haah, Robin Kothari, +1 author Ewin Tang
• Published 27 February 2023
• Computer Science, Mathematics
• ArXiv

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.