# Query-optimal estimation of unitary channels in diamond distance

arXiv: https://arxiv.org/abs/2302.14066 · HTML: https://mlsanity.com/paper/2302.14066/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2023.

## Built on by (most cited first, 8 of 8)

- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Learning the structure of any Hamiltonian from minimal assumptions](https://mlsanity.com/paper/2410.21635/) (2024)
- [Learning Quantum Processes with Quantum Statistical Queries](https://mlsanity.com/paper/2310.02075/) (2023)
- [Adversarial Robustness of Partitioned Quantum Classifiers](https://mlsanity.com/paper/2502.20403/) (2025)
- [Mildly-Interacting Fermionic Unitaries are Efficiently Learnable](https://mlsanity.com/paper/2504.11318/) (2025)
- [Parameterized quantum comb and simpler circuits for reversing unknown qubit-unitary operations](https://mlsanity.com/paper/2403.03761/) (2024)
- [Efficient learning of bosonic Gaussian unitaries](https://mlsanity.com/paper/2510.05531/) (2025)
- [Noise-tolerant learnability of shallow quantum circuits from statistics and the cost of quantum pseudorandomness](https://mlsanity.com/paper/2405.12085/) (2024)

## Abstract (arXiv)

We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a textsfd-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(textsfd2/varepsilon) applications of the unknown channel and only one qudit. This improves over prior results, which use O(textsfd3/varepsilon2) [via standard process tomography] or O(textsfd2.5/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(textsfd2/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.