# Query-optimal estimation of unitary channels in diamond distance

## TL;DR

This paper presents an optimal algorithm for estimating unknown unitary quantum channels with minimal applications, improving efficiency over previous methods and establishing matching lower bounds.

## Contribution

The authors develop a query-optimal algorithm for unitary channel estimation in diamond norm, achieving Heisenberg scaling with minimal resource usage.

## Key findings

- Achieves $ extsf{d}^2/ extsf{ε}$ applications for $ extsf{d}$-dimensional unitaries.
- Introduces a bootstrap technique to improve estimation accuracy.
- Proves a matching lower bound confirming optimality.

## 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.