Unitarity Estimation for Quantum Channels

Abstract

The unitarity is a measure giving information on how much a quantum channel is unitary. Learning the unitarity of an unknown quantum channel $\mathcal{E}$ is a basic and important task in quantum device certification and benchmarking. Generally, this task can be performed with either coherent or incoherent access. For coherent access, there are no restrictions on learning algorithms; while for incoherent access, at each time, after preparing the input state and applying $\mathcal{E}$, the output is measured such that no coherent quantum information can survive or be acted upon by $\mathcal{E}$ again. Quantum algorithms with only incoherent access allow practical implementations without the use of persistent quantum memory, and thus is more suitable for near-term devices. In this paper, we study unitarity estimation in both settings. For coherent access, we provide an ancilla-efficient algorithm that uses $O(\epsilon^{-2})$ calls to $\mathcal{E}$ where $\epsilon$ is the required precision; we show that this algorithm is query-optimal, giving a matching lower bound $\Omega(\epsilon^{-2})$. For incoherent access, we provide a non-adaptive, non-ancilla-assisted algorithm that uses $O(\sqrt{d}\cdot \epsilon^{-2})$ calls to $\mathcal{E}$, where $d$ is the dimension of the system that $\mathcal{E}$ acts on; we show that this algorithm cannot be substantially improved, giving an $\Omega(\sqrt{d}+\epsilon^{-2})$ lower bound, even if adaptive strategies and ancilla systems are allowed. As part of our results, we settle the query complexity of the distinguishing problem for depolarizing and unitary channels with incoherent access by giving a matching lower bound $\Omega(\sqrt{d})$, improving the prior best lower bound $\Omega(\sqrt[3]{d})$ by Aharonov, Cotler, and Qi (Nat. Commun. 2022) and Chen, Cotler, Huang, and Li (FOCS 2021).