Unitarity Estimation for Quantum Channels
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Abstract
Estimating the unitarity of an unknown quantum channel <inline-formula> <tex-math notation="LaTeX">$\mathcal {E}$ </tex-math></inline-formula> provides information on how much it is unitary, which is a basic and important problem in quantum device certification and benchmarking. Unitarity estimation can be performed with either coherent or incoherent access, where the former in general leads to better query complexity while the latter allows more practical implementations. In this paper, we provide a unified framework for unitarity estimation, which induces ancilla-efficient algorithms that use <inline-formula> <tex-math notation="LaTeX">$O(\epsilon ^{-2})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$O(\sqrt {d}\cdot \epsilon ^{-2})$ </tex-math></inline-formula> calls to <inline-formula> <tex-math notation="LaTeX">$\mathcal {E}$ </tex-math></inline-formula> with coherent and incoherent accesses, respectively, where <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> is the dimension of the system that <inline-formula> <tex-math notation="LaTeX">$\mathcal {E}$ </tex-math></inline-formula> acts on and <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> is the required precision. We further show that both the <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>-dependence and <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>-dependence of our algorithms are optimal. 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 <inline-formula> <tex-math notation="LaTeX">$\Omega (\sqrt {d})$ </tex-math></inline-formula>, improving the prior best lower bound <inline-formula> <tex-math notation="LaTeX">$\Omega (\sqrt [{3}]{d})$ </tex-math></inline-formula> by (Aharonov et al., 2022) and (Chen et al., FOCS 2021).