Quantum algorithms for estimating quantum entropies

Abstract

The von Neumann and quantum R\'enyi entropies characterize fundamental properties of quantum systems and lead to theoretical and practical applications in many fields. Quantum algorithms for estimating quantum entropies, using a quantum query model that prepares the purification of the input state, have been established in the literature. {However, constructing such a model is almost as hard as state tomography.} In this paper, we propose quantum algorithms to estimate the von Neumann and quantum $\alpha$-R\'enyi entropies of an $n$-qubit quantum state $\rho$ using independent copies of the input state. We also show how to efficiently construct the quantum circuits for {quantum entropy estimation} using primitive single/two-qubit gates. We prove that the number of required copies scales polynomially in $1/\epsilon$ and $1/\Lambda$, where $\epsilon$ denotes the additive precision and $\Lambda$ denotes the lower bound on all non-zero eigenvalues. Notably, our method outperforms previous methods in the aspect of practicality since it does not require any quantum query oracles, which are usually necessary for previous methods. Furthermore, we conduct experiments to show the efficacy of our algorithms to single-qubit states and study the noise robustness. We also discuss the applications to some quantum states of practical interest as well as some meaningful tasks such as quantum Gibbs state preparation and entanglement estimation.