Quantum Algorithms for Estimating Quantum Entropies

Abstract

The von Neumann and quantum R\'enyi entropies characterize fundamental properties of quantum systems and lead to many theoretical and practical applications. Quantum algorithms using a purified quantum query model can speed up quantum entropy estimation, while little is known about the complexity of using identical copies of the quantum state. This paper presents quantum entropy estimation algorithms with a cost of copies scaling polynomially in the rank of the state. In contrast to current methods that depend on the dimension of the system, our methods could provide exponential resource savings in the scenario of low-rank states. Furthermore, we show how to construct quantum circuits using primitive single-qubit or two-qubit gates efficiently and thus provide practical methods for estimating quantum entropies of quantum systems. We also conduct simulation experiments to show the effectiveness and noise robustness of our algorithms.