Estimation of mutual information via quantum kernel methods

Abstract Recently, the importance of analyzing data and collecting valuable insight efficiently has been increasing in various fields. Estimating mutual information~(MI) plays a critical role in investigating the relationship among multiple random variables with a nonlinear correlation. Particularly, the task to determine whether they are independent or not is called the independence test, whose core subroutine is estimating MI from given data. It is a fundamental tool in statistics and data analysis that can be applied in a wide range of applications such as hypothesis testing, causal discovery and more. In this paper, we propose a method for estimating mutual information using the quantum kernel. We investigate the performance under various problem settings, such as different sample sizes or the shape of the probability distribution. As a result, the quantum kernel method showed higher performance than the classical one under the situation that the number of samples is small, the variance is large or the variables possess highly non-linear relationships. We discuss this behavior in terms of the central limit theorem and the structure of the corresponding quantum reproducing kernel Hilbert space.