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arXiv:2310.12396 · Quantum Machine Intelligence

Estimation of mutual information via quantum kernel methods

Yota Maeda, Hideaki Kawaguchi, Hiroyuki Tezuka

Published Feb 25, 2025• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s42484-025-00263-7

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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.
Two-sample tests have been extensively employed in various scientific fields and machine learning to discriminate whether two sets of samples come from the same distribution or not. Kernel-based procedures for hypothetical testing have been proposed to efficiently disentangle high-dimensional complex structures in data to obtain accurate results in a model-free way by embedding the data into the reproducing kernel Hilbert space (RKHS). While the choice of kernels plays a crucial role for their performance, little is understood about how to choose kernel especially for small datasets. Here we construct a hypothetical test which can be effective even for small datasets, based on the theoretical foundation of kernel-based tests using maximum mean discrepancy, which is called MMD-FUSE. We enhance the MMD-FUSE framework by incorporating quantum kernels and propose a novel hybrid testing strategy that fuses classical and quantum kernels. This approach creates a powerful and adaptive test by combining the domain-specific inductive biases of classical kernels with the unique expressive power of quantum kernels. We evaluate our method on various synthetic and real-world clinical datasets, and our experiments reveal two key findings: 1) With appropriate hyperparameter tuning, MMD-FUSE with quantum kernels consistently improves test power over classical counterparts, especially for small and high-dimensional data. 2) The proposed hybrid framework demonstrates remarkable robustness, adapting to different data characteristics and achieving high test power across diverse scenarios. These results highlight the potential of quantum-inspired and hybrid kernel strategies to build more effective statistical tests, offering versatile tools for data analysis where sample sizes are limited.
Graph kernels are essential for learning the properties of graph data, a fundamental task in the analysis of such data. However, a major limitation of current graph kernel methods is their high computational cost, which becomes prohibitive when dealing with large-scale graphs. In order to address this bottleneck, we propose a quantum algorithm designed to analyze the intrinsic structural information within graph data through an efficient quantum kernel method. The core idea of our kernel method involves encoding pertinent information from graphs into quantum states, utilizing quantum kernel functions to represent the similarities among various graphs. This process is facilitated through the application of quantum walks and quantum signal processing techniques. Building upon this framework, we propose a quantum kernel ridge regression algorithm that consists of two key steps. First, we employ our quantum kernel method to prepare a quantum state, in which the amplitudes encode the kernel functions. Second, we implement quantum feature mapping in parallel to encode the corresponding kernel matrix into the quantum circuit. Crucially, we demonstrate that the computational complexity of our regression algorithm scales primarily as O(M3/2)O({M}^{3/2})O(M3/2) with respect to the size of the graph set MMM. This suggests that our quantum kernel method could process graph data with resource requirements scaling approximately as O(polylog(N))O(\text{poly}log(N))O(polylog(N)), where NNN denotes the number of nodes in the graph. Accordingly, this underscores the advantage of our approach over classical methods in the context of large-scale graph analysis. To validate our approach, we conduct numerical experiments on chemical molecular datasets, demonstrating the method's effectiveness and advantages. In summary, this work represents a significant advancement in quantum graph learning, enhancing its potential for analyzing large-scale graph data.
Kernel method in machine learning consists of encoding input data into a vector in a Hilbert space called the feature space and modeling the target function as a linear map on the feature space. Given a cost function, computing such an optimal linear map requires computation of a kernel matrix whose entries equal the inner products of feature vectors. In the quantum kernel method it is assumed that the feature vectors are quantum states in which case the quantum kernel matrix is given in terms of the overlap of quantum states. In practice, to estimate entries of the quantum kernel matrix one should apply, e.g., the SWAP-test and the number of such SWAP-tests is a relevant parameter in evaluating the performance of the quantum kernel method. Moreover, quantum systems are subject to noise, so the quantum states as feature vectors cannot be prepared exactly and this is another source of error in the computation of the quantum kernel matrix. Taking both the above considerations into account, we prove a bound on the performance (generalization error) of the quantum kernel method.
Abstract Quantum kernel methods are considered a promising avenue for applying quantum computers to machine learning problems. However, recent results overlook the central role hyperparameters play in determining the performance of machine learning methods. In this work we identify the hyperparameter controlling the bandwidth of a quantum kernel and show that it controls the expressivity of the resulting model. We use extensive numerical experiments with multiple quantum kernels and classical datasets to show consistent change in the model behavior from underfitting (bandwidth too large) to overfitting (bandwidth too small), with optimal generalization in between. We draw a connection between the bandwidth of classical and quantum kernels and show analogous behavior in both cases. Furthermore, we show that optimizing the bandwidth can help mitigate the exponential decay of kernel values with qubit count, which is the cause behind recent observations that the performance of quantum kernel methods decreases with qubit count. We reproduce these negative results and show that if the kernel bandwidth is optimized, the performance instead improves with growing qubit count and becomes competitive with the best classical methods.
The problem of estimating a parameter of a quantum system through a series of measurements performed sequentially on a quantum probe is analyzed in the general setting where the underlying statistics is explicitly non-i.i.d. We present a generalization of the central limit theorem in the present context, which under fairly general assumptions shows that as the number N of measurement data increases the probability distribution of functionals of the data (e.g., the average of the data) through which the target parameter is estimated becomes asymptotically normal and independent of the initial state of the probe. At variance with the previous studies (Guţă M 2011 Phys. Rev. A 83 062324 ; van Horssen M and Guţă M 2015 J. Math. Phys. 56 022109 ) we take a diagrammatic approach, which allows one to compute not only the leading orders in N of the moments of the average of the data but also those of the correlations among subsequent measurement outcomes. In particular our analysis points out that the latter, which are not available in usual i.i.d. data, can be exploited in order to improve the accuracy of the parameter estimation. An explicit application of our scheme is discussed by studying how the temperature of a thermal reservoir can be estimated via sequential measurements on a quantum probe in contact with the reservoir.
As quantum computers become increasingly practical, so does the prospect of using quantum computation to improve upon traditional algorithms. Kernel methods in machine learning is one area where such improvements could be realized in the near future. Paired with kernel methods like support-vector machines, small and noisy quantum computers can evaluate classically-hard quantum kernels that capture unique notions of similarity in data. Taking inspiration from techniques in classical machine learning, this work investigates simulated quantum kernels in the context of multiple kernel learning (MKL). We consider pairwise combinations of several classical-classical, quantum-quantum, and quantum-classical kernels in an empirical investigation of their classification performance with support-vector machines. We also introduce a novel approach, which we call QCC-net (quantum-classical-convex neural network), for optimizing the weights of base kernels together with any kernel parameters. We show this approach to be effective for enhancing various performance metrics in an MKL setting. Looking at data with an increasing number of features (up to 13 dimensions), we find parameter training to be important for successfully weighting kernels in some combinations. Using the optimal kernel weights as indicators of relative utility, we find growing contributions from trainable quantum kernels in quantum-classical kernel combinations as the number of features increases. We observe the opposite trend for combinations containing simpler, non-parametric quantum kernels.