Quantum-Enhanced Causal Discovery with Small Sample Data
This project pioneers the quantum-inspired application to causal discovery, a fundamental task in data science, economics, biology, and many other fields. Traditional causal inference methods, such as the classical Peter-Clark (PC) algorithm (Spirtes et al., 2000; Zhang et al., 2012), often struggle with nonlinear relationships and limited sample sizes, which are common in real-world datasets.
To address these challenges, we developed and validated a novel quantum Peter-Clark (qPC) algorithm by extending the functionality of OSS library (Zheng et al., 2024). The qPC algorithm leverages quantum kernel methods, embedding classical data into quantum states to perform conditional independence tests within a reproducing kernel Hilbert space (RKHS) defined by quantum circuits. This approach enables the algorithm to infer causal relationships from observed data without assuming any specific model structure or data distribution.
A key innovation of our work is a systematic optimization strategy for quantum kernel hyperparameters based on Kernel Target Alignment (KTA). This method objectively tunes the quantum circuits, significantly reducing the risk of false positives in causal discovery and enhancing the reliability of the results.
Extensive experiments on both synthetic and real-world datasets, including the Boston Housing dataset, demonstrate that the qPC algorithm outperforms classical methods—especially when only small sample sizes are available. The quantum approach is particularly effective in uncovering complex, nonlinear causal relationships that are difficult for conventional algorithms to detect.
Our findings reveal that quantum circuit-based causal discovery methods can empower classical algorithms, enabling robust and accurate inference even in challenging scenarios. This research opens new avenues for practical quantum machine learning applications, especially in domains where data is scarce or highly complex.
The discovery of causal relationships from observed data has attracted significant interest from disciplines such as economics, social sciences, epidemiology, and biology. In practical applications, considerable knowledge of the underlying systems is often unavailable, and real data are often associated with nonlinear causal structures, which make the direct use of most conventional causality analysis methods difficult. This study proposes a novel quantum Peter-Clark (qPC) algorithm for causal discovery that does not assume any underlying model structures. Based on the independence conditional tests in a class of reproducing kernel Hilbert spaces characterized by quantum circuits, the proposed qPC algorithm can explore causal relationships from the observed data drawn from arbitrary distributions. We conducted systematic experiments on fundamental graph parts of causal structures, demonstrating that the qPC algorithm exhibits a significantly better performance, particularly with smaller sample sizes compared to its classical counterpart. Furthermore, we proposed a novel optimization approach based on Kernel Target Alignment (KTA) for determining hyperparameters of quantum kernels. This method effectively reduced the risk of false positives in causal discovery, enabling more reliable inference. Our theoretical and experimental results demonstrate that the proposed quantum algorithm can empower classical algorithms for robust and accurate inference in causal discovery, supporting them in regimes where classical algorithms typically fail. Additionally, the effectiveness of this method was validated using the Boston Housing dataset as a real-world application. These findings demonstrate the new potential of quantum circuit-based causal discovery methods in addressing practical challenges, particularly in small-sample scenarios where traditional approaches have shown limitations.
@misc{maedaQuantumenhancedCausalDiscovery2025,title={Quantum-Enhanced Causal Discovery for a Small Number of Samples},author={Terada, Yu and Arai, Ken and Tanaka, Yu and Maeda, Yota and Ueno, Hiroshi and Tezuka, Hiroyuki},year={2025},month=jan,number={arXiv:2501.05007},eprint={2501.05007},primaryclass={quant-ph},publisher={arXiv},doi={10.48550/arXiv.2501.05007},urldate={2025-05-30},archiveprefix={arXiv},langid={english},keywords={Computer Science - Artificial Intelligence,Computer Science - Machine Learning,Quantum Physics,Statistics - Methodology},file={files/1056/Maeda et al. - 2025 - Quantum-enhanced causal discovery for a small numb.pdf},category={peer-reviewed-abstract},booktitle={arXiv quant-ph},}
2024
Causal-learn: Causal discovery in python
Yujia Zheng, Biwei Huang, Wei Chen, and 6 more authors
Journal of Machine Learning Research, Jan 2024
Quantum PC Algorithm: Data-Efficient and Nonlinear Causal Discovery
Yota Maeda, Ken Arai, Yu Tanaka, and 3 more authors
2024 IEEE International Conference on Quantum Computing and Engineering (QCE), Sep 2024
Causal discovery is the task of finding causal relationships between random variables from observed data. Typically, one assumes that the causal relationships can be represented by a directed acyclic graph (DAG), and makes additional assumptions to ensure that the DAG can be recovered from the observed data. In this study, we propose the quantum Peter-Clark (qPC) algorithm for nonlinear causal discovery based on quantum kernel methods. The qPC algorithm takes advantage of the quantum kernel-based conditional independent test. Through the synthetic data experiment, we show that the qPC algorithm outperforms the classical method in the regime of a small number of samples. This suggests that the kernel-based causal discovery can significantly improve performance under such conditions. Our experiments highlight that the proposed algorithm can accurately support classical algorithms in causal discovery, paving the way for future advances with the utilization of quantum computation.
@article{maedaQuantumPCAlgorithm2024,title={Quantum {{PC Algorithm}}: {{Data-Efficient}} and {{Nonlinear Causal Discovery}}},author={Maeda, Yota and Arai, Ken and Tanaka, Yu and Terada, Yu and Ueno, Hiroshi and Tezuka, Hiroyuki},year={2024},month=sep,journal={2024 IEEE International Conference on Quantum Computing and Engineering (QCE)},pages={2},doi={10.1109/QCE60285.2024.10322},urldate={2025-05-30},category={peer-reviewed},}
2012
Kernel-based Conditional Independence Test and Application in Causal Discovery
Kun Zhang, Jonas Peters, Dominik Janzing, and 1 more author
