Quantum-enhanced causal discovery for a small number of samples

The discovery of causal relations from observed data has attracted significant interest from disciplines such as economics, social sciences, and biology. In practical applications, considerable knowledge of the underlying systems is often unavailable, and real data are usually associated with nonlinear causal structures, which makes 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 require any assumptions about the underlying model structures. Based on conditional independence tests in a class of reproducing kernel Hilbert spaces characterized by quantum circuits, the proposed algorithm can explore causal relations from the observed data drawn from arbitrary distributions. We conducted systematic experiments on fundamental graphs of causal structures, demonstrating that the qPC algorithm exhibits 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 quantum algorithm can empower classical algorithms for accurate inference in causal discovery, supporting them in regimes where classical algorithms typically fail. In addition, the effectiveness of this method was validated using the datasets on Boston housing prices, heart disease, and biological signaling systems as real-world applications. These findings highlight the potential of quantum-based causal discovery methods in addressing practical challenges, particularly in small-sample scenarios, where traditional approaches have shown significant limitations.

💡 Research Summary

The paper introduces a quantum‑enhanced version of the Peter‑Clark (PC) causal discovery algorithm, called qPC, which replaces the classical kernel‑based conditional independence test (KCIT) with a quantum kernel constructed from parameterized quantum circuits. In the traditional PC algorithm, causal structure is inferred by iteratively testing unconditional and conditional independencies among variables and then orienting the remaining edges to obtain a Completed Partially Directed Acyclic Graph (CPDAG). The quality of the independence test is crucial; classical kernels (Gaussian, polynomial, etc.) often require large sample sizes to reliably detect nonlinear dependencies, and their performance is highly sensitive to hyper‑parameter choices.

The quantum kernel is defined as the fidelity between two quantum states generated by encoding data vectors into a quantum circuit Uθ(x) acting on n qubits initialized in the |0⟩⊗n state. The kernel value kQ(x, x′)=Tr