Quantum Causal Discovery via Amplitude Estimation of Kullback-Leibler Divergence

Causal discovery from observational data underpins applications in finance, climate modeling, and machine learning. Constraint-based causal discovery reduces structure learning to a sequence of conditional independence (CI) tests, where each test decides independence by estimating conditional mutual information $I(X;Y\mid Z)$ to additive precision $\tau$ and thresholding against it. Classically this requires $\mathrm{\Theta }(1\mathrm{/}{\tau }^2)$ samples per test, a cost that dominates in the high-precision regime typical of weak dependencies. We present QKLA (Quantum Kullback–Leibler Amplitude estimation), a quantum algorithm that encodes a clipped log-density ratio as a bounded amplitude and applies amplitude estimation to recover a clipped KL expectation. Given coherent oracle access to the relevant distributions and a reversible log-ratio arithmetic oracle, QKLA achieves a quadratic precision improvement, needing only $\mathcal{O}((L\mathrm{/}\tau )\log (1\mathrm{/}\delta ))$ queries, where $L$ is the log-ratio clip bound. Under per-stratum conditional-oracle access and a margin assumption for CI decisions, embedding this estimator in the PC algorithm compounds to an $\tilde{\mathrm{\Omega }}(1\mathrm{/}(L\tau ))$ reduction in total oracle queries. We validate the theory in three experiments. A gate-level state-vector simulation of the full QKLA circuit confirms the predicted $\mathcal{O}(1\mathrm{/}M)$ error decay. Across $K=20$ random binary distributions, classical and quantum error scalings match theory to within $0.01$ in slope. In an oracle-model benchmark inside PC on two networks ($\text{textsc}Asia$, 8 nodes; $\text{textsc}Synthetic-12$, 12 nodes), the quantum CI subroutine reaches comparable skeleton-recovery $F_1$ while using $2.7$–$3.2\times$ fewer oracle queries at $\tau =5\cdot {10}^{-3}$ bits and $4.0$–$7.4\times$ fewer at $\tau ={10}^{-3}$ bits.