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 τ and thresholding against it. Classically this requires Θ\(1/τ²\) 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 the KL divergence. Given coherent oracle access to the joint distribution, QKLA achieves a quadratic precision improvement, needing only mathcal{O}((L/τ)log(1/δ)) queries, where L is the log-ratio clip bound. Embedded in the PC algorithm, this compounds to an widetildeΩ(1/(Lτ)) reduction in total queries for the full causal discovery procedure. We validate the theory in three experiments. A gate-level state-vector simulation of the full QKLA circuit confirms the predicted mathcal{O}(1/M) error decay. Across K=20 random binary distributions, classical and quantum error scalings match theory to slope accuracy pm 0.005. On two benchmark networks textsc{Asia}, 8 nodes; textsc{Synthetic-12}, 12 nodes, quantum PC matches classical skeleton-recovery F1 while using 2.5--3.0times fewer oracle queries at τ= 5cdot 10^-3 bits and up to 12times fewer at τ= 10^-3 bits.