Quantum Causal Discovery via Amplitude Estimation of Kullback-Leibler Divergence

TL;DR

This paper introduces a quantum algorithm, QKLA, that improves the efficiency of causal discovery by reducing the number of oracle queries needed for high-precision conditional independence tests.

Contribution

The authors develop QKLA, a quantum amplitude estimation method for KL divergence, achieving quadratic query complexity improvement over classical approaches in causal discovery.

Findings

QKLA achieves a quadratic precision improvement in estimating KL divergence.

Simulation results confirm the theoretical error decay and quantum-classical error scaling match.

Quantum CI subroutine reduces oracle queries by up to 7.4 times in benchmark causal discovery tasks.

Abstract

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 $\Theta(1/\tau^{2})$ 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 $\mathcal{O}((L/\tau)\log(1/\delta))$ 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 $\widetilde{\Omega}(1/(L\tau))$ 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 $\mathcal{O}(1/M)$ 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 (\textsc{Asia}, 8 nodes; \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.