Quantum Algorithms for Causal Estimands

TL;DR

This work tackles scalable causal effect estimation by introducing quantum-accelerated, kernel-based estimators. It develops a hybrid quantum-classical approach that uses quantum linear system solvers to invert kernel matrices, preserving uniform consistency and convergence guarantees while offering potential exponential speedups under suitable error scaling, particularly for the dose-response estimand $\hat{\theta}(a)= n^{-1} Y^T (K_{AA} \odot K_{XX} + n\lambda I)^{-1} (K_{Aa} \odot \sum_{x_i} K_{Xx_i})$ with $A = K_{AA} \odot K_{XX} + n\lambda I$. The method extends to distribution shift, ATT, and CATE through kernel mean embeddings, and includes analyses of quantum measurement error, showing that uniform consistency can persist when the readout error scales as $\epsilon_k = O(1/\text{polylog}(n))$. While promising, the approach also highlights readout and conditioning tradeoffs, since quantum speedups rely on efficient state preparation and controlled conditioning via the regularization parameter $\lambda$. Overall, the paper demonstrates that quantum linear algebra can enhance the scalability of advanced causal estimators without sacrificing theoretical guarantees, motivating further exploration of quantum-enabled causal inference in broader graphical models.

Abstract

Modern machine learning (ML) methods typically fail to adequately capture causal information. Consequently, such models do not handle data distributional shifts, are vulnerable to adversarial examples, and often learn spurious correlations. Causal ML, or causal inference, aims to solve these issues by estimating the expected outcome of counterfactual events, using observational and/or interventional data, where causal relationships are typically depicted as directed acyclic graphs. It is an open question as to whether these causal algorithms provide opportunities for quantum enhancement. In this paper we consider a recently developed family of non-parametric, continuous causal estimators and derive quantum algorithms for these tasks. Kernel evaluation and large matrix inversion are critical sub-routines of these classical algorithms, which makes them particularly amenable to a quantum treatment. Unlike other quantum ML algorithms, closed form solutions for the estimators exist, negating the need for gradient evaluation and variational learning. We describe several new hybrid quantum-classical algorithms and show that uniform consistency of the estimators is retained. Furthermore, if one is satisfied with a quantum state output that is proportional to the true causal estimand, then these algorithms inherit the exponential complexity advantages given by quantum linear system solvers.

Figures (9)

• Figure 1: Three node causal Directed Acyclic Graph (DAG). Here $X$ represents confounding covariates: any variable that has a causal effect on both the treatment $A$ and the outcome $Y$. There is an assumed causal relationship between $A$ and $Y$.
• Figure 2: Empirical demonstration of the sensitive dependence between condition number $\kappa$ and regularisation $\lambda$ on Job Corp and Colangelo datasets for a small neighbourhood around $\lambda$. This indicates that one can use regularisation to tune the complexity scaling of the matrix inversion.
• Figure 3: Illustration of the convergence rate function class against sample size. We see that within the blue region---assuming a suitable downstream observable --- one can achieve exponential speedup with $\log(1/\epsilon) = \mathcal{O}(\text{polylog} (n))$, which means that the error $\epsilon = \Omega(e^{-\text{polylog} (n)})$. This provides a lower bound on the error to maintain exponential speedup. In the green, one achieves fast convergence rate of $n^{-1/4}$ as determined via the best case classical uniform consistency convergence rate. In the purple region, we can achieve both a speedup and fast convergence as the additive error decreases fast enough. This is what is meant by the non empty subset where the functions classes overlap in Eq. (\ref{['eq:NonZeroOverlap']}). Hence we can say that for large enough $n$, the algorithm maintains the same convergence rate and an exponential speedup.
• Figure 4: Illustration of the convergence rate function class against sample size that also considers sampling error. We see that the region for quantum exponential speedup (blue) is now disjoint from the classical fast convergence rate (green). This is because the minimum error we can permit with exponential speedup (blue) is always greater and decays slower than the maximum bound on classical fast convergence rate (green). Note this is for the best case classical algorithm where we can assume strong smoothness properties of the function to be learned and low effective RKHS dimension. This may not be true when this smoothness assumption is invalid.
• Figure 5: 3 node confounder causal directed acyclic graph. Here $X$ represents the confounders, all random variables that have a causal effect on both the treatment $A$ and the outcome $Y$. There is also a causal link between $A$ and $Y$. These confounder structures frequently result in Simpson's paradox.

Theorems & Definitions (7)

• Theorem 1
• proof
• Theorem 2
• proof
• Definition 1
• Lemma 1
• Lemma 2