Quantum oracles give an advantage for identifying classical counterfactuals
Ciarán M. Gilligan-Lee, Yìlè Yīng, Jonathan Richens, David Schmid
TL;DR
The paper shows that quantum oracles enable identification of classical counterfactuals in discrete causal models by learning the distribution $p(F)$ over functional dependences that govern $Y$ given $X$. It first proves non-identifiability of counterfactuals under classical data, then demonstrates that coherent quantum queries can recover the full causal-parameter distribution in the binary case, and extends the analysis to higher cardinalities where two-way counterfactuals are quantum-identifiable but higher-order counterfactuals face limits. The work connects quantum oracle problems to causal inference and reveals that quantum advantages can tighten bounds on multi-way counterfactuals, while also showing that in the binary case similar benefits can arise in classically-explainable theories like Spekkens’ toy theory. This suggests a nuanced role for nonclassicality in quantum causal inference and raises questions about the resource requirements for identifying causal parameters beyond simple function-identification tasks.
Abstract
We show that quantum oracles provide an advantage over classical oracles for answering classical counterfactual questions in causal models, or equivalently, for identifying unknown causal parameters such as distributions over functional dependences. In structural causal models with discrete classical variables, observational data and even ideal interventions generally fail to answer all counterfactual questions, since different causal parameters can reproduce the same observational and interventional data while disagreeing on counterfactuals. Using a simple binary example, we demonstrate that if the classical variables of interest are encoded in quantum systems and the causal dependence among them is encoded in a quantum oracle, coherently querying the oracle enables the identification of all causal parameters -- hence all classical counterfactuals. We generalize this to arbitrary finite cardinalities and prove that coherent probing 1) allows the identification of all two-way joint counterfactuals p(Y_x=y, Y_{x'}=y'), which is not possible with any number of queries to a classical oracle, and 2) provides tighter bounds on higher-order multi-way counterfactuals than with a classical oracle. This work can also be viewed as an extension to traditional quantum oracle problems such as Deutsch--Jozsa to identifying more causal parameters beyond just, e.g., whether a function is constant or balanced. Finally, we raise the question of whether this quantum advantage relies on uniquely non-classical features like contextuality. We provide some evidence against this by showing that in the binary case, oracles in some classically-explainable theories like Spekkens' toy theory also give rise to a counterfactual identifiability advantage over strictly classical oracles.