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Implementing Pearl's $\mathcal{DO}$-Calculus on Quantum Circuits: A Simpson-Type Case Study on NISQ Hardware

Pilsung Kang

TL;DR

This work establishes a concrete pathway by which causal graphs and Pearl-style interventions can be represented, executed, and empirically tested within the formalism of quantum circuits.

Abstract

Distinguishing correlation from causation is a central challenge in machine intelligence, and Pearl's $\mathcal{DO}$-calculus provides a rigorous symbolic framework for reasoning about interventions. A complementary question is whether such intervention logic can be given \emph{executable semantics} on physical quantum devices. Our approach maps causal networks onto quantum circuits, where nodes are encoded in qubit registers, probabilistic links are implemented by controlled-rotation gates, and interventions are realized by a structural remodeling of the circuit -- a physical analogue of Pearl's ``graph surgery'' that we term \emph{circuit surgery}. We show that, for a family of 3-node confounded treatment models (including a Simpson-type reversal), the post-surgery circuits reproduce exactly the interventional distributions prescribed by the corresponding classical $\mathcal{DO}$-calculus. We then demonstrate a proof-of-principle experimental realization on an IonQ Aria trapped-ion processor and a 10-qubit synthetic healthcare model, observing close agreement between hardware estimates and classical baselines under realistic noise. We do not claim quantum speedup; instead, our contribution is to establish a concrete pathway by which causal graphs and Pearl-style interventions can be represented, executed, and empirically tested within the formalism of quantum circuits.

Implementing Pearl's $\mathcal{DO}$-Calculus on Quantum Circuits: A Simpson-Type Case Study on NISQ Hardware

TL;DR

This work establishes a concrete pathway by which causal graphs and Pearl-style interventions can be represented, executed, and empirically tested within the formalism of quantum circuits.

Abstract

Distinguishing correlation from causation is a central challenge in machine intelligence, and Pearl's -calculus provides a rigorous symbolic framework for reasoning about interventions. A complementary question is whether such intervention logic can be given \emph{executable semantics} on physical quantum devices. Our approach maps causal networks onto quantum circuits, where nodes are encoded in qubit registers, probabilistic links are implemented by controlled-rotation gates, and interventions are realized by a structural remodeling of the circuit -- a physical analogue of Pearl's ``graph surgery'' that we term \emph{circuit surgery}. We show that, for a family of 3-node confounded treatment models (including a Simpson-type reversal), the post-surgery circuits reproduce exactly the interventional distributions prescribed by the corresponding classical -calculus. We then demonstrate a proof-of-principle experimental realization on an IonQ Aria trapped-ion processor and a 10-qubit synthetic healthcare model, observing close agreement between hardware estimates and classical baselines under realistic noise. We do not claim quantum speedup; instead, our contribution is to establish a concrete pathway by which causal graphs and Pearl-style interventions can be represented, executed, and empirically tested within the formalism of quantum circuits.

Paper Structure

This paper contains 32 sections, 1 theorem, 7 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Quantum Causal Modeling
  4. Born-rule Generative Models (Born Machines)
  5. Quantum Analogues of Simpson's Paradox
  6. Methodology
  7. Causal Framework and the $\mathcal{DO}$-calculus
  8. Quantum Implementation of Causal Interventions
  9. Specific Circuit Models for Simpson's Paradox
  10. 3-Qubit Foundational Model
  11. 10-Qubit Scalability Model
  12. Circuit-Based Resolution of Simpson's Paradox
  13. Experiments and Results
  14. Experimental Setup
  15. Result 1: Simpson's Paradox on a 3-Qubit Model: Simulation and Experimental Validation
  16. ...and 17 more sections

Key Result

Proposition 1

For the circuit $\mathcal{C}(G,\Theta)$ constructed above, the following hold.

Figures (4)

  • Figure 1: The causal directed acyclic graphs (DAGs) for the 3-qubit Simpson's Paradox model. (a) The observational model, where G (Gender) is a common-cause confounder for T (Treatment) and O (Outcome). (b) The interventional model for $\mathcal{DO}(T=t)$, where the confounding path $G \rightarrow T$ has been surgically removed.
  • Figure 2: Quantum circuits for the 3-qubit Simpson model. (a) The observational circuit encodes all causal links. The dashed box highlights the gates implementing the confounding path ($G\!\to\!T$) targeted for removal via circuit surgery. (b) The resulting interventional circuit, where the confounding path has been removed and $T$ is deterministically prepared to sample from $P(O,G\mid \mathcal{DO}(T{=}1))$. All operations act within the block-diagonal (classical) sector, ensuring Born probabilities coincide with classical SCM semantics.
  • Figure 3: Comparison of 3-qubit Simpson's Paradox results from ideal simulation (blue bars) and the IonQ Aria QPU (orange bars). Both the simulation and the QPU experiment demonstrate the core phenomenon: the treatment effect is positive within the male and female subgroups, but reverses to become negative when aggregated ('Overall (Obs.)'). In both cases, the quantum causal intervention ($\mathcal{DO}$-operation) successfully resolves the paradox by revealing a robustly positive true causal effect ('Overall (Causal)'). Error bars represent 95% confidence intervals, calculated over 30 trials for the simulation and 3 successful trials for the QPU experiment. The QPU results qualitatively reproduce the simulation's trend, validating the method on real hardware, while the larger error bars on the QPU data reflect the impact of physical noise.
  • Figure 4: Quantifying confounding bias in the 10-qubit healthcare simulation. This figure compares the treatment effect ($\Delta P$) calculated from raw observational data against results from conventional stratification and our quantum causal intervention. The observational data ('Overall (Obs.)') significantly underestimates the true causal effect, yielding an effect of +0.377, whereas the true effect revealed by the intervention ('Causal Intervention') is +0.486. Simple stratification by a single variable provides only a partial and inconsistent correction; stratifying by Region yields an effect of +0.406, while stratifying by the stronger confounder, Age, yields +0.497.

Theorems & Definitions (2)

  • Proposition 1: Correctness of circuit surgery for the compiled family
  • proof : Proof sketch