How Quantum Circuits Actually Learn: A Causal Identification of Genuine Quantum Contributions

Abstract

Attributing performance gains in quantum machine learning to genuine quantum resources rather than to classical architectural scaling remains an open methodological challenge. We address this by introducing a counterfactual causal mediation framework that decomposes inter-architectural performance differences into direct effects, attributable to circuit parameterization and expressivity, and indirect effects mediated by quantum information-theoretic observables: entanglement entropy, purity, linear entropy, and quantum mutual information. Applying this framework to five circuit topologies and three benchmark datasets (across 43 validated configurations) reveals that direct architectural contributions systematically exceed quantum-mediated effects, with a mean ratio of 13.1:1 and a mean indirect contribution of 0.82%. These results suggest that current variational quantum circuits operate substantially below their quantum potential, and that principled resource-aware circuit design represents a tractable path toward measurable quantum-mediated performance gains.

Figures (12)

• Figure 1: Counterfactual causal mediation framework for quantum machine learning architectures. (A) Conceptual causal graph representing the learning pipeline as a counterfactual causal system. An architectural intervention $t$ contrasts two complete quantum learning pipelines: a baseline shallow circuit ($t=0$) and an enhanced, deeper and more entangling circuit ($t=1$). The intervention influences predictive outcomes $Y$ both directly, through non-quantum architectural factors (e.g., parameterization, expressivity, optimization geometry), and indirectly via a set of quantum-information-theoretic mediators $M$. These mediators, computed post hoc from the trained quantum states, quantify architecture-induced variations in entanglement entropy ($S_A$), purity ($\gamma_A$), linear entropy ($L_A$), and quantum mutual information ($I(A\!:\!B)$). (B) Counterfactual decomposition of the performance contrast between architectures. The causal estimand is the inter-architectural counterfactual difference $Y(t{=}1)-Y(t{=}0)$, evaluated for identical test samples. Within a structural causal model, this total effect decomposes additively into a direct architectural effect (DE), denoted $\tau$, and an indirect quantum-mediated effect (IDE), given by $\sum_k \alpha_k \beta_k$. This decomposition enables a non-redundant attribution of performance changes to genuine quantum resource utilization versus classical architectural scaling, without modeling internal circuit dynamics.
• Figure 2: Dataset-dependent causal decomposition reveals architectural dominance over quantum-mediated pathways. Violin plots display total effects ($\Delta Y$), direct effects ($\tau$, purple), and indirect effects ($\sum\alpha\beta$, orange) across 43/90 configurations (architecture_{n_qubit}_{n_layers}) validating both assumptions (Diabetes n=26, Breast Cancer n=12, Ionosphere n=5). Direct architectural contributions systematically dominate (mean ratio 13.1): Diabetes 3.39$\pm$3.11% direct vs. $-0.34\pm0.65$% indirect; Breast Cancer 5.14$\pm$3.17% vs. $-0.37\pm0.67$%; Ionosphere 8.46$\pm$8.83% vs. $-0.28\pm0.57$%. Quantum-mediated effects account for only 11.1%, 7.8%, and 3.5% of total variation respectively. Ring and deep topologies achieve strongest direct effects (6 - 17%), while pairwise exhibits negligible contributions across datasets. Framework successfully distinguishes classical scaling from quantum enhancement, with 93% configurations classified as Neutral regime.
• Figure 3: Mediator-specific contributions reveal task-dependent quantum resource utilization. Mean Absolute Mediated Contribution (MAMC) quantifies each quantum mediator's causal influence. Purity ($\gamma_A$) dominates in Diabetes (1.82%) and Breast Cancer (0.86%), exceeding entanglement entropy ($S_A$: 0.20%, 0.29%) by factors of 9.1 and 3.0 respectively, indicating coherence preservation as primary quantum pathway. Ionosphere shows $S_A$ (0.42%) as sole quantifiable mediator. Linear entropy ($L_A$) contributes substantially in Diabetes (1.58%), while mutual information ($I(A:B)$) remains weak due to functional dependence on $S_A$. Cross-dataset patterns reveal coherence-based mediation dominates over entanglement in current NISQ implementations, providing actionable targets for architecture-task alignment through mediator amplification strategies.cerezo2025does
• Figure 4: FIG. 4. Regime classification reveals absence of quantum advantage under rigorous causal identifiability criteria. The $43/90$ configurations satisfying all mediation assumptions are positioned in the indirect effect performance space $(\Sigma\alpha\beta, \Delta Y)$ relative to the $\pm10.6\%$ data-driven significance threshold (maximum across all architecture dataset pairs; individual thresholds vary by configuration, see Methods \ref{['sec:threshold']}). Each point represents one architecture dataset configuration; the threshold defines the boundary between statistically negligible and substantive causal contributions. Regime distribution: $93.0\%$ Neutral (both direct and indirect effects below threshold) and $7.0\%$ Classical-Scalable (significant direct gains, negligible quantum mediation). Zero configurations reach the Quantum-Advantage regime ($\tau >$ threshold, $\sum\alpha\beta >$threshold, $\Delta Y >$ threshold), confirming that quantum-mediated contributions remain uniformly below significance thresholds across all circuit topologies and datasets. Even the highest-performing configuration (Ionosphere deep_4_3, $\Delta Y \approx 17\%$) exhibits indirect effects below $1\%$, demonstrating that performance gains are driven entirely by direct architectural scaling.
• Figure 5: Architecture-specific decomposition exposes topology-dependent mediation patterns across all three datasets. Bar charts display the total performance change ($\Delta Y$, bar height) and quantum-mediated indirect effects ($\sum\alpha\beta$, orange markers) with causal regime classification, restricted to configurations satisfying both identification assumptions. The vertical gap between $\Delta Y$ and $\sum\alpha\beta$ quantifies the direct architectural contribution $\tau$. (A) Breast Cancer: Deep_4_3 achieves the largest total effect ($\Delta Y \approx 5\%$) driven entirely by direct pathways ($\tau \approx 5.4\%$, indirect $\approx -0.5\%$); pairwise configurations remain below $2\%$ across all layers. (B) Diabetes: Ring and deep topologies achieve the strongest gains (total $6$ - $8\%$); the largest indirect effect observed in this dataset (full_4_6: $\Sigma\alpha\beta \approx 2.3\%$) remains below the significance threshold, classifying all configurations as Neutral or Classical-Scalable.(C) Ionosphere: Deep_4_3 exhibits the highest total effect across all datasets ($\Delta Y \approx 14$ - $17\%$) with indirect effects below $1\%$, confirming exclusive direct-pathway dominance. Pairwise universally fails ($\Delta Y < 1\%$) across all datasets, consistent with its localized bipartite entanglement structure. These topology-dependent patterns suggest actionable modifications targeted mediator amplification via purity-preserving encodings or entanglement-stabilizing ansätze to transition configurations from the Neutral toward the Quantum-Advantage regime.

Theorems & Definitions (29)

• Theorem 1: Counterfactual Causal Decomposition of Architectural Effects
• Definition 1: Mean Absolute Mediated Contribution
• Definition 2: Relative Quantum Contribution
• Remark 1: Pure State Relations
• Definition 3: Potential Mediators
• Definition 4: Potential Outcomes
• Definition 5: Observed Quantities
• Definition 6: Sample-Level Total Effect
• Definition 7: Average Total Effect (ATE)
• Definition 8: Structural Parameters