Quantum machine learning performance evaluation for air pollution assessment with benchmarking and comparative analysis using multi-metric scoring and quantum circuit expressibility

Technical Summary: Quantum Machine Learning Performance Evaluation for Air Pollution Assessment

Problem Statement
Air quality degradation necessitates immediate remedial action, prompting the exploration of Machine Learning (ML), Deep Learning (DL), and Quantum Machine Learning (QML) for predicting the Air Quality Index (AQI). While various studies have applied quantum-inspired models or hybrid quantum-classical approaches to air pollution data, there is a lack of a unified, multi-metric framework that rigorously benchmarks classical and quantum models against one another. Specifically, existing literature often fails to account for the structural complexities of quantum circuits—such as expressibility and entangling power—and their relationship to generalization gaps and optimization pathologies like barren plateaus under Noisy Intermediate-Scale Quantum (NISQ) constraints. This study addresses the gap by proposing a comparative evaluation of ML, DL, and QML models using a multi-metric hierarchy that incorporates quantum-specific structural penalties.

Methodology
The study utilizes a dataset of 257,834 observations from 2015 to 2020, sourced from the Central Pollution Control Board (India), featuring 13 predictors (including PM2.5, PM10, NOx, SO2, O3, and volatile organic compounds) to predict the AQI. The data underwent preprocessing involving z-score normalization, outlier removal via Isolation Forest, imputation, and Yeo-Johnson transformation to address skewness. Features were reduced to four principal components (retaining 96.5% variance) for quantum encoding.

The evaluation framework, termed MDQ-3L (Machine Learning, Deep Learning, Quantum - 3-Level), integrates three distinct levels of assessment:

1. Level 1 (Predictive Accuracy): Aggregates normalized regression metrics (MAE, MSE, RMSE, MAPE, RMSLE) and the coefficient of determination ($R^2$).
2. Level 2 (Generalization): Measures the gap between training and testing RMSE ($\Delta_m$).
3. Level 3 (Quantum Complexity): Applies a structural regularizer ($\mathcal{R}_m^{(Q)}$) specific to QML models, penalizing high expressibility ($E_m$) and entangling power ($P_m$) to account for NISQ limitations and barren plateau risks.

The unified performance functional ($J_m$) combines these levels with weighted coefficients ($\alpha, \beta, \gamma$), prioritizing predictive accuracy (80%) while incorporating complexity penalties (20%). An extended version, MDQ-3L+$\mathcal{B}$, includes a benchmarking term ($\mathcal{B}_m^{(exp)}$) that penalizes deviations of a circuit's expressibility and entangling power from target values.

The study evaluated:

• 17 ML Regressors: Including CatBoost, XGBoost, Random Forest, and various linear models.
• 9 DL Models: Including MLP, CNN, LSTM, GRU, DBN, and Transformers.
• 7 QML Circuits: Simulated on noiseless backends with 4 qubits, including Random Layers Regressor (RLR), RYCZ, Shifted Entangled Linear Regressor (SELR), Hardware Efficient Ansatz (HEA), Quantum Convolutional Neural Network (QCNN), Tree Tensor Network (TTN), and Quantum Attention (QAttn).

Key Results

• Classical Performance: Tree-based ensemble methods, specifically CatBoost and Extra Trees, and the Deep Belief Network (DBN) emerged as top performers. CatBoost achieved an $R^2$ of 0.68 with an RMSE of 72.41, while DBN achieved the highest $R^2$ (0.72) and lowest MSE (4807.5) among all models.
• Quantum Performance: All QML circuits exhibited negative $R^2$ values (ranging from -1.2 to -7.9), indicating performance worse than a constant-mean baseline. While some circuits like QAttn and RYCZ showed competitive raw error metrics (e.g., QAttn MSE $\approx$ 0.25), their negative variance explanation and structural penalties resulted in significantly higher composite scores compared to classical models.
• Structural Penalties: Circuits with high expressibility (QCNN, TTN with $E \approx 2.0$) and high entangling power (RLR with $P \approx 0.40$) incurred substantial penalties. The MDQ-3L framework revealed that raw circuit capacity does not translate to predictive utility under current constraints.
• Benchmarking Impact: The inclusion of the benchmarking term ($\mathcal{B}$) reduced penalties for architectures like QCNN and TTN by approximately 30% when aligned with target expressibility/entanglement, yet they still trailed classical models by 25-40% in composite scores.
• Feature Importance: SHAP analysis confirmed that PM2.5 is the dominant driver of AQI, followed by PM10, CO, and O3, with the models relying heavily on particulate matter concentrations.

Significance and Claims
The paper claims to provide an interpretable, multi-metric tool for hybrid environmental modeling that advances Quantum Machine Intelligence benchmarks. Its primary contributions include:

1. Unified Benchmarking: The MDQ-3L framework successfully aggregates heterogeneous metrics and quantum structural properties into a single, comparable score, preventing models from being selected based on a single favorable metric while failing in others.
2. Exposing QML Limitations: The study quantitatively demonstrates that under NISQ constraints (4-6 qubits, shallow depth), current QML circuits suffer from generalization gaps and optimization pathologies (barren plateaus), resulting in performance deficits of 25-40% compared to classical ML/DL baselines.
3. Guidance for Future Architectures: By penalizing high expressibility and entanglement, the framework suggests that low-entanglement ansätze (such as SELR and HEA) with moderate expressibility may offer a 10-15% potential gain in future iterations by better aligning with NISQ trainability requirements.
4. Methodological Rigor: The work moves beyond qualitative discussions of entanglement and expressibility by embedding them as quantitative regularization coordinates, offering a rigorous criterion for model selection that balances statistical accuracy with hardware-aligned complexity.

The authors conclude that while quantum circuits are not yet superior for this specific regression task, the proposed framework provides a necessary structure for evaluating future hybrid models, ensuring that improvements in one dimension do not obscure pathologies in another.