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Machine Learning Framework for Efficient Prediction of Quantum Wasserstein Distance

Changchun Feng, Xinyu Qiu, Laifa Tao, Lin Chen

TL;DR

The paper tackles the computational bottleneck of the quantum Wasserstein distance $W_1$ by introducing a data-driven framework that leverages physically meaningful features from quantum state pairs. It systematically compares neural networks and tree-based methods, with Random Forests delivering near-perfect predictions (e.g., $R^2=0.9999$ and MAE~$10^{-5}$ for 3-qubit systems) and tree ensembles outperforming other approaches across 2–4 qubits. The authors validate theoretical propositions in quantum information theory using ML-predicted $W_1$ distances, demonstrating reliable, scalable verification of measurement-probability bounds and gate-error-rate bounds, thereby enabling real-time quantum circuit assessment in NISQ devices. Beyond prediction, the work provides insights into feature importance (Pauli-based measurements dominate) and highlights the potential of ML-guided theory validation and error-correction design. This framework paves the way for fast, scalable quantification of local distinguishability in quantum operations, with practical impact on quantum circuit optimization and fault-tolerant protocol development.

Abstract

The quantum Wasserstein distance (W-distance) is a fundamental metric for quantifying the distinguishability of quantum operations, with critical applications in quantum error correction. However, computing the W-distance remains computationally challenging for multiqubit systems due to exponential scaling. We present a machine learning framework that efficiently predicts the quantum W-distance by extracting physically meaningful features from quantum state pairs, including Pauli measurements, statistical moments, quantum fidelity, and entanglement measures. Our approach employs both classical neural networks and traditional machine learning models. On three-qubit systems, the best-performing Random Forest model achieves near-perfect accuracy ($R^2 = 0.9999$) with mean absolute errors on the order of $10^{-5}$. We further validate the framework's practical utility by successfully verifying two fundamental theoretical propositions in quantum information theory: the bound on measurement probability differences between unitary operations and the $W_1$ gate error rate bound. The results establish machine learning as a viable and scalable alternative to traditional numerical methods for W-distance computation, with particular promise for real-time quantum circuit assessment and error correction protocol design in NISQ devices.

Machine Learning Framework for Efficient Prediction of Quantum Wasserstein Distance

TL;DR

The paper tackles the computational bottleneck of the quantum Wasserstein distance by introducing a data-driven framework that leverages physically meaningful features from quantum state pairs. It systematically compares neural networks and tree-based methods, with Random Forests delivering near-perfect predictions (e.g., and MAE~ for 3-qubit systems) and tree ensembles outperforming other approaches across 2–4 qubits. The authors validate theoretical propositions in quantum information theory using ML-predicted distances, demonstrating reliable, scalable verification of measurement-probability bounds and gate-error-rate bounds, thereby enabling real-time quantum circuit assessment in NISQ devices. Beyond prediction, the work provides insights into feature importance (Pauli-based measurements dominate) and highlights the potential of ML-guided theory validation and error-correction design. This framework paves the way for fast, scalable quantification of local distinguishability in quantum operations, with practical impact on quantum circuit optimization and fault-tolerant protocol development.

Abstract

The quantum Wasserstein distance (W-distance) is a fundamental metric for quantifying the distinguishability of quantum operations, with critical applications in quantum error correction. However, computing the W-distance remains computationally challenging for multiqubit systems due to exponential scaling. We present a machine learning framework that efficiently predicts the quantum W-distance by extracting physically meaningful features from quantum state pairs, including Pauli measurements, statistical moments, quantum fidelity, and entanglement measures. Our approach employs both classical neural networks and traditional machine learning models. On three-qubit systems, the best-performing Random Forest model achieves near-perfect accuracy () with mean absolute errors on the order of . We further validate the framework's practical utility by successfully verifying two fundamental theoretical propositions in quantum information theory: the bound on measurement probability differences between unitary operations and the gate error rate bound. The results establish machine learning as a viable and scalable alternative to traditional numerical methods for W-distance computation, with particular promise for real-time quantum circuit assessment and error correction protocol design in NISQ devices.

Paper Structure

This paper contains 18 sections, 2 theorems, 19 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Methods
  4. Feature Extraction
  5. Data Generation
  6. Model Architectures
  7. Training and Evaluation
  8. Numerical Results
  9. Application
  10. Machine learning validation of the difference of unitary operations
  11. A machine learning verification of error rate propositions in $W_1$
  12. Example: Noise Sensitivity Analysis for Common Quantum Gates
  13. Discussion
  14. Conclusion
  15. Acknowledgments
  16. ...and 3 more sections

Key Result

Proposition 1

PhysRevA.110.012412 For two unitary operations $U$ and $V$ acting on the same initial state $|\psi\rangle$, and a POVM element $M_m$ with maximal eigenvalue $\lambda_{\mathop{\rm max}}(M_m) \in (0,1]$, the measurement probability difference is upper bounded by where $D(U,V)$ is the quantum $W_1$ distance between $U$ and $V$.

Figures (5)

  • Figure 1: Model performance comparison across MSE, MAE, and $R^2$ metrics. The horizontal bar charts illustrate the performance hierarchy, with tree-based models (Random Forest, Decision Tree, Gradient Boosting) achieving superior performance compared to linear models and neural networks.
  • Figure 2: Top 20 features most correlated with the target W-distance. The bar chart displays the absolute Pearson correlation coefficients between each feature and the W-distance, ranked in descending order. All top 20 features are derived from Pauli matrix measurements, specifically expectation value differences and product terms, demonstrating that local measurement statistics are the most predictive features for W-distance estimation.
  • Figure 3: Machine learning validation of Proposition \ref{['prop:operations']}. (a) Scatter plot comparing model-predicted $D(U,V)$ versus true $D(U,V)$. (b) Histogram of inequality ratios for both predicted (orange) and true (blue) distances. The red vertical line marks the theoretical boundary (ratio = 1.0). Both distributions are concentrated well below 1.0, confirming that the ML model preserves the theoretical guarantees.
  • Figure 4: Machine learning validation of Proposition \ref{['prop:w1_error_rate']}. (a) Scatter plot comparing true error rate $e(U,V)$ versus upper bound computed using true distances. (b) Scatter plot comparing true error rate versus upper bound computed using ML-predicted distances. (c) Histogram of inequality ratios for true distances. (d) Histogram of inequality ratios for ML-predicted distances. The red dashed lines in (a) and (b) indicate the boundary $y=x$, and the red vertical lines in (c) and (d) mark the theoretical boundary (ratio = 1.0). All ratios are concentrated well below 1.0, confirming that the machine learning model preserves the theoretical guarantees of Proposition \ref{['prop:w1_error_rate']}.
  • Figure 5: $W_1$ gate error rates $e(U, V)$ for different quantum gates under various noise channels, computed using our ML model. The results reveal gate-specific noise sensitivity: Swap gate is more robust to bit-flip noise but sensitive to phase errors, while CZ gate shows the opposite pattern. Multi-qubit gates (Toffoli, Fredkin) exhibit distinct error rate profiles.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2