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Quantum Earth Mover's Distance: A New Approach to Learning Quantum Data

Bobak T. Kiani, Giacomo De Palma, Milad Marvian, Zi-Wen Liu, Seth Lloyd

Published Jan 8, 2021• 4 citations
OpenAlex

Abstract

Quantifying how far the output of a learning algorithm is from its target is an essential task in machine learning. However, in quantum settings, the loss landscapes of commonly used distance metrics often produce undesirable outcomes such as poor local minima and exponentially decaying gradients. As a new approach, we consider here the quantum earth mover's (EM) or Wasserstein-1 distance, recently proposed in [De Palma et al., arXiv:2009.04469] as a quantum analog to the classical EM distance. We show that the quantum EM distance possesses unique properties, not found in other commonly used quantum distance metrics, that make quantum learning more stable and efficient. We propose a quantum Wasserstein generative adversarial network (qWGAN) which takes advantage of the quantum EM distance and provides an efficient means of performing learning on quantum data. Our qWGAN requires resources polynomial in the number of qubits, and our numerical experiments demonstrate that it is capable of learning a diverse set of quantum data.

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Quantifying how far the output of a learning algorithm is from its target is an essential task in machine learning. However, in quantum settings, the loss landscapes of commonly used distance metrics often produce undesirable outcomes such as poor local minima and exponentially decaying gradients. As a new approach, we consider here the quantum earth mover's (EM) or Wasserstein-1 distance, recently proposed in [De Palma et al., arXiv:2009.04469] as a quantum analog to the classical EM distance. We show that the quantum EM distance possesses unique properties, not found in other commonly used quantum distance metrics, that make quantum learning more stable and efficient. We propose a quantum Wasserstein generative adversarial network (qWGAN) which takes advantage of the quantum EM distance and provides an efficient means of performing learning on quantum data. Our qWGAN requires resources polynomial in the number of qubits, and our numerical experiments demonstrate that it is capable of learning a diverse set of quantum data.
Abstract Quantifying how far the output of a learning algorithm is from its target is an essential task in machine learning. However, in quantum settings, the loss landscapes of commonly used distance metrics often produce undesirable outcomes such as poor local minima and exponentially decaying gradients. To overcome these obstacles, we consider here the recently proposed quantum earth mover’s (EM) or Wasserstein-1 distance as a quantum analog to the classical EM distance. We show that the quantum EM distance possesses unique properties, not found in other commonly used quantum distance metrics, that make quantum learning more stable and efficient. We propose a quantum Wasserstein generative adversarial network (qWGAN) which takes advantage of the quantum EM distance and provides an efficient means of performing learning on quantum data. We provide examples where our qWGAN is capable of learning a diverse set of quantum data with only resources polynomial in the number of qubits.
The study of quantum generative models is well-motivated, not only because of its importance in quantum machine learning and quantum chemistry but also because of the perspective of its implementation on near-term quantum machines. Inspired by previous studies on the adversarial training of classical and quantum generative models, we propose the first design of quantum Wasserstein Generative Adversarial Networks (WGANs), which has been shown to improve the robustness and the scalability of the adversarial training of quantum generative models even on noisy quantum hardware. Specifically, we propose a definition of the Wasserstein semimetric between quantum data, which inherits a few key theoretical merits of its classical counterpart. We also demonstrate how to turn the quantum Wasserstein semimetric into a concrete design of quantum WGANs that can be efficiently implemented on quantum machines. Our numerical study, via classical simulation of quantum systems, shows the more robust and scalable numerical performance of our quantum WGANs over other quantum GAN proposals. As a surprising application, our quantum WGAN has been used to generate a 3-qubit quantum circuit of ~50 gates that well approximates a 3-qubit 1-d Hamiltonian simulation circuit that requires over 10k gates using standard techniques.
Quantum computers have the potential to outperform classical counterparts by leveraging quantum algorithms; however, achieving this advantage requires the efficient encoding of generic data representations into qubit states. Hybrid Quantum-Classical Generative Adversarial Networks (qGANs) have shown potential for learning generic probability distributions from real datasets and loading them efficiently into qubits. This work extends previous approaches to qGANs by implementing a configurable framework for generating multidimensional datasets with specific distributions, training hybrid qGANS to reproduce those distributions, and evaluating the performance of the trained model. The framework supports the configuration of multiple loss functions. In the paper, we evaluate the framework by comparing the training performance when using Binary Cross-Entropy (BCE) and Wasserstein distance as loss functions during the training of qGANS on one, two, and three-dimensional datasets with different distributions. The paper paves the way for future experimental research on implementing efficient quantum machine learning algorithms based on properties of the Wasserstein distance.
Anomaly detection in time series is constrained by the scarcity and unpredictability of real anomalous events. Classical supervised and generative approaches struggle to capture the statistical complexity of rare, high-impact anomalies while maintaining realistic normal dynamics. This work introduces a Quantum Wasserstein Generative Adversarial Network (QWGAN) for synthetic anomaly generation that separates the modeling of normal and anomalous behaviors using quantum parameterized circuits. Leveraging quantum superposition and entanglement, the QWGAN achieves a richer generative capacity with an order-of-magnitude reduction in parameters compared to classical counterparts. The model is trained and evaluated on cyber security datasets, demonstrating lower Earth Mover’s Distance and volatility error while preserving temporal structure and distributional fidelity. A windowed synthesis mechanism enables controlled injection of variable-length point and contextual anomalies. Our results demonstrate that QWGAN provides a scalable, parameter-efficient approach for generating high-fidelity synthetic anomalous time series data to enable effective benchmarking and augmentation of anomaly detection models under limited anomaly data.
Abstract Quantum machine learning offers powerful advantages in processing high-dimensional data, but loading classical probability distributions into quantum states remains a major challenge. This work extends quantum generative adversarial networks (qGANs) to overcome two critical limitations of early approaches: poor robustness to noise on near-term devices and limited capacity to model complex, multi-modal distributions. We introduce hybrid qGAN architectures that combine Wasserstein GAN with gradient penalty (WGAN-GP) and maximum mean discrepancy (MMD) losses with expressive quantum circuits, including quantum convolutional neural networks (QCNNs) and EfficientSU2 ansätze. Training is performed using seamless PyTorch–Qiskit integration, enabling stable optimization on both simulators and real quantum hardware.Experiments on 2D Gaussian and log-normal distributions—directly relevant to financial modeling—show faster convergence and higher fidelity than prior qGAN designs, with up to 80% lower Wasserstein distance under 5% depolarizing noise. The framework is further validated through European call option pricing via quantum amplitude estimation (QAE), achieving pricing errors below 1% on IBM’s 20-qubit superconducting systems. Theoretical analysis confirms gradient stability under noise, while empirical results include comprehensive noise sweeps, t-SNE latent space visualizations, and rigorous statistical tests (KL divergence, KS tests). This work establishes a practical path toward quantum advantage in generative modeling, with fully open-source code for reproducibility.
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 (R2=0.9999R^2 = 0.9999R2=0.9999) with mean absolute errors on the order of 10−510^{-5}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 W1W_1W1​ 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.