Regularized Quantum Wasserstein GAN

• The paper introduces a novel formulation of quantum Wasserstein metrics by integrating entropic regularization to enable smooth, gradient-based training.
• It adapts classical GAN techniques to the quantum realm using penalty measures and manifold regularization that ensure stability and convergence.
• Empirical results demonstrate that these models can efficiently learn both pure and mixed quantum states, while also achieving significant quantum circuit compression.

A Regularized Quantum Wasserstein GAN (QWGAN) extends adversarial generative modeling to quantum probability spaces by minimizing a quantum analog of the Wasserstein distance between data and generator distributions, while incorporating explicit regularization at the level of quantum optimal transport. This paradigm adapts both the statistical and geometric regularization strategies developed for classical Wasserstein GANs, introducing quantum metrics, entropic penalties, and scalable optimization methods suitable for learning quantum states or channels. Below, the mathematical foundations, regularization techniques, optimization algorithms, theoretical guarantees, applied architectures, and quantum-specific challenges are detailed, referencing direct results from the literature.

1. Quantum Wasserstein Metric Formulation and Regularization

The standard classical setting defines the Wasserstein distance between two probability distributions μ and ν as:

$$W_1(\mu, \nu) = \inf_{\pi \in \Pi(\mu, \nu)} \int_{\mathcal{X} \times \mathcal{X}} \|x - y\| \, d\pi(x, y)$$

where Π(μ, ν) is the set of couplings. In a quantum context, distributions are replaced by density operators ρ, σ over Hilbert space $\mathcal{H}$; a coupling π is represented by a joint density on $\mathcal{H} \otimes \mathcal{H}$. The quantum Wasserstein metric takes the form (Chakrabarti et al., 2019):

$$qW(\rho, \sigma) = \min_{\pi \in D(\mathcal{H} \otimes \mathcal{H})} \mathrm{Tr}[\pi C] \quad \text{subject to} \quad \mathrm{Tr}_1(\pi) = \rho,\ \mathrm{Tr}_0(\pi) = \sigma$$

with a quantum cost matrix $C = (I - \text{SWAP})/2$ ensuring sensitivity to off-diagonal components. Regularization is introduced—following classical analogs—by adding a quantum relative entropy term (entropic regularization):

$$qW_\lambda(\rho, \sigma) = \min_{\pi} \left\{ \mathrm{Tr}[\pi C] + \lambda\, S(\pi \| \rho \otimes \sigma) \right\}$$

where $S(\pi \| \rho \otimes \sigma)$ denotes quantum relative entropy. This smoothing guarantees differentiability for gradient-based learning, stabilizes updates, and controls expressivity.

2. Duality and Regularized Quantum Optimal Transport

The classical Kantorovich duality is extended to quantum spaces with regularization:

$$\max_{\phi, \psi} \left[ \mathrm{Tr}(\sigma \psi) - \mathrm{Tr}(\rho \phi) - \mathbb{E}_{\rho \otimes \sigma}[\xi_R] \right]$$

with $$\xi_R = (\lambda/e) \exp\left( \frac{-C - \phi \otimes I + I \otimes \psi}{\lambda} \right)$$. This unconstrained dual replaces the difficult 1-Lipschitz constraint with soft penalties, paralleling techniques in regularized classical optimal transport (Sanjabi et al., 2018). For efficient computation, dual variables are parameterized as either Hermitian matrices (quantum circuits), sums over Pauli strings, or hybrid classical-quantum networks (Jurasz et al., 2023). The adversarial optimization then alternates between maximizing this regularized dual over the critic (discriminator) and minimizing it over the generator.

3. Regularization Techniques: Entropic, Penalty Measures, and Manifolds

• Entropic Regularization: Quantum analogs of classical entropic regularization employ quantum relative entropy as a convex penalty, smoothing the landscape and simplifying gradient estimation (Chakrabarti et al., 2019, Sanjabi et al., 2018). This regularization is essential for tractable optimization in high-dimensional quantum settings.
• Penalty Measures and Gradient Penalties: Classical GANs use penalties on the gradient norm of the discriminator (SGP $\mu$-WGAN (Kim et al., 2018)); quantum generalizations can regularize observable gradients with respect to circuit parameters or enforce constraints on measurement outcomes (Gemici et al., 2018).
• Manifold Regularization: When training quantum GANs on states defined by physical manifolds (e.g., quantum phases), adding manifold regularizers defined via a function $\psi$ over measurement expectation vectors improves stability, equilibrium, and avoids mode collapse (Li et al., 2018). For quantum datasets, these penalties ensure generator outputs respect underlying quantum geometric constraints.

4. Optimization Algorithms for Regularized QWGANs

Incorporating regularization enables the use of first-order stochastic optimization methods:

• Oracle-based SGD: Approximate solutions to the regularized dual are used to supply gradients; SGD (or Adam) updates parameters (Sanjabi et al., 2018, Ballu et al., 2020).
• Proximal Methods: Wasserstein-proximal updates based on the Riemannian geometry of probability distributions are adapted to parameter space using semi-backward Euler discretizations and affine test functions (Lin et al., 2021). For quantum settings, similar proximal penalties may stabilize parameter updates.
• Linear Programming: In qWGANs constrained to Pauli strings, the critic optimization is recast as a scalable LP under quantum Lipschitz constraints (Jurasz et al., 2023, Kiani et al., 2021).

5. Theoretical Properties: Convergence, Generalization, Stability

Regularized QWGANs benefit from:

• Smoothness: Adding convex (e.g., KL or quantum relative entropy) regularizers renders the objective differentiable with respect to generator parameters $\theta$, allowing provable convergence to stationary points. For example, with a regularization weight $\lambda>0$, the generator objective $h_\lambda(\theta)$ is $L$-smooth (Sanjabi et al., 2018).
• Statistical Bounds: Sample complexity and generalization error rates are established for regularized estimators. For regularization parameter $\delta$ and sample size $n$, the estimator's deviation from the true quantum Wasserstein cost $$\leq \epsilon(n, \rho, ..., K_\lambda) + \lambda \delta$$ (Mahdian et al., 2019).
• Stability: Penalty measures applied along quantum data or sample manifolds maintain local stability in adversarial training, preventing instability and collapse (Kim et al., 2018).
• Manifold Equilibrium: Manifold regularization ensures approximate equilibrium in adversarial dynamics, essential for reliable state preparation at unseen parameter points in quantum phase diagrams (Jurasz et al., 2023).

6. Empirical Performance and Quantum Applications

Regularized QWGANs have demonstrated:

• Robust, Scalable Training: Simulations show stable learning for pure and mixed quantum states up to 8 qubits (Chakrabarti et al., 2019), and for state preparation across the phase diagram for quantum systems (Jurasz et al., 2023).
• Efficient Circuit Compression: In one application, a qWGAN learned a 50-gate quantum circuit approximating the dynamics of a 3-qubit Hamiltonian simulation that would natively require over 10,000 gates (Chakrabarti et al., 2019).
• Generalization to Unseen States: By coupling classical interpolation of measurement expectation vectors with quantum adversarial training, new quantum states can be generated at arbitrary points throughout a phase diagram—not limited to those present in the training data (Jurasz et al., 2023).

Regularization Type	Quantum Analog	Empirical Impact
Entropic (KL)	Quantum relative entropy	Smooth gradient, scalable updates
Gradient penalty	Observable/circuit gradient control	Stability in high dimensions
Manifold regularizer	Expectation interpolation/embedding	Generalization, avoids collapse

7. Extensions, Limitations, and Future Directions

• Metric Generalization: The choice of quantum metric—e.g., quantum Earth Mover's (Wasserstein-1) distance (Kiani et al., 2021)—can be tuned for task-specific sensitivity (e.g., local operations, entanglement, coherence).
• Optimization Scaling: Stochastic gradient and linear programming approaches have enabled practical training for moderate qubit counts; further advances in quantum hardware or algorithms are required for large systems.
• Hybrid Classical–Quantum Architectures: Combining classical probabilistic models (GANs) for interpolation or meta-learning of measurement expectations with quantum state generation enhances model flexibility (Jurasz et al., 2023).
• Open Questions: Direct adaptation of Riemannian/proximal regularization (Wasserstein geometry) to quantum spaces, the design of quantum critics with structured measurements, and the effect of regularization on NISQ noise robustness remain subjects for future research.

Regularized Quantum Wasserstein GANs redefine generative modeling for quantum data by merging statistical optimal transport regularization, manifold constraints, and quantum-adapted loss functions with scalable optimization methods. This synthesis of theory and implementation underpins current empirical success in robust quantum state and channel learning, circuit compression, and generalization beyond the original training set.