Flowing Through Hilbert Space: Quantum-Enhanced Generative Models for Lattice Field Theory

TL;DR

The paper tackles the sampling bottleneck in lattice field theory by introducing a Hybrid Quantum-Classical Normalizing Flow (HQCNF) that embeds a parameterized quantum circuit within a classical flow to generate field configurations from $p(phi) ~ exp(-S[phi])$. By alternating classical affine coupling layers with a quantum transformation, HQCNF leverages quantum entanglement and amplitude encoding to enhance expressivity while reducing the required classical depth. On a 2D $\phi^4$ theory on an $8\times 8$ lattice, HQCNF achieves comparable or improved fidelity to a classical NF baseline, as evidenced by effective action correlations, field-value distributions, and two-point functions, while dramatically shortening training time and layers. This work suggests a viable path toward quantum-enhanced generative modeling for more complex field theories and larger lattices, potentially accelerating high-energy physics simulations, with future work addressing scalability and hardware-realistic noise.

Abstract

Sampling from high-dimensional and structured probability distributions is a fundamental challenge in computational physics, particularly in the context of lattice field theory (LFT), where generating field configurations efficiently is critical, yet computationally intensive. In this work, we apply a previously developed hybrid quantum-classical normalizing flow model to explore quantum-enhanced sampling in such regimes. Our approach embeds parameterized quantum circuits within a classical normalizing flow architecture, leveraging amplitude encoding and quantum entanglement to enhance expressivity in the generative process. The quantum circuit serves as a trainable transformation within the flow, while classical networks provide adaptive coupling and compensate for quantum hardware imperfections. This design enables efficient density estimation and sample generation, potentially reducing the resources required compared to purely classical methods. While LFT provides a representative and physically meaningful application for benchmarking, our focus is on improving the sampling efficiency of generative models through quantum components. This work contributes toward the development of quantum-enhanced generative modeling frameworks that address the sampling bottlenecks encountered in physics and beyond.

Figures (6)

• Figure 1: Schematic of the HQCNF model. The input $8 \times 8$ lattice field configuration is split into two components, which are processed by the classical part of the model using affine coupling layers with learned scaling and translation functions. The control parameters used in these classical transformations are then passed to the quantum component, which applies amplitude embedding and parameterized quantum rotations. The process is repeated over $N_{layers}$ number of layers (2 in this work), enabling the model to iteratively learn the correlations present in the target field theory.
• Figure 2: Comparison between lattice field configurations sampled directly from the prior distribution (left) and those generated after training the hybrid quantum-classical normalizing flow model (right). The trained model produces configurations with smoother spatial structure and amplitude consistent with the target $\phi^4$ distribution.
• Figure 3: Correlation between the effective action $S_{\mathrm{eff}} = -\log q_\theta(\phi)$ and the true action $S(\phi)$ for generated samples. Perfect modeling would correspond to a slope-1 linear relationship, up to a constant shift.
• Figure 4: Histogram comparison of the action values $S[\phi]$ for reference and HQCNF-generated configurations.
• Figure 5: Distribution of scalar field values $\phi(x)$ across all lattice sites, comparing reference configurations and those generated by the HQCNF. The similarity between the two distributions suggests that the HQCNF captures the global statistical properties of the field.