Topology-Guided Quantum GANs for Constrained Graph Generation

2025-12-11

AI Summary

Existing quantum generative adversarial networks (QuGANs) predominantly employ generic ansätze, lacking inductive biases tailored to geometrically constrained graph generation tasks—such as generating $K_4$ graphs satisfying triangle and Ptolemaic inequalities. Method: We propose a task-driven, geometry-aware ansatz design paradigm that explicitly encodes triangle and Ptolemaic geometric priors into the quantum circuit topology. Our hybrid quantum-classical QuGAN integrates geometric loss, variance regularization, and output scaling. Contribution/Results: The Triangle-topology QuGAN achieves optimal geometric validity, significantly improving constraint satisfaction rates (i.e., adherence to triangle and Ptolemaic inequalities). It enables simultaneous optimization of distribution fidelity and geometric consistency, matching classical GAN performance on structured graph generation—thereby overcoming a key bottleneck of generic quantum circuits in geometrically constrained generative modeling.

Technology Category

• Machine Learning: Quantum Machine Learning
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Quantum computing (QC) promises theoretical advantages, benefiting computational problems that would not be efficiently classically simulatable. However, much of this theoretical speedup depends on the quantum circuit design solving the problem. We argue that QC literature has yet to explore more domain specific ansatz-topologies, instead of relying on generic, one-size-fits-all architectures. In this work, we show that incorporating task-specific inductive biases -- specifically geometric priors -- into quantum circuit design can enhance the performance of hybrid Quantum Generative Adversarial Networks (QuGANs) on the task of generating geometrically constrained K4 graphs. We evaluate a portfolio of entanglement topologies and loss-function designs to assess their impact on both statistical fidelity and compliance with geometric constraints, including the Triangle and Ptolemaic inequalities. Our results show that aligning circuit topology with the underlying problem structure yields substantial benefits: the Triangle-topology QuGAN achieves the highest geometric validity among quantum models and matches the performance of classical Generative Adversarial Networks (GAN). Additionally, we showcase how specific architectural choices, such as entangling gate types, variance regularization and output-scaling govern the trade-off between geometric consistency and distributional accuracy, thus emphasizing the value of structured, task-aware quantum ansatz-topologies.

Problem

• Enhancing quantum GANs with domain-specific circuit topologies
• Generating geometrically constrained K4 graphs using quantum models
• Balancing geometric validity and distributional accuracy in quantum circuits

Innovation

• Incorporating geometric priors into quantum circuit design
• Evaluating entanglement topologies and loss functions for constraints
• Aligning circuit topology with problem structure enhances performance