Topology-Guided Quantum GANs for Constrained Graph Generation

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.

💡 Research Summary

This paper investigates how domain‑specific inductive biases, in the form of geometric priors, can be embedded into the architecture of hybrid quantum generative adversarial networks (QuGANs) to improve the generation of geometrically constrained K₄ graphs. The authors focus on the task of producing weighted complete graphs with six edges that represent Euclidean distances. Such graphs must satisfy two families of constraints: the triangle inequality for every triple of vertices and the Ptolemaic inequality for every quadruple, ensuring that the generated edge weights correspond to a realizable embedding in three‑dimensional space.

To address this, the authors design a family of parametrized quantum circuits (PQC) that serve as the generator. Each of the six edges is mapped to a dedicated qubit in a six‑qubit register. A latent Gaussian vector is encoded via single‑qubit rotations (RX, RY, RZ). The core of the study is the exploration of five entanglement topologies:

1. Ring – nearest‑neighbor connections forming a closed loop, providing a local translation‑invariant bias.
2. All‑to‑All – every qubit pair is entangled, representing a maximally expressive baseline.
3. Triangle – qubits corresponding to the three edges of each of the four triangular sub‑graphs of K₄ are fully entangled, directly mirroring the triangle structure of the problem.
4. Opposite – only qubit pairs that correspond to empirically anti‑correlated edges in the training data are entangled, capturing data‑driven global correlations.
5. Combined – the union of Triangle and Opposite couplings, aiming to encode both geometric and statistical relationships.

All generators consist of five variational layers, each containing single‑qubit rotations followed by the chosen two‑qubit entangling pattern, yielding a total of 90 trainable parameters. The discriminator is a classical three‑layer MLP kept identical across all experiments to isolate the effect of the generator’s architecture.

Beyond topology, the authors introduce two model‑enhancement mechanisms. First, a variance regularization term (L_variance) is added to the generator loss to penalize deviations between the batch‑wise standard deviation of generated edge weights and that of the real data. This addresses a systematic under‑dispersion observed across all QuGAN variants. Second, an output‑scaling layer is applied after measurement: the raw expectation values (in