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.

翻译：量子计算（QC）在理论上具有优势，能够为无法高效经典模拟的计算问题带来益处。然而，这种理论加速在很大程度上取决于解决问题的量子电路设计。我们认为，量子计算领域尚未充分探索更具领域特异性的拟设拓扑结构，而仍依赖于通用的、一刀切的架构。在本研究中，我们证明将任务特定的归纳偏置——特别是几何先验——融入量子电路设计，能够提升混合量子生成对抗网络（QuGAN）在生成几何约束的K4图任务上的性能。我们评估了一系列纠缠拓扑和损失函数设计，以分析它们对统计保真度和几何约束（包括三角不等式和托勒密不等式）合规性的影响。结果表明，使电路拓扑与底层问题结构对齐可带来显著优势：三角拓扑QuGAN在量子模型中实现了最高的几何有效性，并与经典生成对抗网络（GAN）的性能相当。此外，我们展示了特定架构选择（如纠缠门类型、方差正则化和输出缩放）如何调控几何一致性与分布准确性之间的权衡，从而强调了结构化、任务感知的量子拟设拓扑的价值。