Quantum generative models, in providing inherently efficient sampling strategies, show promise for achieving a near-term advantage on quantum hardware. Nonetheless, important questions remain regarding their scalability. In this work, we investigate the barriers to the trainability of quantum generative models posed by barren plateaus and exponential loss concentration. We explore the interplay between explicit and implicit models and losses, and show that using implicit generative models (such as quantum circuit-based models) with explicit losses (such as the KL divergence) leads to a new flavour of barren plateau. In contrast, the Maximum Mean Discrepancy (MMD), which is a popular example of an implicit loss, can be viewed as the expectation value of an observable that is either low-bodied and trainable, or global and untrainable depending on the choice of kernel. However, in parallel, we highlight that the low-bodied losses required for trainability cannot in general distinguish high-order correlations, leading to a fundamental tension between exponential concentration and the emergence of spurious minima. We further propose a new local quantum fidelity-type loss which, by leveraging quantum circuits to estimate the quality of the encoded distribution, is both faithful and enjoys trainability guarantees. Finally, we compare the performance of different loss functions for modelling real-world data from the High-Energy-Physics domain and confirm the trends predicted by our theoretical results.

翻译：量子生成模型凭借其内在的高效采样策略，在近期量子硬件上展现出实现优势的潜力。尽管如此，其可扩展性仍存在重要难题。本研究深入探究了贫瘠高原与指数级损失集中现象对量子生成模型可训练性构成的障碍。我们解析了显式与隐式模型及损失函数之间的相互作用，并证明将隐式生成模型（如基于量子电路的模型）与显式损失（如KL散度）结合会催生新型贫瘠高原。与此形成对比的是，最大平均差异作为一种典型的隐式损失，可被视为特定观测量（其局域性或全局性取决于核函数选择）的期望值，从而呈现可训练性或不可训练性特征。但与此同时，我们指出可训练性所需的低阶损失通常无法区分高阶相关性，这导致指数级集中与伪极小值涌现之间存在根本性矛盾。为此，我们提出一种新型局部量子保真度损失函数，通过利用量子电路评估编码分布质量，该函数兼具保真度优势与可训练性保证。最后，我们通过高能物理领域真实数据建模实验，对比了不同损失函数的性能表现，验证了理论推导所揭示的规律。