Instantaneous quantum polynomial quantum circuit Born machines (IQP-QCBMs) have been proposed as quantum generative models with a classically tractable training objective based on the maximum mean discrepancy (MMD) and a potential quantum advantage motivated by sampling-complexity arguments, making them an exciting model worth deeper investigation. While recent works have further proven the universality of a (slightly generalized) model, the next immediate question pertains to its trainability, i.e., whether it suffers from the exponentially vanishing loss gradients, known as the barren plateau issue, preventing effective use, and how regimes of trainability overlap with regimes of possible quantum advantage. Here, we provide significant strides in these directions. To study the trainability at initialization, we analytically derive closed-form expressions for the variances of the partial derivatives of the MMD loss function and provide general upper and lower bounds. With uniform initialization, we show that barren plateaus depend on the generator set and the spectrum of the chosen kernel. We identify regimes in which low-weight-biased kernels avoid exponential gradient suppression in structured topologies. Also, we prove that a small-variance Gaussian initialization ensures polynomial scaling for the gradient under mild conditions. As for the potential quantum advantage, we further argue, based on previous complexity-theoretic arguments, that sparse IQP families can output a probability distribution family that is classically intractable, and that this distribution remains trainable at initialization at least at lower-weight frequencies.

翻译：瞬时量子多项式量子电路玻恩机（IQP-QCBM）作为一种量子生成模型被提出，其基于最大均值差异（MMD）的经典可处理训练目标，以及由采样复杂性论证所激发的潜在量子优势，使其成为一个值得深入研究的令人兴奋的模型。虽然近期研究进一步证明了（略微广义化的）该模型的普适性，但随之而来的直接问题涉及其可训练性，即它是否遭受指数级消失的损失梯度（称为贫瘠高原问题）从而阻碍有效使用，以及可训练性区域如何与可能的量子优势区域重叠。在此，我们于这些方向上取得了重要进展。为研究初始化时的可训练性，我们解析推导了MMD损失函数偏导数方差的闭式表达式，并给出了通用的上界和下界。在均匀初始化下，我们证明了贫瘠高原的存在取决于生成器集合以及所选核的谱。我们识别了在结构化拓扑中，低权重偏置核可以避免指数梯度抑制的区域。此外，我们证明了在温和条件下，小方差高斯初始化能确保梯度具有多项式缩放。关于潜在的量子优势，我们进一步基于先前的复杂性理论论证指出，稀疏IQP族能够输出一个经典难处理的概率分布族，并且该分布在初始化时至少在低权重频率下仍然是可训练的。