Quantum circuit Born machines based on instantaneous quantum polynomial-time (IQP) circuits are natural candidates for quantum generative modeling, both because of their probabilistic structure and because IQP sampling is provably classically hard in certain regimes. Recent proposals focus on training IQP-QCBMs using Maximum Mean Discrepancy (MMD) losses built from low-body Pauli-$Z$ correlators, but the effect of initialization on the resulting optimization landscape remains poorly understood. In this work, we address this by first proving that the MMD loss landscape suffers from barren plateaus for random full-angle-range initializations of IQP circuits. We then establish lower bounds on the loss variance for identity and an unbiased data-agnostic initialization. We then additionally consider a data-dependent initialization that is better aligned with the target distribution and, under suitable assumptions, yields provable gradients and generally converges quicker to a good minimum (as indicated by our training of circuits with 150 qubits on genomic data). Finally, as a by-product, the developed variance lower bound framework is applicable to a general class of non-linear losses, offering a broader toolset for analyzing warm-starts in quantum machine learning.

翻译：基于瞬时量子多项式时间（IQP）电路的量子电路玻恩机是量子生成建模的自然候选方案，这既源于其概率结构，也因为在特定条件下IQP采样被证明是经典计算难以模拟的。近期研究主要关注使用基于低阶泡利-$Z$关联算符构建的最大平均差异（MMD）损失函数来训练IQP-QCBM，但初始化对优化景观的影响仍缺乏深入理解。本工作首先证明了对于IQP电路的全角度随机初始化，MMD损失景观会陷入贫瘠高原现象。随后，我们针对恒等初始化与无偏数据不可知初始化建立了损失方差的下界。此外，我们还研究了一种与目标分布更匹配的数据依赖初始化策略，在适当假设下，该策略能产生可证明的梯度并通常更快收敛至良好极小值（正如我们在基因组数据上对150量子位电路的训练结果所示）。最后，作为副产品，所建立的方差下界分析框架可推广至一类广义非线性损失函数，为分析量子机器学习中的热启动策略提供了更广泛的工具集。