这篇arXiv论文给出了浅层量子生成模型的一个严格分离证明：加一点经典随机性就能生成纯幺正模型复制不了的分布，还给了MBQC实现方案。
该论文证明在固定浅层深度下，共享经典随机性足以让通道量子生成模型严格超越对应的浅层幺正Born模型。具体做法是用单一经典采样随机比特控制空间分离的局域泡利操作，生成经典输出分布中的长程关联。对于一维最近邻架构，纯幺正模型复现这类分布最坏需要深度Ω(N)。作者进一步用基于测量量子计算（MBQC）实现了所需共享经典随机性，数值实验支持理论结果。
Representational separation between unitary and channel quantum generative models via shared classical randomness at shallow depth
Near-term quantum hardware limits circuit depth and often imposes geometrically local connectivity for quantum generative models, restricting the output distributions accessible to shallow unitary Born models. Introducing stochasticity into a unitary quantum Born model can improve the empirical generative performance of the resulting channel model and, for a restricted small-scale architecture, has been proven to represent a strictly larger family of distributions than its unitary counterpart. However, whether such randomness provides a provable separation at fixed shallow depth for arbitrarily large systems has remained open. Here, we show that shared classical randomness, a comparatively weak resource from entanglement theory, is sufficient to establish such a strict scalable representational separation over the corresponding shallow unitary Born model. More specifically, we augment bounded-connectivity shallow unitary circuits, followed by computational-basis measurements, with spatially separated local Pauli operations, whose joint application is controlled by a single classically sampled random bit. The resulting shallow-depth channel model generates long-range correlations in the classical output distribution that no purely unitary shallow-depth model with bounded connectivity can reproduce. For one-dimensional nearest-neighbour architectures, reproducing such distributions with a purely unitary model can require depth $Ω(N)$ in the worst case. We further show that measurement-based quantum computation (MBQC) provides a natural implementation of the required shared classical randomness through suitable adaptation of the random measurement outcomes. Numerical experiments on MBQC-based generative models support the analytical results.