Quantum Generative Modelling (QGM) relies on preparing quantum states and generating samples from these states as hidden - or known - probability distributions. As distributions from some classes of quantum states (circuits) are inherently hard to sample classically, QGM represents an excellent testbed for quantum supremacy experiments. Furthermore, generative tasks are increasingly relevant for industrial machine learning applications, and thus QGM is a strong candidate for demonstrating a practical quantum advantage. However, this requires that quantum circuits are trained to represent industrially relevant distributions, and the corresponding training stage has an extensive training cost for current quantum hardware in practice. In this work, we propose protocols for classical training of QGMs based on circuits of the specific type that admit an efficient gradient computation, while remaining hard to sample. In particular, we consider Instantaneous Quantum Polynomial (IQP) circuits and their extensions. Showing their classical simulability in terms of the time complexity, sparsity and anti-concentration properties, we develop a classically tractable way of simulating their output probability distributions, allowing classical training to a target probability distribution. The corresponding quantum sampling from IQPs can be performed efficiently, unlike when using classical sampling. We numerically demonstrate the end-to-end training of IQP circuits using probability distributions for up to 30 qubits on a regular desktop computer. When applied to industrially relevant distributions this combination of classical training with quantum sampling represents an avenue for reaching advantage in the NISQ era.

翻译：量子生成建模依赖于制备量子态并从这些态中作为隐藏或已知的概率分布生成样本。由于某些量子态（电路）类别的分布本身在经典采样上具有困难性，量子生成建模为量子霸权实验提供了绝佳测试平台。此外，生成任务在工业机器学习应用中日益重要，因此量子生成建模是展示实用量子优势的有力候选方案。然而，这需要训练量子电路以表示工业相关分布，且当前量子硬件在实际中的训练阶段成本高昂。本研究提出了基于特定类型电路的QGM经典训练协议，该类电路在保持难以采样的同时允许高效梯度计算。具体而言，我们考虑了瞬时量子多项式电路及其扩展。通过展示其在时间复杂度、稀疏性和反集中特性方面的经典可模拟性，我们开发了模拟其输出概率分布的经典可处理方式，从而实现对目标概率分布的经典训练。与经典采样不同，IQP的相应量子采样可高效执行。我们通过常规台式计算机上最多30量子比特的概率分布，数值演示了IQP电路的端到端训练。当应用于工业相关分布时，这种经典训练与量子采样的组合为在NISQ时代实现优势提供了途径。