The search for an application of near-term quantum devices is widespread. Quantum machine learning is touted as a potential utilisation of such devices, particularly those out of reach of the simulation capabilities of classical computers. In this work, we study such an application in generative modelling, focussing on a class of quantum circuits known as Born machines. Specifically, we define a subset of this class based on Ising Hamiltonians and show that the circuits encountered during gradient-based training cannot be efficiently sampled from classically up to multiplicative error in the worst case. Our gradient-based training methods use cost functions known as the Sinkhorn divergence and the Stein discrepancy, which have not previously been used in the gradient-based training of quantum circuits, and we also introduce quantum kernels to generative modelling. We show that these methods outperform the previous standard method, which used maximum mean discrepancy (MMD) as a cost function, and achieve this with minimal overhead. Finally, we discuss the ability of the model to learn hard distributions and provide formal definitions for ‘quantum learning supremacy’. We also exemplify the work of this paper by using generative modelling to perform quantum circuit compilation.

寻找近期量子装置的应用是广泛的。量子机器学习被认为是对此类设备的潜在利用，特别是那些传统计算机无法实现的仿真能力。在这项工作中，我们研究了这种在生成建模中的应用，重点研究了一类称为Born机器的量子电路。具体来说，我们基于Ising Hamiltonian定义了此类的子集，并表明在最坏的情况下，无法有效地对从基于梯度的训练过程中遇到的电路进行经典采样直至乘法误差。我们基于梯度的训练方法使用了称为Sinkhorn散度和Stein差异的成本函数，这在以前基于量子电路的基于梯度的训练中尚未使用，并且我们还将量子核引入到生成建模中。我们证明了这些方法优于以前的标准方法，后者使用最大平均差异（MMD）作为成本函数，并以最小的开销实现了这一目标。最后，我们讨论该模型学习硬分布的能力，并为“量子学习至上性”提供正式定义。我们还通过使用生成模型执行量子电路编译来举例说明本文的工作。