Born-rule generative modeling, a central task in quantum machine learning, seeks to learn probability distributions that can be efficiently sampled by measuring complex quantum states. One hope is for quantum models to efficiently capture probability distributions that are difficult to learn and simulate by classical means alone. Quantum Boltzmann machines were proposed about one decade ago for this purpose, yet efficient training methods have remained elusive. In this paper, I overcome this obstacle by proposing a practical solution that trains quantum Boltzmann machines for Born-rule generative modeling. Two key ingredients in the proposal are the Donsker-Varadhan variational representation of the classical relative entropy and the quantum Boltzmann gradient estimator of [Patel et al., arXiv:2410.12935]. I present the main result for a more general ansatz known as an evolved quantum Boltzmann machine [Minervini et al., arXiv:2501.03367], which combines parameterized real- and imaginary-time evolution. I also show how to extend the findings to other distinguishability measures beyond relative entropy. Finally, I present four different hybrid quantum-classical algorithms for the minimax optimization underlying training, and I discuss their theoretical convergence guarantees.

翻译：玻恩规则生成建模是量子机器学习中的核心任务，旨在学习可通过测量复杂量子态高效采样的概率分布。其愿景在于量子模型能够高效捕捉仅依靠经典方法难以学习和模拟的概率分布。约十年前提出的量子玻尔兹曼机即为此目标而设计，但高效训练方法一直难以实现。本文通过提出一种实用方案克服了这一障碍，该方案可训练量子玻尔兹曼机以进行玻恩规则生成建模。该方案的两个关键要素是经典相对熵的Donsker-Varadhan变分表示，以及[Patel等人，arXiv:2410.12935]提出的量子玻尔兹曼梯度估计器。我将主要结果呈现于一种更广义的拟设——演化量子玻尔兹曼机[Minervini等人，arXiv:2501.03367]，该模型结合了参数化的实时间与虚时间演化。我还展示了如何将研究结果推广至相对熵以外的其他可区分性度量。最后，我提出了四种不同的混合量子-经典算法用于支撑训练的最小最大优化，并讨论了它们的理论收敛保证。