Equilibrium propagation is a recently introduced method to use and train
artificial neural networks in which the network is at the minimum (more
generally extremum) of an energy functional. Equilibrium propagation has shown
good performance on a number of benchmark tasks. Here we extend equilibrium
propagation in two directions. First we show that there is a natural quantum
generalization of equilibrium propagation in which a quantum neural network is
taken to be in the ground state (more generally any eigenstate) of the network
Hamiltonian, with a similar training mechanism that exploits the fact that the
mean energy is extremal on eigenstates. Second we extend the analysis of
equilibrium propagation at finite temperature, showing that thermal
fluctuations allow one to naturally train the network without having to clamp
the output layer during training. We also study the low temperature limit of
equilibrium propagation.

均衡传播是最近介绍的一种使用和训练人工神经网络的方法，在该方法中，网络处于能量泛函的极小值（更一般地是极值）状态。均衡传播在许多基准任务上显示出良好的性能。本文将均衡传播在两个方面进行了扩展。首先，我们展示了均衡传播的一个自然量子泛化，其中量子神经网络被认为处于网络哈密顿算子的基态（更一般地说是任意的特征态）状态，并利用了均能量在特征态上的极值训练机制。其次，我们扩展了均衡传播的温度有限的分析，显示了热波动使得在训练过程中无需夹紧输出层的网络自然训练。我们还研究了均衡传播的低温极限。

平衡传播：量子和热力学案例

Equilibrium Propagation: the Quantum and the Thermal Cases

Generative models realized with machine learning techniques are powerful
tools to infer complex and unknown data distributions from a finite number of
training samples in order to produce new synthetic data. Diffusion models are
an emerging framework that have recently overcome the performance of the
generative adversarial networks in creating synthetic text and high-quality
images. Here, we propose and discuss the quantum generalization of diffusion
models, i.e., three quantum-noise-driven generative diffusion models that could
be experimentally tested on real quantum systems. The idea is to harness unique
quantum features, in particular the non-trivial interplay among coherence,
entanglement and noise that the currently available noisy quantum processors do
unavoidably suffer from, in order to overcome the main computational burdens of
classical diffusion models during inference. Hence, we suggest to exploit
quantum noise not as an issue to be detected and solved but instead as a very
remarkably beneficial key ingredient to generate much more complex probability
distributions that would be difficult or even impossible to express
classically, and from which a quantum processor might sample more efficiently
than a classical one. Therefore, our results are expected to pave the way for
new quantum-inspired or quantum-based generative diffusion algorithms
addressing more powerfully classical tasks as data generation/prediction with
widespread real-world applications ranging from climate forecasting to
neuroscience, from traffic flow analysis to financial forecasting.

量子驱动扩散模型的量子泛化被提出，并讨论了可在实际量子系统上进行实验测试的三种量子噪声驱动的生成扩散模型。通过利用量子噪声作为生成更复杂的概率分布的重要成分，我们的结果有望为处理从气候预测到神经科学、交通流分析到金融预测等广泛的实际应用任务的新型量子驱动生成扩散算法铺平道路。