In the canonical approach of loop quantum gravity, arguably the most important outstanding problem is finding and interpreting solutions to the Hamiltonian constraint. In this work, we demonstrate that methods of machine learning are in principle applicable to this problem. We consider U(1) BF theory in three dimensions, quantised with loop quantum gravity methods. In particular, we formulate a master constraint corresponding to Hamilton and Gauß constraints using loop quantum gravity methods. To make the problem amenable for numerical simulation we fix a graph and introduce a cutoff on the kinematical degrees of freedom, effectively considering Uq(1) BF theory at a root of unity. We show that the neural network quantum state ansatz can be used to numerically solve the constraints efficiently and accurately. We compute expectation values and fluctuations of certain observables and compare them with exact results or exact numerical methods where possible. We also study the dependence on the cutoff.

中文翻译：

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利用神经网络实现量子引力：求解 U（1） BF 理论的量子哈密尔顿约束


在环量子引力的规范方法中，可以说最重要的悬而未决的问题是找到并解释哈密顿约束的解。在这项工作中，我们证明了机器学习的方法原则上适用于这个问题。我们在三个维度上考虑 U（1） BF 理论，用环量子引力方法进行量化。特别是，我们使用环量子引力方法制定了对应于 Hamilton 和 Gauß 约束的主约束。为了使该问题适合数值模拟，我们固定了一个图形并引入了运动学自由度的截止值，有效地将 Uq（1） BF 理论视为单位根。我们表明，神经网络量子态 ansatz 可用于高效、准确地对约束进行数值求解。我们计算某些可观察对象的期望值和波动，并在可能的情况下将它们与精确结果或精确数值方法进行比较。我们还研究了对临界值的依赖性。