A central problem in quantum mechanics involves solving the Electronic Schrodinger Equation for a molecule or material. The Variational Monte Carlo approach to this problem approximates a particular variational objective via sampling, and then optimizes this approximated objective over a chosen parameterized family of wavefunctions, known as the ansatz. Recently neural networks have been used as the ansatz, with accompanying success. However, sampling from such wavefunctions has required the use of a Markov Chain Monte Carlo approach, which is inherently inefficient. In this work, we propose a solution to this problem via an ansatz which is cheap to sample from, yet satisfies the requisite quantum mechanical properties. We prove that a normalizing flow using the following two essential ingredients satisfies our requirements: (a) a base distribution which is constructed from Determinantal Point Processes; (b) flow layers which are equivariant to a particular subgroup of the permutation group. We then show how to construct both continuous and discrete normalizing flows which satisfy the requisite equivariance. We further demonstrate the manner in which the non-smooth nature ("cusps") of the wavefunction may be captured, and how the framework may be generalized to provide induction across multiple molecules. The resulting theoretical framework entails an efficient approach to solving the Electronic Schrodinger Equation.

翻译：量子力学中的一个核心问题涉及求解分子或材料的电子薛定谔方程。针对该问题的变分蒙特卡洛方法通过采样来近似一个特定的变分目标，然后在选定的参数化波函数族（称为 ansatz）上优化这个近似目标。近年来，神经网络被用作 ansatz 并取得了相应的成功。然而，从这类波函数中采样需要使用马尔可夫链蒙特卡洛方法，这种方法本质上是低效的。在本工作中，我们提出了一种解决此问题的方法，即采用一种采样成本低廉、同时满足必要量子力学性质的 ansatz。我们证明，使用以下两个关键成分的归一化流满足我们的要求：(a) 由行列式点过程构建的基础分布；(b) 对置换群的特定子群具有等变性的流层。随后，我们展示了如何构建满足所需等变性的连续和离散归一化流。我们进一步论证了如何捕捉波函数的非光滑特性（"尖点"），以及如何推广该框架以实现跨多个分子的归纳。所得的理论框架为求解电子薛定谔方程提供了一种高效的方法。