Wasserstein量子蒙特卡洛：一种求解量子多体薛定谔方程的新方法
Wasserstein Quantum Monte Carlo: A Novel Approach for Solving the Quantum Many-Body Schrödinger Equation
AI总结：
本文将QVMC重述为Born分布空间中的投影梯度流，并提出采用Wasserstein度量的WQMC，通过概率质量输运实现分子基态更快收敛。
摘要：
求解量子多体薛定谔方程是量子物理、量子化学和材料科学领域中一个基础性且具有挑战性的问题。解决该问题的常用计算方法之一是量子变分蒙特卡洛（QVMC），该方法通过在受限的参数化波函数族中最小化系统能量来获得基态解。深度学习方法通过以神经网络表示丰富的波函数族，在一定程度上缓解了传统QVMC的局限。然而，QVMC中的优化目标出了名地难以最小化，并且需要自然梯度等二阶优化方法。在本文中，我们首先在与粒子置换（反）对称波函数相对应的Born分布空间中，而非在波函数空间中，重新表述能量泛函最小化。随后，我们将QVMC解释为该分布空间中的Fisher–Rao梯度流，之后再投影到变分流形上。这一视角为我们提供了一个有原则的框架，可通过赋予分布空间更好的度量，并遵循这些度量诱导的投影梯度流来推导新的QMC算法。更具体地说，我们提出了“Wasserstein量子蒙特卡洛”（WQMC），它使用由Wasserstein度量而非Fisher–Rao度量诱导的梯度流，对应于输运概率质量，而非瞬间传送概率质量。我们通过实验证明，WQMC的动力学能够更快地收敛到分子系统的基态。
英文摘要：
Solving the quantum many-body Schrödinger equation is a fundamental and challenging problem in the fields of quantum physics, quantum chemistry, and material sciences. One of the common computational approaches to this problem is Quantum Variational Monte Carlo (QVMC), in which ground-state solutions are obtained by minimizing the energy of the system within a restricted family of parameterized wave functions. Deep learning methods partially address the limitations of traditional QVMC by representing a rich family of wave functions in terms of neural networks. However, the optimization objective in QVMC remains notoriously hard to minimize and requires second-order optimization methods such as natural gradient. In this paper, we first reformulate energy functional minimization in the space of Born distributions corresponding to particle-permutation (anti-)symmetric wave functions, rather than the space of wave functions. We then interpret QVMC as the Fisher-Rao gradient flow in this distributional space, followed by a projection step onto the variational manifold. This perspective provides us with a principled framework to derive new QMC algorithms, by endowing the distributional space with better metrics, and following the projected gradient flow induced by those metrics. More specifically, we propose "Wasserstein Quantum Monte Carlo" (WQMC), which uses the gradient flow induced by the Wasserstein metric, rather than Fisher-Rao metric, and corresponds to transporting the probability mass, rather than teleporting it. We demonstrate empirically that the dynamics of WQMC results in faster convergence to the ground state of molecular systems.
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