Grassmannian Variational Monte Carlo for Simultaneous Multiple State Optimization

Many quantum many-body problems require solving for multiple wave-functions whose optimizations are dependent on one another, creating challenges when using Variational Monte Carlo (VMC) methods. One of the main obstacles that emerges is the apparent necessity of the Gram Matrix (matrix of overlap pairs) which is often very difficult to estimate accurately. Recently VMC methods [Pfau et al Science 385 (6711), eadn0137 2024] have been developed for excited states that bypass this obstacle by using clever sampling schemes, which involve sampling sets of basis states weighted by the determinants of their overlaps with the states. We expand on these methods and the geometric intuitions behind them to create a general VMC framework for optimizing multiple variational wave-functions simultaneously, which can be applied broadly and give a natural higher dimensional extension of standard VMC methods. Using the geometry of Grassmannians, we generalize fundamental VMC methods such as Stochastic Reconfiguration and reconceptualize quantities such as fidelity and operator variance in terms of their higher dimensional analogues. We benchmark our methods using deep neural quantum states and take advantage of the flexibilities provided by the methods to share layers between states.