Accurate Computation of Quantum Excited States with Neural Networks

We present a variational Monte Carlo algorithm for estimating the lowest excited states of a quantum system which is a natural generalization of the estimation of ground states. The method has no free parameters and requires no explicit orthogonalization of the different states, instead transforming the problem of finding excited states of a given system into that of finding the ground state of an expanded system. Expected values of arbitrary observables can be calculated, including off-diagonal expectations between different states such as the transition dipole moment. Although the method is entirely general, it works particularly well in conjunction with recent work on using neural networks as variational Ans\"atze for many-electron systems, and we show that by combining this method with the FermiNet and Psiformer Ans\"atze we can accurately recover vertical excitation energies and oscillator strengths on a range of molecules. Our method is the first deep learning approach to achieve accurate vertical excitation energies, including challenging double excitations, on benzene-scale molecules. Beyond the chemistry examples here, we expect this technique will be of great interest for applications to atomic, nuclear and condensed matter physics.