Automatic differentiation of dominant eigensolver and its applications in quantum physics

We investigate the automatic differentiation of dominant eigensolver where only a small proportion of eigenvalues and corresponding eigenvectors are obtained. Back-propagation through the dominant eigensolver involves solving certain low-rank linear systems without direct access to the full spectrum of the problem. Furthermore, the backward pass can be conveniently differentiated again, which implies that in principle one can obtain arbitrarily higher-order derivatives of the dominant eigendecomposition process. These results allow for the construction of an efficient dominant eigensolver primitive, which has wide applications in quantum physics. As a demonstration, we compute second-order derivative of the ground-state energy and fidelity susceptibility of one-dimensional transverse-field Ising model through the exact diagonalization approach. We also calculate the ground-state energy of the same model in the thermodynamic limit by performing gradient-based optimization of uniform matrix product states. By programming these computational tasks in a fully differentiable way, one can efficiently handle the dominant eigendecomposition of very large matrices while still sharing various advantages of differentiable programming paradigm, notably, the generic nature of the implementation and free of tedious human efforts of deriving gradients analytically.