  We investigate the automatic differentiation of dominant eigensolver where
only a small proportion of eigenvalues and corresponding eigenvectors are
obtained. Backpropagation 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 eigen-decomposition 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 1D
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 computation tasks in a fully differentiable way,
one can efficiently handle the dominant eigen-decomposition 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.
