Implicit differentiation of variational quantum algorithms

We show how to leverage implicit differentiation for gradient computation through variational quantum algorithms. A function defined implicitly as the solution of a quantum algorithm, e.g., a variationally obtained ground- or steady-state, can be differentiated using implicit differentiation while being agnostic to how the solution is computed. Since unrolling all the steps in the quantum algorithm is not required, implicit differentiation can be memory efficient compared to backpropagation. Physical quantities of interest in condensed-matter systems such as generalized susceptibility or nuclear gradients can be computed easily using implicit differentiation as we discuss in this work. We also develop two novel applications of implicit differentiation --- hyperparamter optimization in a quantum machine learning algorithm, and creating entangled quantum states variationally with gradient-based optimization of a geometric measure of entanglement.