  Several quantities important in condensed matter physics, quantum
information, and quantum chemistry, as well as quantities required in
meta-optimization of machine learning algorithms, can be expressed as gradients
of implicitly defined functions of the parameters characterizing the system.
Here, we show how to leverage implicit differentiation for gradient computation
through variational quantum algorithms and explore applications in condensed
matter physics, quantum machine learning, and quantum information. A function
defined implicitly as the solution of a quantum algorithm, e.g., a
variationally obtained ground- or steady-state, can be automatically
differentiated using implicit differentiation while being agnostic to how the
solution is computed. We apply this notion to the evaluation of physical
quantities in condensed matter physics such as generalized susceptibilities
studied through a variational quantum algorithm. Moreover, we develop two
additional applications of implicit differentiation -- hyperparameter
optimization in a quantum machine learning algorithm, and the variational
construction of entangled quantum states based on a gradient-based maximization
of a geometric measure of entanglement. Our work ties together several types of
gradient calculations that can be computed using variational quantum circuits
in a general way without relying on tedious analytic derivations, or
approximate finite-difference methods.
