  Automatic differentiation is an invaluable feature of machine learning and
quantum machine learning software libraries. In this work it is shown how
quantum automatic differentiation can be used to solve the condensed-matter
problem of computing fidelity susceptibility, a quantity whose value may be
indicative of a phase transition in a system. Results are presented using
simulations including hardware noise for small instances of the
transverse-field Ising model, and a number of optimizations that can be applied
are highlighted. Error mitigation (zero-noise extrapolation) is applied within
the autodifferentiation framework to a number of gradient values required for
computation of fidelity susceptibility and a related quantity, the second
derivative of the energy. Such computations are found to be highly sensitive to
the additional statistical noise incurred by the error mitigation method
