  This paper considers adaptive estimation of quadratic functionals in the
nonparametric instrumental variables (NPIV) models. Minimax estimation of a
quadratic functional of a NPIV is an important problem in optimal estimation of
a nonlinear functional of an ill-posed inverse regression with an unknown
operator using one random sample. We first show that a leave-one-out, sieve
NPIV estimator of the quadratic functional proposed by \cite{BC2020} attains a
convergence rate that coincides with the lower bound previously derived by
\cite{ChenChristensen2017}. The minimax rate is achieved by the optimal choice
of a key tuning parameter (sieve dimension) that depends on unknown NPIV model
features. We next propose a data driven choice of the tuning parameter based on
Lepski's method. The adaptive estimator attains the minimax optimal rate in the
severely ill-posed case and in the regular, mildly ill-posed case, but up to a
multiplicative $\sqrt{\log n}$ in the irregular, mildly ill-posed case.
