  We study the regularity of the law of a quadratic form $Q(X,X)$, evaluated in
a sequence $X = (X_{i})$ of independent and identically distributed random
variables, when $X_{1}$ can be expressed as a sufficiently smooth function of a
Gaussian field. This setting encompasses a large class of important and
frequently used distributions, such as, among others, Gaussian, Beta, for
instance uniform, Gamma distributions, or else any polynomial transform of
them.
  Let us present an emblematic application. Take $X = (X_{i})$ a sequence of
independent and identically distributed centered random variables, with unit
variance, following such distribution. Consider also $(Q_{n})$ a sequence of
quadratic forms, with associated symmetric Hilbert--Schmidt operators
$(\mathsf{A}^{(n)})$. Assume that $\operatorname{Tr}[ (\mathsf{A}^{(n)})^{2} ]
= 1/2$, $\mathsf{A}^{(n)}_{ii} =0$, and the spectral radius of
$\mathsf{A}^{(n)}$ tends to $0$. Then, $(Q_{n}(X))$ converges in a strong sense
to the standard Gaussian distribution. Namely, all derivatives of the
densities, which are well-defined for $n$ sufficiently large, converge
uniformly on $\mathbb{R}$ to the corresponding derivatives of the standard
Gaussian density.
  While classical methods, from Malliavin calculus or $\Gamma$-calculus,
generally consist in bounding negative moments of the so-called \emph{carr\'e
du champ} operator $\Gamma(Q(X),Q(X))$, we provide a new paradigm through a
second-order criterion involving the eigenvalues of a Hessian-type matrix
related to $Q(X)$. This Hessian is built by iterating twice a tailor-made
gradient, the \emph{sharp operator} $\sharp$, obtained via a Gaussian
representation of the carr\'e du champ. We believe that this method, recently
developed by the authors in the current paper and in their companion paper [AoP
52 n{\deg}3 (2024)] , is of independent interest and could prove useful in
other settings.
