  We prove the existence of a complex valued $C^2$-function on the unit circle,
a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class
$S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U)
-\frac{d}{dt}\bigl(f(e^{itZ}U)\bigr)_{\vert t=0} $$ does not belong to the
space $S^1$ of trace class operators. This resolves a problem of Peller
concerning the validity of the Koplienko-Neidhardt trace formula for unitaries.
