  Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint
operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$.
In 2012, Potapov and Sukochev \cite{PoSu} obtained a similar formula for pairs
of contractions by answering an open question posed by Gesztesy, Pushnitski,
and Simon in \cite[Open Question 11.2]{GePu}. In this article, we give a still
another proof of the Koplienko trace formula in the case of unitary operators
via linear path by reducing the problem to a finite-dimensional one as in the
proof of Krein's trace formula by Voiculescu \cite{Voi}, Sinha and Mohapatra
\cite{MoSi94,MoSi96}. Consequently, we obtain the Koplienko trace formula for a
class of pairs of contractions using the Sch$\ddot{a}$ffer matrix unitary
dilation. Moreover, we also obtain the Koplienko trace formula for a pair of
self-adjoint operators and maximal dissipative operators using the Cayley
transform.
