  We present a novel $\mathcal{N} = 2 $ $\mathbb{Z}_2^2$-graded supersymmetric
quantum mechanics ($\mathbb{Z}_2^2$-SQM) which has different features from
those introduced so far. It is a two-dimensional (two-particle) system and is
the first example of the quantum mechanical realization of an eight-dimensional
irrep of the $\mathcal{N}=2$ $\mathbb{Z}_2^2$-supersymmetry algebra. The
$\mathbb{Z}_2^2$-SQM is obtained by quantizing the one-dimensional classical
system derived by dimensional reduction from the two-dimensional
$\mathbb{Z}_2^2$-supersymmetric Lagrangian of $\mathcal{N}=1$, which we
constructed in our previous work. The ground states of the $\mathbb{Z}_2^2$-SQM
are also investigated.
