  All quantum gates with one and two qubits may be described by elements of
$Spin$ groups due to isomorphisms $Spin(3) \simeq SU(2)$ and $Spin(6) \simeq
SU(4)$. However, the group of $n$-qubit gates $SU(2^n)$ for $n > 2$ has bigger
dimension than $Spin(3n)$. A quantum circuit with one- and two-qubit gates may
be used for construction of arbitrary unitary transformation $SU(2^n)$.
Analogously, the `$Spin(3n)$ circuits' are introduced in this work as products
of elements associated with one- and two-qubit gates with respect to the
above-mentioned isomorphisms.
  The matrix tensor product implementation of the $Spin(3n)$ group together
with relevant models by usual quantum circuits with $2n$ qubits are
investigated in such a framework. A certain resemblance with well-known sets of
non-universal quantum gates e.g., matchgates, noninteracting-fermion quantum
circuits) related with $Spin(2n)$ may be found in presented approach. Finally,
a possibility of the classical simulation of such circuits in polynomial time
is discussed.
