Quantum Circuits and Spin(3n) Groups

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\(2ⁿ\) 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\(2ⁿ\). 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.