Quantum circuits and Spin(3n) groups (pp0235-0259)
          Alexander Yu. Vlasov
          doi: https://doi.org/10.26421/QIC15.3-4-3
Abstracts: All quantum gates with one and two qubits may be described by elements of Spin groups due to isomorphisms Spin(3) ≃ SU(2) and Spin(6) ≃ 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.
Key words:  quantum computation, matchgates, spin groups, polynomial time