X, Y, and Z Subgroups of the Unitary Group

Abstract

The subgroups XU(2), YU(2), and ZU(2) of the unitary group U(2) allow to find twenty-four different decompositions of an arbitrary [Formula: see text] unitary matrix. Because the group YU(2) has not been considered before, 20 out of the 24 decompositions are new. Introducing this YU(2) group allows to highlight, within quantum circuit design, a symmetry similar to the threefold symmetry of the Pauli matrices of spin systems. Similar subgroups of the unitary group U([Formula: see text]) lead to various new decompositions of an arbitrary [Formula: see text] unitary matrix. Whereas for [Formula: see text] equal 2, this leads to the synthesis of single-qubit quantum gates, for [Formula: see text] equal a power of 2, this leads to new design methods for multiple-qubit circuits.