The decomposition of an arbitrary reversible logic circuit

The (2w)! reversible logic circuits of width w, i.e. reversible logic circuits with w inputs and w outputs, together with the action of cascading, form a group G, isomorphic to the symmetric group . We define two conjugate subgroups G1 and G2. Together they partition the group G into 2w−1 + 1 double cosets. These allow us to decompose an arbitrary member of G into a cascade of three simpler members. This decomposition is a far relative of the well-known LU decomposition of a square matrix.