The roots of the NOT gate

The quantum gates called 'k th root of NOT' and 'controlled k th root of NOT' can be applied to synthesize circuits, both classical reversible circuits and quantum circuits. Such circuits, acting on w qubits, fill a (2(w) -1)(2)-dimensional subspace of the (2(w))(2)-dimensional space U(2(w)) of the 2(w) x 2(w) unitary matrices and thus describe computers situated between classical reversible computers and full quantum computers.