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 × 2^w unitary matrices and thus describe computers situated between classical reversible computers and full quantum computers.