The Roots of the NOT Gate

Abstract

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 (2w-1)2-dimensional subspace of the (2w)2 -dimensional space U(2w) of the 2w × 2w unitary matrices and thus describe computers situated between classical reversible computers and full quantum computers.