  A novel universal and fault-tolerant basis (set of gates) for quantum
computation is described. Such a set is necessary to perform quantum
computation in a realistic noisy environment. The new basis consists of two
single-qubit gates (Hadamard and ${\sigma_z}^{1/4}$), and one double-qubit gate
(Controlled-NOT). Since the set consisting of Controlled-NOT and Hadamard gates
is not universal, the new basis achieves universality by including only one
additional elementary (in the sense that it does not include angles that are
irrational multiples of $\pi$) single-qubit gate, and hence, is potentially the
simplest universal basis that one can construct. We also provide an alternative
proof of universality for the only other known class of universal and
fault-tolerant basis proposed by Shor and by Kitaev.
