  Any quantum computation consists of a sequence of unitary evolutions
described by a finite set of Hamiltonians. When this set is taken to consist of
only products of Pauli operators, we show that the minimal such set generating
$\mathfrak{su}(2^{N})$ contains $2N+1$ elements. We provide a number of
examples of such generating sets and furthermore provide an algorithm for
producing a sequence of rotations corresponding to any given Pauli rotation,
which is shown to have optimal complexity.
