We show that universal quantum computation can be achieved in the standard
pure-state circuit model while, at any time, the entanglement entropy of all
bipartitions is small---even tending to zero with growing system size. The
result is obtained by showing that a quantum computer operating within a small
region around the set of unentangled states still has universal computational
power, and by using continuity of entanglement entropy. In fact an analogous
conclusion applies to every entanglement measure which is continuous in a
certain natural sense, which amounts to a large class. Other examples include
the geometric measure, localizable entanglement, smooth epsilon-measures,
multipartite concurrence, squashed entanglement, and several others. We discuss
implications of these results for the believed role of entanglement as a key
necessary resource for quantum speed-ups
