  We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold.
First, we construct a simple sheaf E of rank 6 as an elementary modification of
an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an
extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a
general deformation of E(1) becomes Ulrich. In particular, this says that
general cubic fourfolds have Ulrich complexity 6.
