  In recent years a new class of supersymmetric lattice theories have been
proposed which retain one or more exact supersymmetries for non-zero lattice
spacing. Recently there has been some controversy in the literature concerning
whether these theories suffer from a sign problem. In this paper we address
this issue by conducting simulations of the N=(2, 2) and N=(8, 8)
supersymmetric Yang--Mills theories in two dimensions for the U(N) theories
with N=2,3,4, using the new twisted lattice formulations. Our results provide
evidence that these theories do not suffer from a sign problem in the continuum
limit. These results thus boost confidence that the new lattice formulations
can be used successfully to explore non-perturbative aspects of
four-dimensional N=4 supersymmetric Yang--Mills theory.
