  We investigate the problem when the tensor functor by a bimodule yields a
singular equivalence. It turns out that this problem is equivalent to the one
when the Hom functor given by the same bimodule induces a triangle equivalence
between the homotopy categories of acyclic complexes of injective modules. We
give conditions on when a bimodule appears in a pair of bimodules, that defines
a singular equivalence with level. We construct an explicit bimodule, which
yields a singular equivalence between a quadratic monomial algebra and its
associated algebra with radical square zero. Under certain conditions which
include the Gorenstein cases, the bimodule does appear in a pair of bimodules
defining a singular equivalence with level.
