  In this paper we will consider quantum aspects of a non-local,
infinite-derivative scalar field theory - a $\it toy \, model$ depiction of a
covariant infinite-derivative, non-local extension of Einstein's general
relativity which has previously been shown to be free from ghosts around the
Minkowski background. The graviton propagator in this theory gets an
exponential suppression making it $\it asymptotically \, free$, thus providing
strong prospects of resolving various classical and quantum divergences. In
particular, we will find that at $1$-loop, the $2$-point function is still
divergent, but once this amplitude is renormalized by adding appropriate
counter terms, the ultraviolet (UV) behavior of all other $1$-loop diagrams as
well as the $2$-loop, $2$-point function remains well under control. We will go
on to discuss how one may be able to generalize our computations and arguments
to arbitrary loops.
