  It is well known that quantum tunneling can be described by instantons in the
imaginary-time path integral formalism. However, its description in the
real-time path integral formalism has been elusive. Here we establish a
statement that quantum tunneling can be characterized in general by the
contribution of complex saddle points, which can be identified by using the
Picard-Lefschetz theory. We demonstrate this explicitly by performing Monte
Carlo simulations of simple quantum mechanical systems, overcoming the sign
problem by the generalized Lefschetz thimble method. We confirm numerically
that the contribution of complex saddle points manifests itself in a complex
``weak value'' of the Hermitian coordinate operator $\hat{x}$ evaluated at time
$t$, which is a physical quantity that can be measured by experiments in
principle. We also discuss the transition to classical dynamics based on our
picture.
