  The path integral formalism is the building block of many powerful numerical
methods for quantum impurity problems. However, existing fermionic path
integral based numerical calculations have only been performed in either the
imaginary-time or the real-time axis, while the most generic scenario
formulated on the L-shaped Kadanoff-Baym contour is left unexplored. In this
work, we extended the recently developed Grassmann time-evolving matrix product
operator (GTEMPO) method to solve quantum impurity problems directly on the
Kadanoff-Baym contour. The resulting method is numerically exact, with only two
sources of numerical errors, e.g., the time discretization error and the matrix
product state bond truncation error, which can both be well controlled. The
accuracy of this method is numerically demonstrated against exact solutions in
the noninteracting case, and against existing calculations on the real- and
imaginary-time axes for the single-orbital Anderson impurity model. Our method
is a perfect benchmarking baseline for its alternatives which often employ
less-controlled approximations, and can also be used as a real-time impurity
solver in dynamical mean field theory and its non-equilibrium extension.
