  Quantum mechanical transition amplitudes directly tells the probability of
each transition and which one is more favourable. Path-integrals offers a
systematic methodology to compute this quantum mechanical process in a
consistent manner. Although it is not complicated in simple quantum mechanical
system but defining path-integral legitimately becomes highly nontrivial in the
context of quantum-gravity, where apart from usual issues of renormalizability,
regularisation, measure, gauge-fixing, boundary conditions, one still has to
define the sensible integration contour for convergence. Picard-Lefschetz (PL)
theory offers a unique way to find a contour of integration based on the
analysis of saddle points and the steepest descent/ascent flow lines in the
complex plane. In this paper we make use of PL-theory to investigate Lorentzian
quantum cosmology where the gravity gets modified in the ultraviolet with the
$R^2$ corrections. We approach the problem perturbatively and compute the
transition amplitude in the saddle point approximation to first order in
higher-derivative coupling. This perturbative approximation is valid in certain
regimes but the approximation cannot be used to address issues of very early
Universe or no-boundary proposal.
