  We investigate the topological-to-non-topological quantum phase transitions
(QPTs) occurring in the Kitaev code under local perturbations in the form of
local magnetic field and spin-spin interactions of the Ising-type using
fidelity susceptibility (FS) and entanglement as the probes. We assume the code
to be embedded on the surface of a wide cylinder of height $M$ and
circumference $D$ with $M\ll D$. We demonstrate a power-law divergence of FS
across the QPT, and determine the quantum critical points (QCPs) via a
finite-size scaling analysis. We verify these results by mapping the perturbed
Kitaev code to the 2D Ising model with nearest- and next-nearest-neighbor
interactions, and computing the single-site magnetization as order parameter
using quantum Monte-Carlo technique. We also point out an odd-even dichotomy in
the occurrence of the QPT in the Kitaev ladder with respect to the odd and even
values of $D$, when the system is perturbed with only Ising interaction. Our
results also indicate a higher robustness of the topological phase of the
Kitaev code against local perturbations if the boundary is made open along one
direction. We further consider a local entanglement witness operator designed
specifically to capture a lower bound to the localizable entanglement on the
vertical non-trivial loop of the code. We show that the first derivative of the
expectation value of the witness operator exhibits a logarithmic divergence
across the QPT, and perform the finite-size scaling analysis. We demonstrate
similar behaviour of the expectation value of the appropriately constructed
witness operator also in the case of locally perturbed color code with open
boundaries.
