  The Adiabatic Gauge Potential (AGP) is the generator of adiabatic
deformations between quantum eigenstates. There are many ways to construct the
AGP operator and evaluate the AGP norm. Recently, it was proposed that a
Gram-Schmidt-type algorithm can be used to explicitly evaluate the expression
of the AGP. We employ a version of this approach by using the Lanczos algorithm
to evaluate the AGP operator in terms of Krylov vectors and the AGP norm in
terms of the Lanczos coefficients. It has the advantage of minimizing
redundancies in evaluating the nested commutators in the analytic expression
for the AGP operator. The algorithm is used to explicitly construct the AGP
operator for some simple systems. We derive an integral transform relation
between the AGP norm and the autocorrelation function of the deformation
operator. We present a modification of the Variational approach to derive the
regulated AGP norm. Using this, we approximate the AGP to varying degrees of
success. Finally, we compare and contrast the quantum-chaos-probing capacities
of the AGP and K-complexity in view of the Operator Growth Hypothesis.
