  Computation of circuit complexity has gained much attention in the
Theoretical Physics community in recent times to gain insights about the
chaotic features and random fluctuations of fields in the quantum regime.
Recent studies of circuit complexity take inspiration from the geometric
approach of Nielsen, which itself is based on the idea of optimal quantum
control in which a cost function is introduced for the various possible path to
determine the optimum circuit. In this paper, we study the relationship between
the circuit complexity and Morse theory within the framework of algebraic
topology using which we study circuit complexity in supersymmetric quantum
field theory describing both simple and inverted harmonic oscillators up to
higher orders of quantum corrections. The expression of circuit complexity in
quantum regime would then be given by the Hessian of the Morse function in
supersymmetric quantum field theory, and try to draw conclusion from their
graphical behaviour. We also provide a technical proof of the well known
universal connecting relation between quantum chaos and circuit complexity of
the supersymmetric quantum field theories, using the general description of
Morse theory.
