  Computation of circuit complexity has gained much attention in the
Theoretical Physics community in recent times to gain insights into the chaotic
features and random fluctuations of fields in the quantum regime. Recent
studies of circuit complexity take inspiration from Nielsen's geometric
approach, which 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, which
will then help us study circuit complexity in supersymmetric quantum field
theory describing both simple and inverted harmonic oscillators up to higher
orders of quantum corrections. We will restrict ourselves to $\mathcal{N} = 1$
supersymmetry with one fermionic generator $Q_{\alpha}$. The expression of
circuit complexity in quantum regime would then be given by the Hessian of the
Morse function in supersymmetric quantum field theory. We also provide
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.
