This research proposal is devoted to an investigation in the context of the holographic duality between a theory of quantum gravity in Anti-de Sitter spacetime and a conformal field theory on its boundary, the AdS/CFT correspondence. I propose quantitative investigations of the holographic duality between bulk and boundary complexity candidates using soluble models. Complexity is a quantity that originates from quantifying the difficulty of carrying out a task in quantum information theory. In the context of AdS/CFT, complexity describes important bulk physics such as the growth of the black hole interior at late times which is inaccessible using existing tools like entanglement entropy. I propose research on four key questions. Firstly, I aim to construct a circuit model for black hole collapse. Secondly, I propose a detailed study of saturation and the switch-back effect in the model of a quantum circuit that implements conformal transformations and whose exact gravity dual is known. Thirdly, I put forward a rigorous verification/falsification study of holographic complexity conjectures by again employing the soluble model of the quantum circuit that implements conformal transformations. Lastly, I propose to formulate circuit complexity in another soluble model in lower dimension.