In this work, an algorithm inspired by graph theory is developed to calculate the minimum number of cuts in a quantum circuit, with the aim of analyzing unitary quantum circuits with a random architecture of local one and two qubits. A uniform probability distribution of the gates is studied, investigating the behavior of entropy as a function of the number of qubits and the circuit depth, additionally, the thermodynamic limit for these parameters is obtained analytically, allowing the study of quantum entanglement growth. Finally, a comparison is made between the numerically exact entropy and the estimation based on the minimum cut. [IN PROCESS] [REF Skinner, B. (2023). Lecture Notes: Introduction to random unitary circuits and the measurement-induced entanglement phase transition. arXiv preprint arXiv:2307.02986.]