This paper aims to consider the importance and use of ZX-calculus taking into account the solid structure of this branch to give a representation of quantum algorithms by finding the function's properties pictorially using logic and category theory. It's desired to show this topic formally, establishing axioms and definitions that support which foundations will be operating, explaining the processes and steps followed. Will start with the notion of what a ZX-diagram is, what spiders are, symmetries, and transposes. After that, get into ZX-calculus regarding spider fusion, $$\pi $$ -commutation, color changing, and algebraic structures; it will show reasoning about the influence of connectivity. After this, their advantages in simplification, and, finally, how a ZX-diagram is translated into a quantum circuit.