https://en.wikipedia.org/wiki/Quantum_circuit Circuits are written much like a video editor or music editor. They are written where the horizontal axis represents time. Each circuit needs to conform to DiVincenzo's criteria
One difference between traditional logic gates is that quantum gates are reversible. You can derive the inputs from the output.
Necessary component is linear algebra. This is because superpositions are represented by linear combinations (kind of like polynomials and factorials)