Quantum circuit tranpilation into rotations and introducing an informative measure of overhead at constant number of Qubits

     It is known that quantum circuit overhead is proportional to the number of gates times the number of qubits, and it is usually used to compare the resource usage of different circuits. Here, I am introducing an informative measure (return a number) for circuits with the same number of qubits that does not only convey the number of gates but also how much it is effective in the circuit. For example, a circuit composed of 6 identities is not the same as a circuit contains 6 Pauli gates. This will be useful in physical realization, as quantum gates are physically realized by rotations, and the new physical circuit will have the same dimensions. However, the same output can be done with different choices of rotations; one can choose many gates of small contributions or many redundant gates with high effect, or just the effective gates. In this work, I am making a function that transpiles the original circuit onto a physical circuit and introduces an informative measure that discriminates the mentioned situations; the measure proposes that the second circuit will have the highest overhead while the last one is the least.

Note : this project is written is self contained way, so the code is meant to be within the notebook.