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I have found the procedure of simulation process as this picture :
Image reference. So in the low-level compilers, all single-qubit gates are approximated by the universal gate set (CNOT, H, T, S).
When implementing arbitrary gates, I want to know if all simulators (like Qiskit) will do the low-level compilers job in this picture with a fixed accuracy. If so, what is the algorithm during this process? Are they using the Solovay-Kitaev Decomposition strategy (or some improved algorithm)?
Generally, a simulator does not have to do any decomposition of gates to hardware-level specifics. Simulators only follow a mathematical model of a gate (described by matrix). Since each algorithm can be described by a matrix, whole simulation can be expressed as $|\psi_1\rangle = U |\psi_0\rangle$, where $|\psi_0\rangle$ is initial state of a quantum computer, $|\psi_1\rangle$ is its final state and $U$ is a matrix describing algorithm. Hence, a simulation is reduced to matrix multiplication.
Some simulators artifically introduce noise present on real quantum hardware. This can be done by perturbation of matrices describing gates by a random variable.
Of course, it is possible to constrain a gate set on a simulator to have it more similar to real quantum processors. However, as I mentioned above, in the end you can have any gate you want, you are not constrained by quantum hardware specifics (unless you want to be) and any decomposition is not necessary.
Note: My answer is concerning gate-based computers. Adiabatic quantum computers case may be different.
