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Is there a clear explaination of how exactly optimization_level choice in qiskit's transpile() method affects on circuit? I expected to see some mathematical operations with the gates that similifies the circuit. However, in the source codes (https://qiskit.org/documentation/stubs/qiskit.transpiler.preset_passmanagers.level_1_pass_manager.html#qiskit.transpiler.preset_passmanagers.level_1_pass_manager, https://qiskit.org/documentation/_modules/qiskit/transpiler/preset_passmanagers/level2.html#level_2_pass_manager, https://qiskit.org/documentation/_modules/qiskit/transpiler/preset_passmanagers/level3.html#level_3_pass_manager) there is nothing connected to that.

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watch Circuit Sessions with Kevin Krsulich: https://youtu.be/2T3163VjvWQ?t=1623 and 1MinuteQiskit for easy explanation: https://www.youtube.com/watch?v=xWJbfyE9Vi4
and look at docs explanation: https://qiskit.org/documentation/tutorials/circuits_advanced/04_transpiler_passes_and_passmanager.html#Preset-Pass-Managers

From Circuit Sessions with Kevin Krsulich: enter image description here

From source code:
in level_1 it will run
_opt = [Optimize1qGatesDecomposition(basis_gates), CXCancellation()]
in level_2 it run

_opt = [
        Optimize1qGatesDecomposition(basis_gates),
        CommutativeCancellation(basis_gates=basis_gates),
    ]

level_3

_opt = [
        Collect2qBlocks(),
        ConsolidateBlocks(basis_gates=basis_gates, target=target),
        UnitarySynthesis(
            basis_gates,
            approximation_degree=approximation_degree,
            coupling_map=coupling_map,
            backend_props=backend_properties,
            method=unitary_synthesis_method,
            plugin_config=unitary_synthesis_plugin_config,
            target=target,
        ),
        Optimize1qGatesDecomposition(basis_gates),
        CommutativeCancellation(),
    ]

For the math part, i would recommend watch Synthesis and Optimization of Quantum Circuits - Circuit Sessions with Dmitri Maslov for how much math is applied in it.

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