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I am currently running a simple algorithm using Qiskit and I am running it for various transpiler optimization levels (0-3). I wish to know what exactly occurs differently when for example I run the algorithm with optimization level 1 as compared to when the algorithm is run with optimization level 3. There is general information available on what a transpiler does, however, I cannot find what exactly happens differently in each optimization level.

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These levels are provided via "preset passmanagers" in Qiskit. These are simple-to-use transpiler pipelines, but you can also build your own passmanager pipeline. You can see what each does by inspecting the documentation for those: https://github.com/Qiskit/qiskit-terra/tree/master/qiskit/transpiler/preset_passmanagers

But briefly, level 0 does no explicit optimization, it will just try to make a circuit runnable by transforming it to match a topology and basis gate set, if necessary.

Level 1, 2 and 3 do light, medium and heavy optimization, using a combination of passes, and by configuring the passes to search for better solutions. Typically more optimality comes at the cost of slower transpilation, as it spends more time searching for a good optimization.

For example if you look at the difference between level 2 and level 3, in level 2 the transpiler does some commutation analysis to see what gates can be collapsed. In level 3, in addition to that, it does peep-hole optimization by combining a chain of gates on the same qubits and re-synthesizing them with better cost. Also the routing stage of level 3 is more expensive than level 2 (it has a larger search space, as set by parameters of the StochasticSwap and LookaheadSwap passes).

It is important to keep in mind that these are heuristics. It is possible that, for a specific circuit, you will not see much improvement by increasing the level.

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