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In the context of quantum control theory, it is common to see references to both quantum control and quantum optimal control (e.g. 0910.2350 or the guide on qutip quantrum control functions). Sometimes it seems like the two terms are used interchangeably, while sometimes they are treated as different things.

For example, from the above link, it seems that quantum optimal control is (unsurprisingly) a special kind of quantum control. It is however not too clear what the exact difference is between the two. For example, are both approaches used to tackle the same classes of problems? Is the only difference in quantum optimal control theory looking for optimal solutions, while quantum control techniques have less strict requirements?

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    $\begingroup$ To be called optimal quantum control, the control strategy (for example a specific NMR pulse sequence) must be optimal for a certain goal, for example to achieve maximum fidelity or minimum time. That's the only difference to (non-optimal) quantum control. $\endgroup$
    – user1039
    Mar 30, 2018 at 11:56
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    $\begingroup$ @pyramids great, that is the answer then! Can you provide some references which make this point explicit? $\endgroup$
    – glS
    Mar 30, 2018 at 12:40
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    $\begingroup$ Take the first sentence in the second paragraph under section 3.2 "Optimal control" from your own reference (0910.2350): "In the optimal control approach, the quantum control problem can be formulated as a problem of seeking a set of admissible controls satisfying the system dynamic equations and simultaneously minimizing a cost functional. " $\endgroup$
    – user1039
    Mar 30, 2018 at 12:58
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    $\begingroup$ @pyramids I still do not understand whether quantum optimal control uses different methodologies, or whether one calls it "optimal control" only when for some reason it is possible to prove that the found strategy is indeed optimal $\endgroup$
    – glS
    Mar 30, 2018 at 13:18
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    $\begingroup$ I think it is both (using different methodologies AND called "optimal control" because its results are optimal in a certain sense). But I'm not enough of an expert to be comfortable answering it as such. $\endgroup$
    – user1039
    Mar 30, 2018 at 13:26

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Indeed,quantum control theory studies whether a system is controllable or not. For example, if you have a Hamiltonian, there are several tests that rely on the Lie algebra, such as the Lie-Trotter theorem (including universal controllability) and the commutator formula. These allow you to check whether your system is controllable or not. However, these methods say nothing about efficiency. They are more useful if the system is not controllable. This is because in the case of a non-controllable system, these methods directly tell you that your system is not controllable. However, if you implement a pulse or gate in your system and have used the commutator formula to verify that your system is controllable, you have no indication of what the efficiency of your gate will be. In this case, quantum optimum theory comes into play, and there are several algorithms for finding optimal pulses, such as the CRAB algorithm. Quantum optimal theory looks for the best method/best pulse for your system (such as time optimal pulses, reshaping pulses etc..) whereas quantum control theory checks if the system is controllable or not.(you can find a solution by using quantum control theory but the result has not to be the best solution)

Some useful readings: quantum optimal control, quantum control theory

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  • $\begingroup$ Well, CRAB and other 1st order methods seem to have been destroyed: arxiv.org/abs/2209.05790 I think this was in this years APS MM. $\endgroup$
    – Marion
    Apr 7, 2023 at 8:25

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