How to sequentially enumerate multiple admissible linear combinations?

Suppose I have some quantum algorithm for which it may be the case that any of zero, one or many possible linear combinations may be the desired result at the measurement of entanglement once the algorithm is run.

 

My thinking is, for such a scenario to be supported, you’d need to have a generic algorithm for sequentially eliciting the linear combinations that have raised probability amplitudes in a series of rounds. Simply doing measurement in each basis isn’t enough, because you don’t want to either treat each qubit as an individual or enumerate the potentially spaces between linear combinations. Therefore, for example,  you want to run the algorithm, and read the ith linear combination with very high probability, regrouping the probability amplitude for any other linear combination with raised probability to that item.

By repeatedly running the algorithm, and simply iterating until either an already seen linear combination is witnessed, or some moniker, such as the zero vector, is read, you can get all possible results.

Does such an algorithm already exist? If so, what is it’s name?

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