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If you did not already, start by having a look to the Deutsch's algorithm, which is the single-qubit version of the Deutsch-Jozsa algorithm you are looking at. Here you have some quick resources: http://www.cs.xu.edu/~kinne/quantum/deutche.html https://www.youtube.com/watch?v=5xsyx-aNClM It is simpler than Deutsch-Jozsa and a good stepping stone.


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Regarding your first question, there are $n$ inputs to $f$ and the task is to determine if $f$ is constant or balanced (under the promise that it is one or the other); you can do this on a quantum computer with a single query to $f$. If $n=1$, then classically you can do as you suggest, input $0$ for one query, and input $1$ for another query, for a total ...


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