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It is known that the Simon Problem lies in $BQP^O$ (oracular problem). Even it proves $\exists O$ $BPP^O\neq BQP^O$. Or It separates the classes in the Oracle/Query model of computation.

Meanwhile, the factoring problem lies in $BQP$ (non-oracular).

I reviewed the FORRELATION problem in the paper link. An oracle for function $f$ is provided as part of the problem.

The k-fold FORRELATION is shown (in the paper) to be (promise) BQP-complete. I find the definition mentioned in section 1.1.3 (page 4) to be oracular.

My query is:

Is the k-fold FORRELATION problem in BQP or $BQP^{O'}$?

Or am I missing some detail/context?

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There is a subtle difference in the terminology used in the paper. The wording of the theorem 5 in the paper makes it more clear.

If you have been given an explicit construction (say, as a circuit) of the functions ($f_i$), then k-fold FORRELATION is a promise BQP-complete problem.

Otherwise, merely an access of black box for $f_i$ don't make it even eligible to be called a BQP problem.

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  • $\begingroup$ Thanks. I missed to notice this difference. $\endgroup$ Commented Mar 26 at 20:14

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