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Let $A$ be a class of quantum circuits such that

\begin{equation} \text{Post}A = \text{Post}BQP, \end{equation} where $\text{Post}$ indicates post-selection. Is only this amount of information sufficient to conclude that it is $\# P$ hard to estimate each output amplitude of a circuit in the class $A$, in the worst case, upto an inverse polynomial multiplicative error?

Note that it should be $\# P$ hard to compute each output amplitude of a circuit in $\text{Post}BQP$ upto inverse polynomial multiplicative error (as it is $\# P$ hard to compute each output amplitude of a circuit in $BQP$ upto the same error).

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