I do think it's kind of implicit in Yao's [Quantum Circuit Complexity](https://dl.acm.org/doi/10.1109/SFCS.1993.366852), especially in the proof of Theorem 1 of that paper; otherwise I think it's a bit of a folk-theorem. Yao speaks of the standard promise gap of $1/3$ vs. $2/3$ for one single qubit in the output, the same as the Wiki article on the AQP problem.

Being (promise) BQP-complete means both being in (promise) BQP and being BQP-hard.

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Although we *could* prove that problems are BQP- or QMA- or QCMA-complete by a reduction to the (quantum) Turing-machine model, nobody does this after Yao because quantum Turing machines are so darn difficult to work with. So mostly it's just done relative to the gate model... that is how, e.g., the Jones polynomial proof goes through most days.