  Near-term quantum computers are likely to have small depths due to short
coherence time and noisy gates, and thus a potential way to use these quantum
devices is using a hybrid scheme that interleaves them with classical
computers. For example, the quantum Fourier transform can be implemented by a
hybrid of logarithmic-depth quantum circuits and a classical polynomial-time
algorithm. Along the line, it seems possible that a general quantum computer
may only be polynomially faster than a hybrid quantum-classical computer. Jozsa
raised the question of whether $BQP = BPP^{BQNC}$ and conjectured that they are
equal, where $BQNC$ means $polylog$-depth quantum circuits. Nevertheless,
Aaronson conjectured an oracle separation for these two classes and gave a
candidate. In this work, we prove Aaronson's conjecture for a different but
related oracle problem. Our result also proves that Jozsa's conjecture fails
relative to an oracle.
