What should I read to understand this problem?The power of small-depth quantum  circuits. Is $BQP = BPP^{BQNC}$? In other  words, can the quantum part of any  quantum algorithm be compressed to  polylog(n) depth, provided we're  willing to do polynomial-time  classical postprocessing? (This is  known to be true for Shor's  algorithm.) If so, building a  general-purpose quantum computer would  be much easier than is generally  believed! Incidentally, it's not hard  to give an oracle separation between  $BQP$ and $BPP^{BQNC}$, but the question is  whether there's any concrete function  instantiating such an oracle. --Scott Aaronson  http://www.scottaaronson.com/writings/qchallenge.html
Reading up on $BQP = BPP^{BQNC}$
cc.complexity theory;reference request;quantum computing
This was conjectured by R. Jozsa in Section 8 of arXiv:quant-ph/0508124. If you are already familiar with quantum computing and quantum complexity theory, you can start by reading that section.An important reading is arXiv:quant-ph/0006004, where Cleve and Watrous show that Shor's algorithm is in that class. 