Reading up on $BQP = BPP^{BQNC}$ 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
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.