  It has previously been established that the logarithmic-depth approximate
quantum Fourier transform (AQFT) provides a suitable replacement for the
regular QFT in many quantum algorithms. Since the AQFT is less accurate by
definition, polynomially many more applications of the AQFT are required to
achieve the original accuracy. However, in many quantum algorithms, the smaller
size of the AQFT circuit yields a net improvement over using the QFT.
  This paper presents a more thorough analysis of the AQFT circuit, which leads
to the surprising conclusion that for sufficiently large input sizes, the
difference between the QFT and the logarithmic-depth AQFT is negligible. In
effect, the AQFT can be used as an direct replacement for the QFT, yielding
improvements in any application which does not require exact quantum
computation.
