  We investigate the cost of three phase estimation procedures that require
only constant-precision phase shift operators. The cost is in terms of the
number of elementary gates, not just the number of measurements. Faster phase
estimation requires the minimal number of measurements with a \log * factor of
reduction when the required precision n is large. The arbitrary
constant-precision approach (ACPA) requires the minimal number of elementary
gates with a minimal factor of 14 of reduction in comparison to Kitaev's
approach. The reduction factor increases as the precision gets higher in ACPA.
Kitaev's approach is with a reduction factor of 14 in comparison to the faster
phase estimation in terms of elementary gate counts.
