  Modular exponentiation (ME) operators are one of the fundamental components
of Shor's algorithm, and the place where most of the quantum resources are
deployed. I propose a method for constructing the ME operators that relies upon
the simple observation that the work register starts in state $\vert 1
\rangle$. Therefore, we do not have to create an ME operator $U$ that accepts a
general input, but rather, one that takes an input from the periodic sequence
of states $\vert f(x) \rangle$ for $x \in \{0, 1, \cdots, r-1\}$, where $f(x)$
is the ME function with period $r$. The operator $U$ can be partitioned into
$r$ levels, where the gates in level $x \in \{0, 1, \cdots, r-1\}$ increment
the state $\vert f(x) \rangle$ to the state $\vert f(x+1) \rangle$. The gates
below $x$ do not affect the state $\vert f(x+1) \rangle$. The obvious problem
with this method is that it is self-defeating: If we knew the operator $U$,
then we would know the period $r$ of the ME function, and there would be no
need for Shor's algorithm. I show, however, that the ME operators are very
forgiving, and truncated approximate forms in which levels have been omitted
are able to extract factors just as well as the exact operators. I demonstrate
this by factoring the numbers $N = 21, 33, 35, 143, 247$ by using less than
half the requisite number of levels in the ME operators. This procedure works
because the method of continued fractions only requires an approximate phase
value. This is the basis for a factorization strategy in which we fill the
circuits for the ME operators with more and more gates, and the correlations
between the various composite operators $U^p$ (where $p$ is a power of two)
compensate for the missing levels.
