To illustrate the algorithm's operation and behavior, consider the
small case of 
with the map starting from level 
and going up to level 
. Suppose that 
3 
and its supersets are the only
nogoods. We begin with all amplitude in the empty set, i.e., with the
state 
. The map from level 0 to 1 gives equal amplitude to
all singleton sets, producing 
. We then introduce a phase for the nogood set, giving
. Finally we use Eq. 16 to map this to the sets at level 2, giving the final
state
At this level, only set 1,2 is good, i.e., a solution. Note that the
algorithm does not make any use of testing the states at the solution
level for consistency.
The probability to obtain a solution when the final measurement is made is determined by the amplitude of the solution set, so in this case Eq. 22 becomes
From this we can see the effect of different methods for selecting the
phase for nogoods. If the phase is selected randomly, 
because the average value of 
is zero. Inverting the phase of the nogood, i.e.,
using 
, gives 
. These probabilities compare with the 1/3 chance of
selecting a solution by random choice. In this case, the optimal choice
of phase is the same as that obtained by simple inversion. However this
is not true in general: picking phases optimally will require knowledge
about the solutions and hence is not a feasible mapping. Note also that
even the optimal choice of phase doesn't guarantee a solution is
found.