Can Schrodinger's Cat Factor Numbers?

Suppose we wish to factor a 500-bit number N (assumed to be a product 
of 2 large primes).  We construct a hard-wired division circuit that 
outputs the remainder R of N when divided by an input D.  We know that 
if N is composite then it has a divisor of 250 bits or less, so our 
circuit only needs to handle 250-bit inputs.

If we flip a fair coin 250 times and use the results to set the input 
bits, then the probability of finding a factor is roughly 1/(2^250). 
This is just equivalent to randomly guessing a number between 1 and 
sqrt(N).  Not very efficient.

But suppose we perform 250 "Schrodinger's Cat" experiments in a giant 
sealed room, and connect the results to the inputs of our division 
circuit (which is also contained in the sealed room).  Now instead of 
a 1/(2^250) chance that the remainder is zero, quantum theory tells 
us that the remainder (from our perspective outside the sealed room) 
is the linear superposition of all 2^250 possible outcomes, one of 
which is R=0. 

I suppose as soon as we observe the result, it takes on just one of the
2^250 possible outcomes, and the probability of the outcome R=0 is just
1/(2^250), so nothing has been gained.

But the idea that, in the unobserved room, the R=0 condition is 
actually THERE seems very tantalizing.  Is it conceivable that some 
sort of feedback amplification could be used to enhance the R=0 
outcome?  We know that seemingly mutually exclusive outcomes of quantum 
processes can actually interfere with each other, as in the "two-slit 
experiments".  Could we similarly detect interference effects of the 
R=0 outcome on the other possible outcomes?