We now have as a primitive operation that the Fourier transform is a fast process on a quantum computer. Note that while the Fast Fourier transform on a conventional computer takes of order steps, on a quantum computer the Fast Fourier transform of the state (note this is of the state, not of a collection of input numbers) takes only of order steps.
The first really significant result of Quantum Computing was Peter Shor's demonstration that factoring could be fast on a quantum computer. He accompished this by noticing that if one could find the period of a certain periodic function depending on the number in questin, one could factor that number. Finding that period made use of the quantum Fourier transform. In order to see how, we need to look at some number theory.
Let us assume that we have some integer , which is the product of two prime factors and . There are then a number of features of numbers which are important.
Definition: mod operation. Given two integers, and , the term means that integer, between and which is the remainder when you divide by . I.e., for some integer .
Now, for some number lying between 1 and , which shares no common factors with , consider . Eventually for some (smallest) value of , this must be equal to . ( since there are only possible numbers that this could be, eventually one power must equal some other power. I.e., which means that divides . Since x has no common factors with , neither does , so must be a multiple of and . Define as the smallest such power.
Then the function is a periodic function in with period . There are now three possibilities for that :
a) is odd;
If is even then we can write , which is a multiple of as . Then, either
b) divides , in which case our was not the smallest possible value of ,
c) divides ,
d) or one, but not both, of the factors of divides ( and thus also .) I.e., and have a common factor.
If two numbers have a common factor, one can rapidly discover what that common factor is using Euclid's algorithm. If two numbers have a common factor, then the remainder when dividing the larger by the smaller also has the same common factor. Successively dividing in this way eventually leads to that unique greatest common factor in a time less than of order .
Thus, if we can find the period of the function , we can test whether this falls into category d, and if it does, we have a factor of . If it does not, (i.e., falls under classes a,b, or c), we try with another value of . For randomly selected x, Shor claimed that there will be at least a 50% probability that the will fall into class d, and that we will have a solution. After testing a small number of randomly chosen , the probability will become overwhelming that we will have found the factor. Note that the Shor algorithm is not a deterministic one. It is one in which the probability of not finding the right answer decreases at least exponentially with the number of trials.
The solution thus hinges on finding the period of the
function, for a given . Finding periods is ideally
suited to a Fourier
Transform. To do this, first design a computer which
takes a state and calculates
This need not even be a
reversible computer as long as nothing in the environment becomes correlated
with . This can be done
in a number of operations of order where is a number like
3. Then feed into this computer the state
is the first integer larger than -- i.e., .
The maximum value of is thus greater than .
This will produce the state
. Measure the
output state, (the bits which
correspond to the output, but not the input state. Assume
that this measurement gave the value (it does not matter what this value
actually is. It plays no role. This measurement need not even be carried out at
this point, since those output bits will play no further role in the calculation
and could also be measured a the end of the experiment). This
will leave the system in the state
We have a state in which the amplitudes are periodic and can do a Fourier
transform of these amplitudes. The resultant state will be
, where the
are the Fourier transform of the
A measurement of will then give a us a number which is a multiple of . Unless and the multiple have a common factors (a rare situation), one can uniquely determine from the measured value of . One can then use that to try to find the common factors of and . If one finds a common factor, one has a factor of . If one does not, one tries again with a different value of . The probability is thus very high that in only a few attempts one will have found a factors .