# how to make barcode in c#.net Finding such a root is reduced to computing the order of a random integer modulo N in Software Generation QR Code in Software Finding such a root is reduced to computing the order of a random integer modulo N

Finding such a root is reduced to computing the order of a random integer modulo N
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103 The quantum Fourier transform
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Recall the fast Fourier transform (FFT) from 2 It takes as input an M -dimensional, complex-valued vector (where M is a power of 2, say M = 2 m ), and outputs an M -dimensional complex-valued vector : 1 1 1 1 1 2 M 1 0 0 1 2 4 2(M 1) 1 1 1 2 2 , = M 1 j 2j (M 1)j M 1 M 1 1 (M 1) 2(M 1) (M 1)(M 1) where is a complex M th root of unity (the extra factor of M is new and has the effect of ensuring that if the | i |2 add up to 1, then so do the | i |2 ) Although the preceding equation suggests an O(M 2 ) algorithm, the classical FFT is able to perform this calculation in just O(M log M ) steps, and it is this speedup that has had the profound effect of making digital signal processing practically feasible We will now see that quantum computers can implement the FFT exponentially faster, in O(log 2 M ) time! But wait, how can any algorithm take time less than M , the length of the input The point is that we can encode the input in a superposition of just m = log M qubits: after all, this superposition consists of 2m amplitude values In the notation we introduced earlier, we would write the superposition as = M 1 j j where i is the amplitude of the m-bit j=0 binary string corresponding to the number i in the natural way This brings up an important 296
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Figure 104 The classical FFT circuit from 2 Input vectors of M bits are processed in a sequence of m = log M levels
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