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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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point: the j notation is really just another way of writing a vector, where the index of each entry of the vector is written out explicitly in the special bracket symbol Starting from this input superposition , the quantum Fourier transform (QFT) manipulates it appropriately in m = log M stages At each stage the superposition evolves so that it encodes the intermediate results at the same stage of the classical FFT (whose circuit, with m = log M stages, is reproduced from 2 in Figure 104) As we will see in Section 105, this can be achieved with m quantum operations per stage Ultimately, after m such stages and m2 = log 2 M elementary operations, we obtain the superposition that corresponds to the desired output of the QFT So far we have only considered the good news about the QFT: its amazing speed Now it is time to read the ne print The classical FFT algorithm actually outputs the M complex numbers 0 , , M 1 In contrast, the QFT only prepares a superposition = M 1 j j=0 And, as we saw earlier, these amplitudes are part of the private world of this quantum system Thus the only way to get our hands on this result is by measuring it! And measuring the state of the system only yields m = log M classical bits: speci cally, the output is index j with probability | j |2 So, instead of QFT, it would be more accurate to call this algorithm quantum Fourier sampling Moreover, even though we have con ned our attention to the case M = 2 m in this section, the algorithm can be implemented for arbitrary values of M , and can be summarized as follows: 297
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Input: A superposition of m = log M qubits, =
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