In the special case of a periodic function with period R, where are the nonzero amplitudes located after applying the QFT and how many nonzero amplitudes are there?
The Quantum Fourier Transform (QFT) is a fundamental operation in quantum information processing that plays a important role in quantum algorithms, such as Shor's algorithm for factoring large numbers and the quantum phase estimation algorithm. The QFT is a quantum analogue of the classical discrete Fourier transform, and it enables the efficient computation of the Fourier transform of a quantum state.
In the special case of a periodic function with period R, the QFT can be used to determine the amplitudes of the Fourier components of the function. The QFT maps the input state, which encodes the function values at equidistant points on the interval [0, R), to the output state, which encodes the Fourier coefficients of the function.
After applying the QFT to a periodic function with period R, the nonzero amplitudes are located at specific positions in the output state. These positions correspond to the frequencies of the Fourier components of the function. More precisely, the nonzero amplitudes are located at positions k, where k is an integer between 0 and R-1. Each position k corresponds to a specific frequency, given by k/R.
The number of nonzero amplitudes in the output state after applying the QFT depends on the function being transformed. In general, if the function has M distinct frequencies, then the number of nonzero amplitudes in the output state will be M. However, it is important to note that the QFT can also introduce additional amplitudes due to the superposition of different frequency components. Therefore, the number of nonzero amplitudes in the output state can be greater than M.
To illustrate this, let's consider a simple example. Suppose we have a periodic function with period R=4, and the function has two distinct frequencies: f1=1 and f2=3. After applying the QFT, the nonzero amplitudes will be located at positions k=1 and k=3 in the output state. These positions correspond to the frequencies f1=1/4 and f2=3/4, respectively. Thus, in this example, there are two nonzero amplitudes in the output state.
After applying the QFT to a periodic function with period R, the nonzero amplitudes are located at positions k, where k is an integer between 0 and R-1. The number of nonzero amplitudes in the output state depends on the function being transformed and can be greater than the number of distinct frequencies in the function.
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