花开花落花无悔,缘来缘去缘如水。
花谢为花开,花飞为花悲。
花悲为花泪,花泪为花碎。
花舞花落泪,花哭花瓣飞。
花开为谁谢,花谢为谁悲。
Author: Malcolm Low
Dr. Malcolm Low is currently an Associate Professor with the Singapore Institute of Technology (SIT).
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Quantum Fourier Transform (QFT) of a Single Qubit is Hadamard Transform
Below is the definition of QFT as illustrated in the YouTube lecture by Abraham Asfaw.
The LaTex code for the equation is as follows and also available here.
Latex
| \tilde{x} \rangle \equiv ~ QFT ~ |x \rangle ~ \equiv \frac{1}{\sqrt{N}}\sum_{y=0}^{N-1}{e^{\frac{2\pi ix y}{N}}} ~| y \rangle
For the one qubit case, N = 21 = 2:
Latex
| \tilde{x} \rangle \equiv ~ QFT ~ |x \rangle ~ \equiv \frac{1}{\sqrt{}N}\sum_{y=0}^{N-1}{e^{\frac{2\pi ix y}{N}}} ~| y \rangle
Latex
\frac{1}{\sqrt{2}}\sum_{y=0}^{1}{e^{\pi ix y}} ~| y \rangle = \frac{1}{\sqrt{2}}[~e^{i \pi x 0}~ | 0 \rangle ~ + ~ e^{i \pi x 1}~| 1 \rangle] = \frac{1}{\sqrt{2}}[~|0\rangle ~+~e^{i \pi x}~|1 \rangle~]
Latex
QFT~| 0 \rangle = \frac{1}{\sqrt{2}}[~|0\rangle ~+~e^{i \pi 0}~|1 \rangle~] = \frac{1}{\sqrt{2}}[~| 0 \rangle + |1 \rangle~] = |+\rangle
Latex
QFT~| 1 \rangle = \frac{1}{\sqrt{2}}[~|0\rangle ~+~e^{i \pi 1}~|1 \rangle~] = \frac{1}{\sqrt{2}}[~| 0 \rangle - |1 \rangle~] = |-\rangle
Hence the QFT of a single qubit is essentially the Hadamard transform.
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