Tag: mathematics

Mathematical Patterns: The Curious Case of 1/998001

Mathematics reveals elegant patterns in unexpected places. Consider 1/998001, which equals 0.000001002003004… containing every three-digit number in sequence.

This occurs because 998001 = 999². Similar patterns emerge in related fractions:

  • 1/9 = 0.111111…
  • 1/99 = 0.010101…
  • 1/999 = 0.001001001…

These numerical sequences demonstrate that mathematics is not merely computational but reveals fundamental structures underlying our universe. Such patterns have practical applications in algorithm development, cryptography, and data analysis.

The ordered nature of these mathematical curiosities reminds us that even within apparent complexity, we can discover remarkable simplicity and structure.

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~]

When x = 0:

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

When x = 1:


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.