Exploring Quantum Fourier Transform and the Golden Ratio

Exploring Quantum Fourier Transform and the Golden Ratio

In this experiment, we explore the relationship between the Quantum Fourier Transform (QFT), prime numbers, Fibonacci numbers, and the Golden Ratio.

Background

The Golden Ratio is a unique mathematical constant, approximately equal to \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618. It appears in various areas of mathematics, art, and nature. Our goal was to see if applying QFT on a superposition of prime and Fibonacci numbers could reveal a pattern that approximates this special ratio.

Methodology

We used 6 qubits to allow us to work with numbers up to 64. We created a superposition that included both prime numbers and Fibonacci numbers within this range. The following primes and Fibonacci numbers were used:

  • Prime Numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61
  • Fibonacci Numbers: 1, 2, 3, 5, 8, 13, 21, 34, 55

The superposition was passed through the QFT, which was applied iteratively three times to amplify any potential patterns.

Results

After the third iteration of the QFT, we analyzed the amplitude ratios of consecutive states in the resulting quantum state. We found two values that closely approximate the Golden Ratio:

  • Ratio 1: 1.6147
  • Ratio 2: 1.6270

Both values are remarkably close to the Golden Ratio (\phi \approx 1.618), suggesting a possible relationship between the QFT, primes, Fibonacci numbers, and this special constant.

Conclusion

Through iterative application of the QFT on a combined superposition of prime and Fibonacci numbers, we were able to approximate the Golden Ratio. This suggests that further exploration of quantum algorithms and number theory might uncover deeper connections between quantum mechanics and fundamental mathematical constants like \phi.

Further research could involve testing with larger qubit systems, additional mathematical constants, or exploring other quantum transformations beyond the QFT.

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