6.13.6. Differences Between QFT and Classical Fourier Transform

6.13.6.1. Parallel Processing Capability

In the classical Fourier transform, each frequency component is calculated separately. QFT, however, processes all frequency components simultaneously using quantum superposition.

6.13.6.2. Complexity

The classical Fourier transform has a time complexity of \( O(n2^n) \), while QFT operates with a time complexity of \( O(n^2) \), allowing quantum computers to handle large datasets much faster.

6.13.7. Applications of QFT

QFT is not limited to Shor’s Algorithm and is used in other quantum algorithms as well, including:

• Quantum Simulations: Used in the frequency analysis of molecular and physical systems.
• Spectral Analysis: Applied in quantum signal processing and spectral analysis.
• Quantum Image Processing: Utilized in analyzing frequency components in image processing algorithms.

6.13.8. Example: QFT Computation

Let us examine how QFT works with a simple example. Apply QFT to the state \( |01\rangle \) in a 2-qubit system.

Step 1: Initial State

\( |01\rangle \)

Step 2: Apply Hadamard Gate (Qubit 0)

\( H|0\rangle = \frac{|1\rangle + |0\rangle}{\sqrt{2}} \)

After this operation, the system state becomes:

\( \frac{|0\rangle + |1\rangle}{\sqrt{2}} \otimes |1\rangle = \frac{|01\rangle + |11\rangle}{\sqrt{2}} \)

Step 3: Apply Controlled \( R_2 \) Gate (Qubit 1 as Control)

\( R_2|1\rangle = e^{2\pi i / 4}|1\rangle = i|1\rangle \)

After this operation, the system state becomes:

\( \frac{|0\rangle|1\rangle + i|1\rangle|1\rangle}{\sqrt{2}} = \frac{|01\rangle + i|11\rangle}{\sqrt{2}} \)

Step 4: Swap Gate (Swap Qubit 0 and Qubit 1)

\( \frac{|10\rangle + i|11\rangle}{\sqrt{2}} \)

Step 5: Final State

\( \frac{|10\rangle}{\sqrt{2}} + \frac{i|11\rangle}{\sqrt{2}} = \frac{1}{\sqrt{2}}|10\rangle + \frac{i}{\sqrt{2}}|11\rangle \)

This example illustrates the basic steps of QFT and how quantum superposition is utilized in the process.