Building a Quantum Fourier Transform Circuit

The Quantum Fourier Transform (QFT) is a fundamental quantum algorithm that performs a Fourier transform on quantum amplitudes. This guide will walk you through creating a QFT circuit in LeafSheep QVU.

Circuit Overview

The QFT circuit shown above consists of:

• 3 qubits
• Hadamard gates (H)
• Controlled Phase gates (P)
• Measurements on all qubits

Step-by-Step Construction

1. Create the Basic Circuit Structure

1. Open LeafSheep QVU
2. Add 3 qubits using the '+' button
3. Enable measurements on all qubits

2. Adding Gates

1. Drag the required gates from the Gate Palette in this order:

For Qubit 0 (top line):

• First, add a Hadamard (H) gate
• Then add two Phase (P) gates after the H gate

For Qubit 1 (middle line):

• Add a Hadamard (H) gate, positioned after the second P gate of Qubit 0
• Add one Phase (P) gate after the H gate

For Qubit 2 (bottom line):

• Add a Hadamard (H) gate, positioned after the P gate of Qubit 1

note

The order and positioning of gates is important for the QFT algorithm to work correctly. Make sure each gate is placed after the previous operations as described above.

3. Add Controlled Phase Gates

Now we'll convert the P Gates into Controlled-P Gates by adding control points:

For the P Gates on Qubit 0 (top line):

• First P Gate: Add control to Qubit 1 (middle line)
• Second P Gate: Add control to Qubit 2 (bottom line)

For the P Gate on Qubit 1 (middle line):

• Add control to Qubit 2 (bottom line)

To add controls:

• Hover on any P Gate to reveal control points
• Click and drag from the control point to the target qubit line
• A black dot and vertical line will appear, showing the control connection

4. Configure Phase Gates

Now we'll set the phase angles for each Controlled-P gate:

For the P Gates on Qubit 0 (top line):

• First P gate (controlling Qubit 1): Set phase to 0.5 (equivalent to π/2)
• Second P gate (controlling Qubit 2): Set phase to 0.25 (equivalent to π/4)

For the P Gate on Qubit 1 (middle line):

• P gate (controlling Qubit 2): Set phase to 0.5 (equivalent to π/2)

To set the phase parameters:

• Hover over a P gate to open its parameter panel
• Enter the value as a decimal (the value will be multiplied by π)
• Example: entering 0.5 means 0.5π or π/2

caution

The phase values are crucial for the QFT algorithm. Make sure each gate has the correct phase value before running the circuit.

Running the Circuit

1. Ensure all measurement buttons are enabled
2. Open the Console tab by clicking on the console button () on the right of the screen
3. Set the number of shots
4. Click "RUN" to execute the simulation

Expected Results

When you run this circuit, you should see:

• Probability distribution across all possible states
• You can see the Measurement counts by turning on the count switch on results panel

Common Issues and Tips

tip

• Ensure precise phase angles for accurate results
• Check that all control connections are properly set
• Verify that all measurements are enabled
• Use sufficient shots for reliable statistics

Theoretical Background

The Quantum Fourier Transform (QFT) is a quantum analogue of the classical discrete Fourier transform. It's a linear transformation on quantum bits that plays a crucial role in many quantum algorithms, including:

• Shor's algorithm for factoring
• Quantum phase estimation
• Hidden subgroup problems
• Fast quantum arithmetic operations

Mathematical Definition

For an n-qubit system (N = 2ⁿ), the QFT transforms a quantum state |x⟩ into:

$$QFT:|x⟩ \mapsto \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\omega_N^{xk}|k⟩$$

where $ωₙ = e^{2πi/N}$ is an $N$th root of unity.

Why Phase Gates (P Gates)?

In our circuit implementation, we use P gates (controlled-phase gates) to create the required phase relationships. Here's why:

1. The QFT can be decomposed into a product of simpler operations:

   • Hadamard gates (H) create superpositions
   • Phase gates (P) apply the necessary phase factors

2. The phase angles in our implementation correspond to the powers of ωₙ in the mathematical formula:

   • First P gate: phase = π/2 (equivalent to ω₄)
   • Second P gate: phase = π/4 (equivalent to ω₈)
   • And so on...

3. When controlled, these P gates create the entanglement necessary for the QFT's quantum parallelism.

tip

Understanding the mathematical foundation helps in:

• Verifying circuit correctness
• Choosing appropriate phase angles
• Extending to larger numbers of qubits

Next Steps

Now that you've built your first QFT circuit, you can:

1. Experiment with Different Sizes

   • Add more qubits to see how the circuit scales
   • Observe how the phase angles follow a pattern

2. Modify Parameters

   • Try different phase angles to see their effect
   • Observe how measurement results change