QFT

This section provides an overview of the Quantum Fourier Transform (QFT) in the LogosQ Library.

Quantum Fourier Transform (QFT)

Mathematical Definition

The Quantum Fourier Transform is defined as the following unitary transformation on the computational basis states:

∣j⟩↦1N∑k=0N−1e2πijk/N∣k⟩|j\rangle \mapsto \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} e^{2\pi ijk/N} |k\rangle

Where:

  • N = 2^n for an n-qubit system
  • j and k are integers from 0 to N-1
  • e2πijk/Ne^{2\pi ijk/N} represents the complex phase factor

For a general quantum state ∣ψ⟩=∑j=0N−1αj∣j⟩|\psi\rangle = \sum_{j=0}^{N-1} \alpha_j |j\rangle, the QFT produces:

QFT∣ψ⟩=∑j=0N−1αj1N∑k=0N−1e2πijk/N∣k⟩QFT|\psi\rangle = \sum_{j=0}^{N-1} \alpha_j \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} e^{2\pi ijk/N} |k\rangle

This can be rewritten as:

QFT∣ψ⟩=1N∑k=0N−1(∑j=0N−1αje2πijk/N)∣k⟩QFT|\psi\rangle = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} \left(\sum_{j=0}^{N-1} \alpha_j e^{2\pi ijk/N}\right) |k\rangle

The QFT is the quantum analog of the discrete Fourier transform, performing a basis transformation from the computational basis to the Fourier basis.

Implementation in LogosQ

The LogosQ library provides a built-in function to perform the Quantum Fourier Transform on a quantum circuit. The function can be applied to a specified range of qubits within the circuit.

use logosq::algorithms::qft;
use logosq::circuits::Circuit;
use logosq::vis::circuit_text;
use logosq::vis::save_circuit_svg;
use logosq::vis::Visualizable;
use logosq::State;

fn main() {
    let num_qubits = 5; // Example with 5 qubits
    let results = quantum_fourier_transform_example(num_qubits);
    println!("Measurement results after QFT: {:?}", results);
}
// how to combine algorithms
pub fn quantum_fourier_transform_example(num_qubits: usize) -> Vec<usize> {

    let mut state = State::zero_state(num_qubits);
    // Create a circuit that demonstrates QFT
    let mut circuit = Circuit::new(num_qubits);
    circuit.x(0);
    circuit.execute(&mut state);

    // Apply QFT
    qft::apply(&mut state);
    print!("State after QFT: {}\n", state.visualize());
    qft::apply_inverse(&mut state);
    print!("State after inverse QFT: {}\n", state.visualize());
    circuit.execute_and_measure()
}