Exploring Quantum Fourier Transform In Quantum Computing

Introduction to Quantum Fourier Transform

Quantum Fourier Transform (QFT) is an essential algorithm used in many other applications such as phase estimation, order finding, and the quantum approximation algorithm. Just like the classical Fourier Transform changes the basis vectors from the time domain to the frequency domain, its quantum counterpart does the same: it converts the state of a quantum register from the position to the momentum representation.

Quantum Fourier Transform Algorithm

The Quantum Fourier Transform (QFT) of an n-qubit state |j>, with (0 ≤ j < 2^n ), is defined as follows:

|ψin⟩=1/√(2^n) Σ(j=0 to 2^n−1)e^(2πij.sinφ/2^n) |j⟩

Where |ψin⟩is an n-qubit input state and output state |ψout⟩ is given by the QFT of |ψin⟩. The quantum circuit that implements QFT makes use of two types of gates: a single-qubit Hadamard gate and a two-qubit controlled phase rotation gate.

Let's now have a look at a Python code executing a Quantum Fourier Transform using Qiskit, a Python quantum computing SDK.

from qiskit import QuantumCircuit, transpile
from qiskit.extensions import UnitaryGate

def qft_dagger(circ, n):
    """n-qubit QFTdagger using recursion."""
    # Don't forget the Swaps!
    for qubit in range(n//2):
        circ.swap(qubit, n-qubit-1)
    if n == 0:
        return circ
    n -= 1
    circ.h(n)
    for qubit in range(n):
        circ.cp(-np.pi/2**(n-qubit), qubit, n)
    qft_dagger(circ, n)

Circ = QuantumCircuit(3)
qft_dagger(Circ, 3)
Circ.measure_all()
Circ.draw()

In this script, we define a function qft_dagger that uses recursion to generate Quantum Fourier Transform operations on a QuantumCircuit object from qiskit. The function uses several gates including, Hadamard (h) gate and controlled-phase rotation (cp) gate. The circuit measures all the qubits at the end and then we visualize the circuit using the draw() method.

Conclusion

The quantum Fourier transform is a cornerstone of many quantum algorithms and serves as a quantum version of the discrete Fourier transform. It is promising because it is able to compute the discrete Fourier transform of an amplitude-encoded mode in a time that scales like the logarithm of the number of mode components, which is exponentially faster than the best classical algorithms. This extraordinary increase in speed-up paves the way for the exciting and much-anticipated advent of quantum computing.

Remember, this guide is an introduction to some fundamental concepts, and you might enjoy exploring even further nuances and examples of Quantum Fourier Transform in quantum computing.