Understanding Quantum Fourier Transform In Quantum Computing

Introduction

Quantum Fourier Transform (QFT) plays a fundamental role in many quantum algorithms, such as Shor's algorithm for factoring and the quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator.

The Quantum Fourier Transform is the quantum implementation of the discrete Fourier transform over the amplitudes of a wavefunction. It is part of many quantum algorithms and it can offer exponential speedup over the classical discrete Fourier transform.

The Mathematics of Quantum Fourier Transform

The mathematical representation of the quantum Fourier transform (QFT) on a state |j> in an n-qubit quantum system is as follows:

The recipe to build a QFT uses controlled rotations, and their implementation is relatively straightforward.

Quantum Computing Language – Qiskit

To understand the implementation of the QFT, we would heavily use Qiskit, which is an open-source quantum computing software development framework by IBM.

Qiskit allows users to write quantum circuits using Python, and execute them using publicly available quantum computers. Here, we will use Qiskit to implement the quantum Fourier transform.

Here is the Python code for a Qiskit implementation of the QFT:

from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister
from qiskit import execute

def qft(circ, q, n):
    """n-qubit QFT on q in circ."""
    for j in range(n):
        for k in range(j):
            circ.cu1(pi/float(2**(j-k)), q[j], q[k])
        circ.h(q[j])

q = QuantumRegister(3, 'x')
c = ClassicalRegister(3, 'c')
qc = QuantumCircuit(q, c)

qft(qc, q, 3)

print(qc)

Conclusion

Quantum Fourier Transform is a quantum algorithm used to transform state amplitudes in a quantum register that allows the transfer of information between the computational (amplitude) and Fourier (phase) bases.

These concepts and more are part of the exciting field of quantum computing, which combines principles of physics, mathematics, and computer science. This field offers potentially revolutionary technology, and understanding its foundational algorithms in computer applications are important steps towards mastery.