Hello Quantum Enthusiasts,
Today, we're going to dive into the Quantum Fourier Transform (QFT) and its implementation in Quantum Computing. It's a linear transformation on quantum bits and is the quantum analogue of the discrete Fourier transform.
The Quantum Fourier Transform forms the basis of many quantum algorithms such as Shor's Algorithm for integer factorization and Quantum Phase Estimation. The QFT can transform a set of complex numbers into their Fourier series coefficients, which gives us a lot of useful information—like the periodicity of these numbers.
Suppose, we have a state |x⟩ in a 3-qubits system;
|x⟩ = ∑ (from i=0 to 7) x[i] |i⟩
QFT on |x⟩ is;
QFT(|x⟩) = 1/√8 ∑ (from i=0 to 7) ∑ (from j=0 to 7) e^(2πi*j/8) x[i] |j⟩
Let's examine how to implement QFT in a quantum computing environment using Qiskit, a Python library.
We will implement a 3-qubits QFT as a demonstration with Qiskit's tools developed by IBM Quantum for creating, manipulating, and running quantum programs. The quantum circuits can be run either on local simulators or on cloud-based quantum computing devices.
First, let's install Qiskit library:
pip install qiskit
Then we can move on to create a 3-qubit circuit and apply QFT to it.
from qiskit import QuantumCircuit,Aer,execute from qiskit.visualization import plot_bloch_multivector, plot_histogram import numpy as np # Create a 3 qubit circuit qc = QuantumCircuit(3) # Apply H-gate to each qubit: for qubit in range(3): qc.h(qubit) # Apply controlled rotation gates for qubit in range(3): for index in range(qubit): qc.cp(np.pi/2**(qubit-index), index, qubit) # Repeat procedure for reversed qubit order qc.swap(0, 2) # Transpile the circuit qc = transpile(qc) # Visualize Circuit qc.draw(output='mpl') simulator = Aer.get_backend('statevector_simulator') statevector = execute(qc, backend=simulator).result().get_statevector() # Output the state vector print(statevector)
Proficiency in Python and understanding of basic Quantum Computing principles are prerequisites for getting the most out of this tutorial. If you're new to quantum computing, be sure to check out additional resources on the subject, such as Qiskit’s own textbook.
Stay Tuned and Happy Quantum Computing!
P.S. Always remember this is an extremely simplified demonstration and for real-world quantum computing applications, you would need many more qubits, error correction codes, and a lot more complexity!