The Quantum Fourier Transform (QFT) is a fundamental quantum algorithm used in various other quantum algorithms. while QFT is employed in Shor’s algorithm and other quantum algorithms for its ability to efficiently transform between time and frequency domains.
QFT transforms quantum states from one basis to another, specifically between the computational (Z) basis and the Fourier basis. While classical Fourier Transform operates on continuous functions, QFT operates on the amplitudes of quantum states, which can represent multiple possibilities simultaneously due to superposition.
In quantum circuits, the Hadamard gate (H-gate) acts as the single-qubit QFT, converting between the Z-basis and X-basis states. The QFT is also applicable to multi-qubit systems, where it transforms the entire quantum state from the computational basis to the Fourier basis and vice versa.
One of the most prominent applications of QFT is in Shor’s algorithm, a quantum algorithm for integer factorization, where it enables efficient computation of the discrete Fourier transform, a crucial step in the algorithm’s implementation. Additionally, QFT finds applications in quantum phase estimation, quantum error correction, and various quantum machine learning algorithms. Despite its significance, the implementation of QFT in large-scale quantum systems remains a challenge due to the sensitivity of quantum states to errors and decoherence.
Step 1: Apply a Hadamard gate (H-gate) to the first qubit.
- Apply controlled rotations (phase gates) on subsequent qubits, controlled by previous qubits.
Step 2: Controlled Rotations:
For each qubit j from 2 to n:
- Apply Hadamard gate (H-gate) to qubit j.
- Apply controlled rotations (phase gates) on qubit j, controlled by qubits 1 to j-1.
- Our phases will increase π/2**1, π/2**2, till the number of qubits j-1.
Step 3: Swap Gates
- Perform swap gates to reverse the order of the qubits.
This will start with a short video tour of how the circuit looks for the 2-Qubit Selection and the results obtained when submitting the circuit.