Quantum Fourier Transform
The Quantum Fourier Transform (QFT) is a quantum algorithm used to transform a quantum state from the computational basis to the Fourier basis. It plays a crucial role in many quantum algorithms, such as Shor's algorithm for factoring large numbers and quantum phase estimation.
The QFT operates on a quantum register consisting of n qubits, where n is the number of qubits required to represent the desired precision of the transformation. Here's an overview of how the QFT works:
1. Initialization: The input quantum state is prepared in the computational basis, represented as a superposition of all possible states in the register. For example, if the register consists of n qubits, the initial state would be |x⟩ = 1/√(2^n) * Σ(y=0 to 2^n-1) |y⟩, where |y⟩ represents the binary representation of y.
2. Hadamard Transform: A Hadamard gate (H gate) is applied to each qubit in the register. This transforms the basis states |0⟩ and |1⟩ into equal superpositions: H|0⟩ = 1/√2(|0⟩ + |1⟩) and H|1⟩ = 1/√2(|0⟩ - |1⟩). After this step, the state becomes a superposition of all possible states.
3. Controlled Phase Rotations: Controlled phase rotations are applied to the qubits, controlled by the preceding qubits in the register. For each qubit j, a rotation gate Rk is applied, where k depends on the index of the qubit and the target qubit to which the rotation is applied. The rotation angle θ is determined by the binary representation of k/2^n. The controlled phase rotations introduce the desired phase relationships among the basis states.
4. Permutation: Finally, the qubits in the register are permuted. The order of the qubits is reversed compared to their initial order. This step is necessary to align the qubits properly for subsequent calculations.
The output of the QFT is the transformed state in the Fourier basis, representing the amplitudes of the basis states in the original computational basis. The QFT enables the extraction of frequency information from quantum states, making it a fundamental tool for many quantum algorithms.
Implementing the QFT in a quantum computer requires the availability of gate operations that can perform the necessary transformations and controlled rotations with the required precision. The number of qubits and the computational resources required for QFT increase with the desired precision of the transformation, making it more challenging to implement on larger registers or with higher precision.
Nevertheless, the QFT is a powerful algorithm that enables efficient calculations in the Fourier basis and is a key component in numerous quantum algorithms.
