The Quantum Fourier Transform (QFT) is a powerful quantum operation that transforms quantum states into a Fourier basis. It's like a quantum version of the classical Fourier transform, but with some cool quantum twists that make it super efficient.

QFT is a game-changer in quantum computing. It's the secret sauce behind many quantum algorithms, like Shor's factoring algorithm. QFT can process multiple states in parallel, making it exponentially faster than classical Fourier transforms.

Quantum Fourier Transform

Definition of quantum Fourier transform

• Quantum analog of the classical discrete Fourier transform (DFT) performs Fourier transform on quantum states by mapping computational basis states to Fourier basis states
• Unitary operation preserves inner products and norms of quantum states ensuring the transformation is reversible and maintains the quantum properties of the system
• Linearity property allows the QFT to be applied to superpositions of states $QFT(\alpha|x\rangle + \beta|y\rangle) = \alpha QFT(|x\rangle) + \beta QFT(|y\rangle)$ enabling parallel processing of multiple states
• Periodicity property $QFT(|x\rangle) = QFT(|x+2^n\rangle)$ for an n-qubit system implies that the QFT output is periodic with respect to the input state (similar to the classical DFT)

Circuit representation of QFT

• Mathematical representation $|x\rangle \rightarrow \frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^n-1} e^{2\pi i x k / 2^n} |k\rangle$ where $x$ is the input state, $k$ is the output state, and $n$ is the number of qubits describes the transformation of the input state to a superposition of output states with specific amplitudes
• Composed of Hadamard gates and controlled rotation gates where Hadamard gates create superposition and controlled rotation gates apply phase rotations
  • Rotation angles depend on the position of the control and target qubits with smaller angles for qubits further from the most significant qubit
• Apply Hadamard and controlled rotations in reverse order starting with the least significant qubit and moving towards the most significant qubit to ensure the correct phase relationships between the qubits

Implementation with quantum gates

• Hadamard gate ($H$) is a single-qubit gate that creates superposition transforming $|0\rangle$ to $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|1\rangle$ to $\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$
• Controlled rotation gates ($R_k$) are two-qubit gates that apply a phase rotation to the target qubit with the matrix representation $R_k = \begin{pmatrix} 1 & 0 \ 0 & e^{2\pi i / 2^k} \end{pmatrix}$ where $k$ determines the rotation angle and depends on the position of the control and target qubits
• Swap gates are used to reverse the order of qubits after applying the QFT circuit which is necessary to obtain the correct output state order (e.g., $|q_0q_1q_2\rangle$ becomes $|q_2q_1q_0\rangle$)

Complexity analysis vs classical transform

• Quantum Fourier Transform (QFT) requires $O(n^2)$ quantum gates for an n-qubit system with $n$ Hadamard gates and $\frac{n(n-1)}{2}$ controlled rotation gates resulting in an exponentially faster algorithm compared to the classical Fourier transform
• Classical Discrete Fourier Transform (DFT) has a complexity of $O(n2^n)$ for an input of size $2^n$ while the Fast Fourier Transform (FFT) algorithm reduces the complexity to $O(n2^n)$ which is still exponentially slower than the QFT
• QFT provides an exponential speedup over the classical DFT enabling efficient quantum algorithms for problems such as period finding (used in Shor's algorithm for factoring) and quantum phase estimation (used in quantum chemistry simulations)