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Quantum Fourier Transform

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Absolutely! Quantum Fourier Transform (QFT) is a fundamental quantum algorithm that is key to many quantum computing applications, including Shor’s Algorithm for factoring and solving periodicity problems. Let’s break it down clearly.

Here’s a detailed explanation of Quantum Fourier Transform (QFT)—perfect for slides, notes, or teaching materials.

🔄 Quantum Fourier Transform (QFT)

🧠 What is the Quantum Fourier Transform?

The Quantum Fourier Transform (QFT) is the quantum version of the Discrete Fourier Transform (DFT), a mathematical operation that transforms a signal from the time domain to the frequency domain.

In quantum computing, QFT transforms the state of a quantum register into a superposition of frequencies, and it can be done exponentially faster than classical algorithms. It’s a crucial part of many quantum algorithms, such as Shor’s Algorithm for factoring and Quantum Phase Estimation.

🔍 The Classical Fourier Transform

In classical computing, the Discrete Fourier Transform (DFT) of a sequence of NN complex numbers x0,x1,...,xN−1x_0, x_1, ..., x_{N-1} is defined as:

Xk=∑n=0N−1xne−2πikn/NX_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i kn / N}

This converts the data into a frequency domain.

But the classical DFT takes O(N2)O(N^2) time to compute. The Quantum Fourier Transform reduces this to O(Nlog⁡N)O(N \log N), but using quantum parallelism and superposition.

🧩 Quantum Fourier Transform on a Qubit

Let's consider a quantum state of nn qubits:

∣ψ⟩=∑k=0N−1ak∣k⟩|\psi\rangle = \sum_{k=0}^{N-1} a_k |k\rangle

Where the aka_k's are complex amplitudes, and ∣k⟩|k\rangle represents the computational basis states.

The goal of the Quantum Fourier Transform is to transform this quantum state into a new state ∣ψ′⟩|\psi'\rangle, such that:

∣ψ′⟩=∑k=0N−1bk∣k⟩|\psi'\rangle = \sum_{k=0}^{N-1} b_k |k\rangle

where bkb_k is related to the Fourier coefficients of aka_k, but it’s much faster and more efficient using quantum gates.

⚡ How Does the Quantum Fourier Transform Work?

The Quantum Fourier Transform operates on each qubit by applying a series of Hadamard gates and controlled phase shift gates.

Here’s a high-level breakdown of how QFT is implemented on nn qubits:

  1. Hadamard Gate:
    • The first qubit undergoes a Hadamard gate, creating a superposition. This starts the Fourier transformation.
  2. Controlled Phase Shift Gates:
    • For each subsequent qubit, a series of controlled phase shift gates are applied. These gates cause the relative phases between qubits to change, which is essential to the Fourier transform.
  3. Reversing the Qubits:
    • Finally, the qubits are reversed in order. This reversal ensures the final quantum state corresponds to the correct Fourier-transformed state.

🧑‍💻 The Quantum Circuit for QFT

  1. Start with a quantum register initialized to ∣0⟩⊗n|0\rangle^{\otimes n}.
  2. Apply a Hadamard gate to the first qubit.
  3. For each successive qubit:
    • Apply controlled-phase gates to the qubits in positions after the current qubit.
  4. Reverse the qubits in the quantum register.
  5. Measure the resulting quantum state.

💡 Key Gates Used in QFT

  1. Hadamard Gate (H):
    The Hadamard gate creates a superposition of states. It’s a critical part of creating the “spread-out” states that quantum algorithms need to perform efficiently. H∣0⟩=12(∣0⟩+∣1⟩)H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
  2. Controlled Phase Shift Gate (R_k):
    The controlled phase shift gate is used to apply a phase shift on a qubit conditioned on the state of another qubit.
    The phase shift gates RkR_k apply a rotation depending on the relative distance between qubits: Rk∣1⟩=e2πi/2k∣1⟩R_k|1\rangle = e^{2\pi i / 2^k}|1\rangle

📐 Circuit Diagram

Here’s a simplified circuit for QFT with 3 qubits:

  1. Apply a Hadamard gate on the first qubit.
  2. Apply controlled phase shifts between qubits.
  3. Reverse the qubits at the end. 
|0⟩ — H — R(2) — R(3) — Reverse
|0⟩ — H — R(2) — Reverse
|0⟩ — H — Reverse

📊 Example: QFT on 3 Qubits

Suppose you have a 3-qubit state:

∣ψ⟩=a0∣000⟩+a1∣001⟩+a2∣010⟩+a3∣011⟩+a4∣100⟩+a5∣101⟩+a6∣110⟩+a7∣111⟩|\psi\rangle = a_0|000\rangle + a_1|001\rangle + a_2|010\rangle + a_3|011\rangle + a_4|100\rangle + a_5|101\rangle + a_6|110\rangle + a_7|111\rangle

The QFT of this state will perform the following:

  1. Hadamard Gate on the first qubit.
  2. Controlled-phase gates will adjust the relative phases between qubits.
  3. Finally, the qubits are reversed in order, and the resulting state will be the Fourier-transformed state.

This dramatically reduces the number of operations compared to the classical DFT.

⏱️ Time Complexity

  • Classical Discrete Fourier Transform (DFT):
    O(N2)O(N^2)
  • Quantum Fourier Transform (QFT):
    O(n2)O(n^2) where nn is the number of qubits.
  • The exponential speedup is achieved because QFT can be performed in polynomial time on quantum computers.

🌐 Applications of QFT

  • Shor’s Algorithm: QFT is the key quantum tool for finding the period of a function, which is central to integer factorization.
  • Quantum Phase Estimation (QPE): QFT is used in algorithms that estimate the eigenvalues of a unitary operator, which is important for various quantum algorithms.
  • Quantum Simulation: QFT is applied in quantum simulations, where the system's behavior in the frequency domain is critical for accurate results.

✅ Summary

  • The Quantum Fourier Transform (QFT) is a quantum algorithm that efficiently computes the Fourier transform on quantum states.
  • QFT is used in many quantum algorithms, including Shor’s algorithm for factoring large numbers and quantum phase estimation.
  • It operates on qubits by applying Hadamard gates and controlled phase shift gates, and reversing the qubits at the end.
  • The QFT provides polynomial time speedup over classical algorithms, enabling powerful quantum algorithms.

🚀 Want More?

  • A step-by-step example with actual numbers (like QFT on a 2-qubit system)?
  • Visual diagrams showing QFT operations in more detail?
  • Qiskit code to run QFT simulations on a quantum computer?
  • Flashcards or quizzes for reviewing concepts?

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