What is a phase shift gate?

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🔹 What is a Phase Shift Gate?

In quantum computing, a Phase Shift Gate (also called a Phase Gate or R(ϕ) Gate) is a single-qubit quantum gate that changes the phase of the quantum state without altering its probability amplitudes (the magnitudes).

Mathematically, it applies a rotation around the Z-axis of the Bloch sphere by an angle ϕ.

🔹 Matrix Representation

The general form of the Phase Shift Gate is:

$$R_{\phi} = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\phi} \end{bmatrix}$$

• The $|0\rangle$ state remains unchanged.

• The $|1\rangle$ state picks up a phase factor of $e^{i\phi}$.

🔹 Special Cases of Phase Shift Gates

• Z Gate → When $\phi = \pi$

  $$Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

• S Gate (Phase Gate) → When $\phi = \frac{\pi}{2}$

  $$S = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}$$

• T Gate (π/8 Gate) → When $\phi = \frac{\pi}{4}$

  $$T = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{bmatrix}$$

🔹 Effect on Quantum State

If the input state is:

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$$

After applying $R_{\phi}$:

$$|\psi'\rangle = \alpha|0\rangle + \beta e^{i\phi}|1\rangle$$

• The probabilities $|\alpha|^2$ and $|\beta|^2$ remain the same.

• Only the phase of the |1⟩ component changes.

🔹 Real-World Usage

• Essential for quantum algorithms (like Shor’s algorithm, QFT).

• Helps in constructing arbitrary rotations.

• Crucial for error correction and quantum gates universality.

✅ In short:
A Phase Shift Gate rotates the phase of the $|1\rangle$ state by an angle ϕ, leaving probabilities unchanged but altering interference effects—making it fundamental in quantum computing.

Read More :

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Explain the CNOT (Controlled-NOT) gate.

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