The S gate applies a phase shift of \( \frac{\pi}{2} \) (90°) around the Z axis. Mathematically, it is expressed as:

\[ S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \]

This gate adds an \( i \) phase to the \(|1\rangle\) state while leaving the \(|0\rangle\) state unchanged.

Let the state of the qubit be \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\). After applying the S gate, we have:

\[ S |\psi\rangle = \alpha|0\rangle + \beta i |1\rangle \]

This expression indicates that the phase of the \(|1\rangle\) state is shifted by \( \frac{\pi}{2} \), or 90°, while the \(|0\rangle\) state remains unchanged.

The S gate represents a 90° rotation around the Z axis on the Bloch sphere. This gate shifts the phase of superposition states, thereby affecting the interference patterns in quantum algorithms.

The application of the S gate is typically realized in superconducting circuits using microwave signals. In ion traps and optical systems, the S gate can be implemented using laser pulses.

The S gate is used to control phase shifts and manage interference effects. It is particularly useful in quantum algorithms for controlling phase relationships between qubits.

13.1 Important Scenarios for the S Gate

There are three important scenarios where the S gate is essential:

1. Quantum Fourier Transform: The S gate is used to manage phase shifts during the Fourier transform process.

2. Error Correction: The S gate is used to correct phase shift errors in quantum error correction protocols.

3. Entanglement: The S gate is used to create phase relationships between qubits and to generate entanglement.

The S gate plays a crucial role in quantum algorithms by controlling phase shifts and facilitating interference effects. It is essential for creating phase relationships and entanglement, which are key components of many quantum computational processes.