phase gate

A phase gate, often denoted as the S gate or P gate, is a single-qubit quantum gate that shifts the phase of a quantum state. Specifically, it multiplies the |1⟩ state amplitude by *i* (the square root of -1) while leaving the |0⟩ state unchanged. It rotates the qubit's state around the Z-axis of the Bloch sphere by 90 degrees.

The S gate rotates the phase by π/2 (multiplying by *i*), while the T gate rotates the phase by π/4 (multiplying by *e^(iπ/4)*). The T gate is considered a fundamental universal gate, along with the Hadamard (H) gate and the CNOT gate. Together, these gates can approximate any unitary operation on a quantum computer.

While the phase gate does not change the probability of measuring a 0 or a 1, it alters the relative phase between them. This relative phase shift is crucial for creating quantum interference. Interference is the mechanism that allows quantum algorithms to amplify correct answers and cancel out wrong ones.

The standard S gate is not its own inverse; applying it four times in a row returns the qubit to its original state. The inverse of the S gate is the S† (S-dagger) gate, which rotates the phase by -π/2. The Hadamard gate (H gate), however, is an example of a quantum gate that is its own inverse.