P gate applies a phase $e^{i\theta}$ to the $|1\rangle$ state, leaving $|0\rangle$ unchanged. It is a parameterized single-qubit gate that rotates around the z-axis on the Bloch sphere. With specific angle choices, the P gate becomes S ($\theta = \pi/2$), T ($\theta = \pi/4$), or Z ($\theta = \pi$).

Matrix:

$$P(\theta) = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\theta} \end{pmatrix}$$

Action: $P(\theta)|0\rangle = |0\rangle$, $P(\theta)|1\rangle = e^{i\theta}|1\rangle$ (adds phase $\theta$ to $|1\rangle$).

• Parameterized: $\theta$ is an arbitrary angle; different $\theta$ values give different gates
• Diagonal: phase gates do not entangle or mix basis states
• Inverse: $P(\theta)^\dagger = P(-\theta)$
• Composition: $P(\theta_1) P(\theta_2) = P(\theta_1 + \theta_2)$ (phases add)
• Periodicity: $P(\theta + 2\pi) = P(\theta)$ (global phase equivalence)
• Clifford structure: P is Clifford only for specific angles (e.g., $\theta = \pi/2$ (S), $\theta = \pi$ (Z)); non-Clifford for others (e.g., $\theta = \pi/4$ (T))

Common instances of the P gate with specific angles:

• S gate: $P(\pi/2) = S = R_Z(\pi/2)$; applies 90° phase
• T gate: $P(\pi/4) = T = R_Z(\pi/4)$; applies 45° phase
• Z gate: $P(\pi) = Z = R_Z(\pi)$; applies 180° phase (Pauli Z)
• S† gate: $P(-\pi/2) = S^\dagger = R_Z(-\pi/2)$; applies -90° phase
• T† gate: $P(-\pi/4) = T^\dagger = R_Z(-\pi/4)$; applies -45° phase

The P gate is a special case of the RZ rotation gate:

$$P(\theta) = R_Z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix} \times e^{i\theta/2}$$

The difference is a global phase factor that doesn't affect measurement outcomes. On superconducting qubits, P gates are typically implemented as virtual RZ rotations via reference frame adjustment.

• State preparation: rotate basis states to arbitrary phases
• Circuit synthesis: build arbitrary unitaries from phase and bit-flip layers
• Variational algorithms: parameterized phase gates in quantum circuits (VQE, QAOA)
• Algorithm implementation: phase corrections in quantum algorithms
• Bloch sphere rotation: explicit z-axis rotation without physical pulses (superconducting qubits)

• Superconducting qubits: RZ($\theta$) is virtual (reference frame adjustment); essentially free
• Trapped ions: detuned laser pulse applies phase without excitation
• Photonic: optical phase shifter set to angle $\theta$
• Cost: free or near-free on most platforms (no physical pulse required if virtual); specific-angle gates (S, T) may be optimized

• Phase gates: category containing P and other parameterized phase gates
• S gate: special case $P(\pi/2)$
• S† gate: special case $P(-\pi/2)$
• T gate: special case $P(\pi/4)$
• T† gate: special case $P(-\pi/4)$
• Z gate: special case $P(\pi)$; Pauli Z
• RZ gate: P gates are equivalent to RZ rotations (modulo global phase)
• Clifford gates: P is Clifford for specific angles (S, Z) but not others (T)
• Single-qubit gates: fundamental building block