P (phase)¶
The P gate is the general-purpose phase gate: pick any angle \( \theta \), and P applies exactly that much relative phase to \( |1\rangle \). Z, S, and T are all just P with a specific angle plugged in.
| Qubits | 1 |
| Parameters | \( \theta \) - phase angle, in radians |
| Inverse | P(−θ) |
| Also called | Phase gate; sometimes written R1 or U1 |
What it does¶
P(θ) leaves \( |0\rangle \) completely untouched and multiplies \( |1\rangle \) by \( e^{i\theta} \). Setting \( \theta = \pi/4, \pi/2, \pi \) reproduces T, S, and Z exactly.
Matrix¶
Action on basis states¶
- \( |0\rangle \rightarrow |0\rangle \)
- \( |1\rangle \rightarrow e^{i\theta}\,|1\rangle \)
Phase and Bloch-sphere effect¶
P(θ) rotates the Bloch vector by θ about the Z-axis - like RZ, but without RZ's extra overall phase factor (see the note on that page for the exact relationship between the two).
Example¶
H then P with \( \theta = \pi/3 \) on q[0] gives \( \dfrac{1}{\sqrt{2}}(|0\rangle + e^{i\pi/3}|1\rangle) \) - any angle you like, not just the fixed steps S and T provide.
Other notations¶
Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.
| Language | Typical form |
|---|---|
| OpenQASM 2.0 | u1(theta) q[0]; |
| Qiskit | circuit.p[theta, q[0]] |
| Cirq | cirq.ZPowGate(exponent=theta / math.pi)(q[0]) |
| Q# | R1(theta, q[0]); |