RZZ¶
RZZ is the Z-based counterpart of RXX: a continuously tunable two-qubit coupling built from the \( Z \otimes Z \) interaction.
| Qubits | 2 |
| Parameters | \( \theta \) - coupling angle, in radians |
| Inverse | RZZ(−θ) |
| Also called | ZZ-rotation, Ising ZZ coupling gate |
What it does¶
RZZ(θ) is defined as \( e^{-i\theta Z \otimes Z / 2} \). Unlike RXX, it never mixes basis states into each other - it only ever applies a phase, based on whether the two qubits agree (\( |00\rangle, |11\rangle \)) or disagree (\( |01\rangle, |10\rangle \)).
Matrix¶
\[
RZZ(\theta) = \mathrm{diag}\left(e^{-i\theta/2},\ e^{i\theta/2},\ e^{i\theta/2},\ e^{-i\theta/2}\right)
\]
Action on basis states¶
- \( |00\rangle \rightarrow e^{-i\theta/2}\,|00\rangle \)
- \( |01\rangle \rightarrow e^{i\theta/2}\,|01\rangle \)
- \( |10\rangle \rightarrow e^{i\theta/2}\,|10\rangle \)
- \( |11\rangle \rightarrow e^{-i\theta/2}\,|11\rangle \)
Example¶
Applying RZZ to two qubits already in superposition adds a phase that depends on whether the two qubits' bits match - a building block for algorithms like QAOA that encode a cost function into phases.
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 | rzz(theta) q[0], q[1]; |
| Qiskit | circuit.rzz[theta, q[0], q[1]] |
| Cirq | cirq.ZZPowGate(exponent=theta / math.pi)(q[0], q[1]) |
| Q# | - |