quantum gate directory
Givens Rotation
Symbol
$G(\theta)$
Description:
Two-level rotation that mixes the $|01\rangle$ and $|10\rangle$ subspace.
Alternate notations:
- $G(\theta)$
- $G_{ij}(\theta)$
A Givens rotation is a two-qubit gate that performs a real rotation in the single-excitation subspace spanned by $|01\rangle$ and $|10\rangle$, leaving $|00\rangle$ and $|11\rangle$ unchanged.
$$ G(\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta & 0 \\ 0 & \sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Properties
- Conserves excitation number (acts nontrivially only on the single-excitation subspace).
- Real orthogonal rotation on ${|01\rangle, |10\rangle}$.
- Often combined with single-qubit phases to form a general complex Givens rotation.
Usage
- Fermionic simulation and Gaussian/Slater-determinant state preparation.
- Structured ansatze that require pairwise mode rotations (e.g., in quantum chemistry).
Groups
This gate is contained in the following groups:
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