Givens Rotation

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).

The unitary, drawn

Every entry of the matrix as a disc. The area is the transition probability $|U_{jk}|^2$; the colour and the hand are the phase. Rows are outputs $\langle j|$, columns inputs $|k\rangle$, in the computational basis. Each parameter has a slider; play runs through the family along the path of the Quirk link ($\theta = 2\pi t$).

Phase

Disc area $= |U_{jk}|^2$. Hover an entry for its value, or take the JSON.

Weyl chamber

Where this gate sits among all two-qubit gates, up to single-qubit rotations before and after.

Coordinates:

$(a, b, c)$ means the gate equals $e^{i(a\,XX + b\,YY + c\,ZZ)}$ up to single-qubit gates, reduced to the chamber $\pi/4 \ge a \ge b \ge |c|$ (the convention of the canonical gate, Qiskit and Cirq). The $a = \pi/4$ face is a mirror: $(\pi/4, b, c)$ and $(\pi/4, b, -c)$ are the same gate. Drag the figure to rotate it, double-click to reset.

SDK support

How to call this gate in each framework. Hover a ⓘ for a note; the support matrix compares every gate at once.

SDK	Name
Qiskit	— ⓘ
PennyLane	`pennylane.SingleExcitation` ⓘ
Cirq	`cirq.givens` ⓘ
Q#	—
PyQuil	—
Braket	—
BQSKit	—
Qibo	`qibo.gates.GIVENS`
pytket	— ⓘ
Stim	—
OpenQASM	—

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