quantum gate directory

Qubits: $2$ Parameters: $2$

SDK Support

SDK	Name
Qiskit	— ⓘ
PennyLane	— ⓘ
Cirq	cirq.FSimGate
Q#	—
PyQuil	pyquil.simulation.matrices.FSIM ⓘ
Braket	— ⓘ
BQSKit	bqskit.ir.gates.FSIMGate
Qibo	qibo.gates.fSim
pytket	pytket.circuit.OpType.FSim ⓘ
Stim	—
OpenQASM	—

Groups

This gate is contained in the following groups:

• number-preserving

The fermionic simulator gate is a two-parameter family that unifies the iSWAP and CZ interactions.

$$ \mathrm{FSim}(\theta, \phi) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -i\sin\theta & 0 \\ 0 & -i\sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & \mathrm{e}^{-i\phi} \end{bmatrix} $$

Special cases

• $\mathrm{FSim}(\pi/2, 0) = i\mathrm{SWAP}$
• $\mathrm{FSim}(\pi/4, 0) = \sqrt{i\mathrm{SWAP}}$
• $\mathrm{FSim}(\pi/2, \pi) = \mathrm{fSWAP}$
• $\mathrm{FSim}(0, \phi) = \mathrm{C}P(\phi)$
• $\mathrm{FSim}(0, \pi) = \mathrm{CZ}$
• $\mathrm{FSim}(\pi/2, \pi/6)$ approximates the Sycamore gate used in Google's quantum supremacy experiment.

Properties

• The $\theta$ parameter controls the swap angle in the $\{|01\rangle, |10\rangle\}$ subspace.
• The $\phi$ parameter is a controlled phase on $|11\rangle$.
• Reduces to a product of single-qubit gates at $\theta = 0, \phi = 0$.