hardware natives 2 qubits 2 parameters
Fermionic Simulator
Parameterized two-qubit gate combining an $XY$ interaction with a controlled-phase, native to Google superconducting hardware.
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 $\theta = 0, \phi = 0$.
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 ($\theta = 2\pi t,\ \phi = 4\pi t$).
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:
not entangling entangling perfect entangler
$(a, b, c)$ 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 | — ⓘ |
| 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 | — |