swap family 2 qubits
fSWAP
Swaps $|01\rangle$ and $|10\rangle$ while applying a $-1$ phase to $|11\rangle$.
The fermionic SWAP (fSWAP) gate behaves like SWAP on single-excitation states, but adds a phase of $-1$ to the $|11\rangle$ state. The name comes from the fact that exchanging two fermions introduces a $-1$ phase as fermionic wavefunctions are antisymmetric under particle exchange.
$$ \mathrm{fSWAP} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} $$
Properties
- Self-inverse: $\mathrm{fSWAP}^2 = I$.
- Equals SWAP on the $\{|01\rangle, |10\rangle\}$ subspace.
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.
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 | pennylane.FermionicSWAP ⓘ |
| Cirq | — ⓘ |
| Q# | — |
| PyQuil | — |
| Braket | — ⓘ |
| BQSKit | — |
| Qibo | qibo.gates.FSWAP |
| pytket | — ⓘ |
| Stim | CZSWAP ⓘ |
| OpenQASM | — |