iSWAP gate

Definition

$$\mathrm{iSWAP} = \begin{pmatrix} 1&0&0&0\\0&0&i&0\\0&i&0&0\\0&0&0&1 \end{pmatrix}$$

in the $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ basis — a SWAP gate with an extra factor of $i$ attached to the states it actually exchanges.

Action on basis states

$$\mathrm{iSWAP}|00\rangle = |00\rangle,\quad \mathrm{iSWAP}|01\rangle = i|10\rangle,\quad \mathrm{iSWAP}|10\rangle = i|01\rangle,\quad \mathrm{iSWAP}|11\rangle = |11\rangle$$

Unlike plain SWAP, iSWAP entangles superposition inputs: $\mathrm{iSWAP}\,\tfrac{|01\rangle+|10\rangle}{\sqrt2} = i\,\tfrac{|01\rangle+|10\rangle}{\sqrt2}$ (an eigenstate, only a phase), but $\mathrm{iSWAP}\,\tfrac{|01\rangle+|11\rangle}{\sqrt2} = \tfrac{i|10\rangle+|11\rangle}{\sqrt2}$, which is entangled even though the input was a product state.

Physical origin

$\mathrm{iSWAP}$ is exactly the time evolution generated by the isotropic $XY$ exchange interaction:

$$\mathrm{iSWAP} = \exp\!\Big(i\frac{\pi}{4}(X\otimes X + Y\otimes Y)\Big)$$

Schuch and Siewert (2003) analyze this family of gates and show that, unlike an Ising ($ZZ$) coupling which needs single-qubit gates to reach a CNOT-equivalent, the $XY$ interaction alone already yields an entangling gate — but reaching full CNOT-equivalence still needs single-qubit rotations layered around it, since $\mathrm{iSWAP}$ and $\mathrm{CNOT}$ are only locally equivalent, not identical.

Key identities

• $\mathrm{iSWAP}^2 = -\,\mathrm{SWAP}$: two iSWAPs give minus the ordinary swap (the two accumulated factors of $i$ multiply to $-1$).
• $\mathrm{iSWAP} = \mathrm{SWAP}\cdot \mathrm{CZ}$-like phase structure: iSWAP and CNOT are related by a "magic basis" change and are both maximally entangling, but not identical — a $\sqrt{\mathrm{iSWAP}}$ combined with single-qubit gates is itself sometimes used as a native entangling gate on hardware where a full iSWAP pulse is harder to calibrate than a half-angle one.