Table of Contents
iSWAP gate
iSWAP gate is a two-qubit gate that swaps the $\lvert 01\rangle$ and $\lvert 10\rangle$ amplitudes while multiplying each by $i$, and leaves $\lvert 00\rangle$ and $\lvert 11\rangle$ unchanged.
$$\text{iSWAP} = \begin{pmatrix}1&0&0&0\\ 0&0&i&0\\ 0&i&0&0\\ 0&0&0&1\end{pmatrix}$$
$$\text{iSWAP}\lvert 01\rangle = i\lvert 10\rangle \qquad \text{iSWAP}\lvert 10\rangle = i\lvert 01\rangle$$
The factor of $i$ distinguishes iSWAP from the plain SWAP gate. SWAP permutes amplitudes without changing phases and therefore cannot create entanglement from a product state; iSWAP can, because the phase $i$ introduces interference between the swapped components.
Hardware origin
The iSWAP gate arises naturally from the exchange coupling Hamiltonian $H = g(a^\dagger b + ab^\dagger)$, which describes capacitively coupled superconducting qubits. Evolving under this Hamiltonian for time $t = \pi/(2g)$ produces exactly the iSWAP gate. This makes iSWAP a native gate on many superconducting processors — it can be implemented in a single microwave pulse, without decomposing into CX gates.
Universality
iSWAP together with all single-qubit gates is universal for quantum computation. The related $\sqrt{\text{iSWAP}}$ gate (half the evolution time) is also commonly used as a native two-qubit gate on superconducting hardware, and like $\sqrt{\text{SWAP}}$, it directly generates entanglement.