iSWAP Gate

iSWAP swaps two qubits and applies a phase factor. Often a native gate on superconducting qubit systems.

Matrix:

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

Action: $|ab\rangle \to i^{\delta_{ab}} |ba\rangle$ where $\delta_{ab} = 1$ if $a \neq b$, else 0.

Specifically: swaps $|01\rangle \to i|10\rangle$ and $|10\rangle \to i|01\rangle$, while leaving $|00\rangle$ and $|11\rangle$ unchanged.

$$\text{iSWAP} = e^{i\pi/4} \text{SWAP} \cdot \text{(diagonal phase)}$$

iSWAP can be decomposed into SWAP and single-qubit phase gates.

• Self-inverse: $\text{iSWAP}^2 = -\text{SWAP}$ (applies additional phase)
• Entangling: unlike SWAP, iSWAP creates entanglement
• Native on some platforms: superconducting qubits with tunable coupling naturally produce iSWAP

Superconducting qubits (flux-tunable): the parametric coupling between adjacent qubits naturally produces an iSWAP-like gate when tuned appropriately. Gate time: ~20–50 ns.

Trapped ions: engineered via laser pulses.

• Entanglement: more efficient than SWAP for creating entanglement
• Hamiltonian simulation: appears naturally in XX-coupled systems
• Universal gate: iSWAP + single-qubit gates form universal set (though CNOT is standard)

• CNOT-based: $\text{iSWAP} = (I \otimes H) \text{CX}_{01} \text{CZ}_{01} \text{CX}_{01} (I \otimes H)$
• Alternative: SWAP + phase correction
• Advantage: native iSWAP saves gate count and reduces errors vs decomposition

• Superconducting qubits: flux-tunable coupling naturally produces iSWAP; gate time ~20–50 ns
• Trapped ions: engineered via laser pulses
• Native advantage: direct use preferred when available on platform