# 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.

## Relation to SWAP

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

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

## Properties

- **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

## Native Implementation

**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.

## Uses

- **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)

## Decomposition

- **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

## Implementation

- **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