# CiSWAP gate (Controlled-iSWAP)

**CiSWAP** applies iSWAP to two qubits if a third control qubit is $|1\rangle$. Swaps and applies phase. Native on some transmon platforms.

Action: $|c ab\rangle \to |c (a' b')\rangle$ where iSWAP swaps qubits and applies phase $i$ to $|11\rangle$ when control is $|1\rangle$.

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

Basis action: swaps qubits 2 and 3 when qubit 1 is $|1\rangle$, applying phase $i$ to the swapped $|11\rangle$ state. Mapping: $|101\rangle \to i|110\rangle$ and $|110\rangle \to i|101\rangle$.

## Properties

- **Self-inverse**: $(\text{CiSWAP})^4 = I$ (since $(iSWAP)^4 = I$)
- **Asymmetric**: applies phase during swap, unlike CSWAP
- **Phase from iSWAP**: inherits the $i$ phase of iSWAP on the swapped $|11\rangle$ state

## Uses

- **Native gate on transmon qubits**: superconducting platforms with tunable coupling
- **Variational circuits**: QAOA and VQE with native two-qubit interactions
- **Entanglement**: creates controlled entanglement with phase correction
- **Gate decomposition**: alternative to CX chains when iSWAP is cheaper

## Parametric version

- **CiSWAP($\theta$)**: applies $\text{iSWAP}(\theta)$ when control is $|1\rangle$; tunes interaction strength
- **Use**: parametric coupling in variational quantum algorithms

## Implementation

- **Superconducting qubits**: native on transmon qubit systems with tunable coupling; gate time ~20-50 ns
- **Trapped ions**: less common; typically decomposed
- **Photonic**: challenging; usually decomposed

## Relations

- [[quantum-gate-iswap|iSWAP]]: two-qubit version; CiSWAP adds control
- [[quantum-gate-cswap|CSWAP]]: similar controlled swap but without phase
- [[quantum-gate-three-qubit|Three-qubit gates]]: general category