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 $|110\rangle \to i|101\rangle$.

• 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

• 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

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

• 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

• iSWAP: two-qubit version; CiSWAP adds control
• CSWAP: similar controlled swap but without phase
• Three-qubit gates: general category