iSWAP Gate

The iSWAP gate is a two-qubit entangling gate that swaps the $|01\rangle$ and $|10\rangle$ components of a two-qubit state while multiplying each by a factor of $i$. It is a natural gate for superconducting qubit platforms where qubits interact via a transverse (XY) coupling, and it forms part of Google’s native gate set.

Matrix representation

In the computational basis $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$, the iSWAP gate has the matrix:

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

Its action on basis states:

$|00\rangle \to |00\rangle, \quad |01\rangle \to i|10\rangle, \quad |10\rangle \to i|01\rangle, \quad |11\rangle \to |11\rangle$

The $|00\rangle$ and $|11\rangle$ states are unchanged, while $|01\rangle$ and $|10\rangle$ are swapped with an added phase of $i$.

Physical origin

The iSWAP arises naturally from the XY (transverse) coupling Hamiltonian between two qubits:

$H_{XY} = g(X_1 X_2 + Y_1 Y_2)$

where $g$ is the coupling strength. Evolving under this Hamiltonian for a time $t = \pi/(4g)$ produces the iSWAP gate (up to single-qubit phases). This interaction is physically realized when two transmon qubits are brought into resonance, either by tuning their frequencies together or by activating a tunable coupler.

Google’s Sycamore processor uses the $\sqrt{\text{iSWAP}}$ gate (half the interaction time) as its primary entangling gate. The $\sqrt{\text{iSWAP}}$ is often preferred because it produces maximal entanglement with shorter gate times, reducing exposure to decoherence.

Relationship to other gates

The iSWAP is equivalent to a SWAP gate followed by a controlled-Z, up to single-qubit phases:

$\text{iSWAP} = \text{SWAP} \cdot \text{CZ} \cdot (S \otimes S)$

where $S = R_z(\pi/2)$ is the phase gate. Any two-qubit gate can be decomposed into at most three iSWAP gates (or two $\sqrt{\text{iSWAP}}$ gates) plus single-qubit rotations.

A CNOT gate can be constructed from two iSWAP gates:

$\text{CNOT} = (I \otimes S^\dagger)(I \otimes H) \cdot \text{iSWAP} \cdot (S^\dagger \otimes I) \cdot \text{iSWAP} \cdot (H \otimes I)$

This decomposition matters when targeting hardware that uses iSWAP natively, since expressing algorithms in terms of native gates avoids additional transpilation overhead.

The $\sqrt{\text{iSWAP}}$ gate

The square root of iSWAP, obtained by evolving under the XY Hamiltonian for half the time, has the matrix:

$\sqrt{\text{iSWAP}} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1/\sqrt{2} & i/\sqrt{2} & 0 \\ 0 & i/\sqrt{2} & 1/\sqrt{2} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$

This gate creates a maximally entangled state from $|01\rangle$ or $|10\rangle$, making it a natural entangling primitive.

Why it matters for learners

Understanding the iSWAP gate is essential for working with Google’s quantum processors and for appreciating how hardware physics shapes gate set design. The choice between CX-based (IBM) and iSWAP-based (Google) native gates affects transpilation strategies and circuit depths for the same algorithm. When benchmarking or comparing hardware platforms, knowing the native two-qubit gate is necessary to make fair comparisons.

See also

• CNOT Gate
• CZ Gate
• Native Gate Set
• Two-Qubit Gate Fidelity
• Transpilation