Square-root-iSWAP gate

Identifier	Operator	Example statement
SqrtISWAP	\(\sqrt{iSWAP}\)	SqrtISWAP q[0], q[1]

Description

The square-root-iSWAP gate is a two-qubit gate. Whereas the iSWAP gate exchanges the state of two qubits while adding a phase factor, the square-root-iSWAP gate performs half of this exchange, putting pure separable states into a maximally entangled state.

Representation

\[\begin{align} \sqrt{iSWAP} &= \left(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ 0 & \frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right) \end{align}\]

Operation examples

Standard basis

\[\begin{align} \sqrt{iSWAP}\,|00\rangle &= |00\rangle \\ \\ \sqrt{iSWAP}\,|01\rangle &= \tfrac{1}{\sqrt{2}}\,|01\rangle + \tfrac{i}{\sqrt{2}}\,|10\rangle \\ \\ \sqrt{iSWAP}\,|10\rangle &= \tfrac{i}{\sqrt{2}}\,|01\rangle + \tfrac{1}{\sqrt{2}}\,|10\rangle \\ \\ \sqrt{iSWAP}\,|11\rangle &= |11\rangle \\ \end{align}\]

Qubit state ordering convention and matrix representation

Note that qubits in a ket are ordered with qubit indices decreasing from left to right, i.e.,

\[|\psi\rangle = \sum c_i~|q_nq_{n-1}~...q_1q_0\rangle_i\]

Note that for matrices a reversed basis ordering convention is adopted, as is done in most textbooks. For instance, in the case of a two-qubit control gate, the matrix is represented such that \(q_0\) is the control qubit and \(q_1\) is the target qubit; the state on which the matrix is applied should then effectively be written as \(|q_0\rangle \otimes |q_1\rangle\).