Inverse-square-root-SWAP gate

Identifier	Operator	Example statement
InvSqrtSWAP	\(\sqrt{SWAP}^\dagger\)	InvSqrtSWAP q[0], q[1]

Description

The inverse-square-root-SWAP gate is a two-qubit gate. It reverses the action of the square-root-SWAP gate.

Representation

\[\begin{align} \sqrt{SWAP}^\dagger &= \left(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{2} \left ( i-1 \right ) & \frac{1}{2} \left ( i+1 \right ) & 0 \\ 0 & \frac{1}{2} \left ( i+1 \right ) & \frac{1}{2} \left ( i-1 \right ) & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right) \end{align}\]

Operation examples

Standard basis

\[\begin{align} \sqrt{SWAP}^\dagger\,|00\rangle &= |00\rangle \\ \\ \sqrt{SWAP}^\dagger\,|01\rangle &= \tfrac{1}{2} \left ( i+1 \right )\,|10\rangle + \tfrac{1}{2} \left ( i-1 \right )\,|01\rangle \\ \\ \sqrt{SWAP}^\dagger\,|10\rangle &= \tfrac{1}{2} \left ( i-1 \right )\,|10\rangle + \tfrac{1}{2} \left ( i+1 \right )\,|01\rangle \\ \\ \sqrt{SWAP}^\dagger\,|11\rangle &= |11\rangle \\ \end{align}\]

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 \(|q_0\rangle \otimes |q_1\rangle\).