Square-root-SWAP gate

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

Description

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

Representation

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

Operation examples

Standard basis

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

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