Double-CNOT gate

Identifier	Operator	Example statement
DCNOT	\(DCNOT\)	DCNOT q[0], q[1]

Description

The double-CNOT, or DCNOT, gate is a two-qubit gate. It is defined as a sequence of two anti-parallel CNOT gates, where the first CNOT gate takes the first qubit as the control qubit and the second CNOT gate is anti-parallel and takes the second qubit as the control qubit.

Properties

• Involutory operation (its own inverse);
• Perfect Entangler (maximally entangles specific product states).

Representation

\[\begin{align} DCNOT &= \left(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{matrix}\right) \end{align}\]

Operation examples

Standard basis

\[\begin{align} DCNOT\,|00\rangle &= |00\rangle \\ \\ DCNOT\,|01\rangle &= |10\rangle \\ \\ DCNOT\,|10\rangle &= |11\rangle \\ \\ DCNOT\,|11\rangle &= |01\rangle \\ \end{align}\]

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