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CV gate

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

Description

The CV gate is a two-qubit gate. Note that \(V = X^{1/2}\), i.e., CV is the controlled-square-root-of-X (CSX) gate. It performs an X90 gate on the second qubit, conditional on the state of the first qubit.

Aliases

Also known as the, controlled-V, controlled-X90, or controlled-square-root-of-X gate.

Properties

Representation

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

which is equal to:

\[CV = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes V.\]

Operation examples

Standard basis

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

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