CY gate
| Identifier | Operator | Example statement |
|---|---|---|
| CY | \(CY\) | CY q[0], q[1] |
Description
The CY gate is a two-qubit gate. It is the controlled-Y gate. It performs a Y gate on the second qubit, conditional on the state of the first qubit.
Properties
- Clifford gate;
- Controlled gate;
- Ising gate.
Representation
\[\begin{align}
CY &= \left(\begin{matrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & -i\\
0 & 0 & i & 0
\end{matrix}\right)
\end{align}\]
which is equal to:
\[CY = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1| \otimes Y.\]
Operation examples
Standard basis
\[\begin{align}
CY\,|00\rangle &= |00\rangle \\
\\
CY\,|01\rangle &= i\,|11\rangle \\
\\
CY\,|10\rangle &= |10\rangle \\
\\
CY\,|11\rangle &= -i\,|01\rangle \\
\end{align}\]
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 \(|q_0\rangle \otimes |q_1\rangle\).