YGate
Contents
5.7. YGate#
This gate is not popular since it can be replaced with other common operators. Hence, this section only briefly covers its properties.
5.7.1. Definition#
Transformation
\[ Y |0\rangle = i|1\rangle, \qquad Y |1\rangle = -i |0\rangle \]
Matrix expression
(5.5)#\[\begin{split} Y \doteq \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \end{split}\]
U gate expression
(5.6)#\[ Y = U\left(\pi,\frac{\pi}{2},\frac{\pi}{2}\right) \]
R gate expression
\[ Y = i R_y(\pi) \]
The qiskit circuit code symbol is y and it appears in quantum circuit as
from qiskit import QuantumCircuit
qc=QuantumCircuit(1)
qc.y(0)
qc.draw('mpl')
or
qc.draw()
┌───┐ q: ┤ Y ├ └───┘
5.7.2. Acting on a superposition state#
When Ygate is applied to a super position state the coefficient is swapped. That is
(5.7)#\[ Y \left (c_0 |0\rangle + c_1 |1\rangle\right) = -i (c_1 |0\rangle - c_0 |1\rangle) \]
If the global phase factor is omitted, we have
(5.8)#\[ Y \left (c_0 |0\rangle + c_1 |1\rangle\right) \simeq c_1 |0\rangle - c_0 |1\rangle \]
Like Xgate, the coefficients are swapped. In addition the relative phase of \(\pi\) is applied.
5.7.3. Important Properties#
\(Y^2 = I\)
This means that
\(Y^2\) does not do any thing on the qubit.
\(Y\) is self-inverse, that is \(Y^{-1} = Y\).
\(Y\) is self-adjoint (\(Y^\dagger = Y\)) since \(Y\) is unitary (\(Y^\dagger = Y^{-1}\)) by definition.
Exercise 5.7.1 Show that \(Y|+\rangle = -i |-\rangle\) and \(Y|-\rangle = i |+\rangle\). Apart from the phase factor, \(Y\) flips \(|\pm\rangle\).
Last modified: 08/31/2022