YGate

5.8. YGate#

This gate is not popular since it can be replaced with other common operators. Hence, this section only briefly covers its properties.

API References: YGate

5.8.1. 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

Hide code cell source

 
from qiskit import *
qc=QuantumCircuit(1)
qc.y(0)
qc.draw('mpl')
../_images/79bc0d1942d6bcdcf96fd5e4f2dacef4862cbb78de49fab5fd8ad3b29d670d14.png

or

Hide code cell source

 
qc.draw()

   ┌───┐
q: ┤ Y ├
   └───┘

5.8.2. 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.

5.8.3. Important Properties#

\(Y^2 = I\)

This means that

  1. \(Y^2\) does not do any thing on the qubit.

  2. \(Y\) is self-inverse, that is \(Y^{-1} = Y\).

  3. \(Y\) is self-adjoint (\(Y^\dagger = Y\)) since \(Y\) is unitary (\(Y^\dagger = Y^{-1}\)) by definition.

Exercise 5.8.1  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