Ex5 4

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 5.4}$

Prove that the following are decompositions for some of the standard gates.

(1)\begin{align} I = K(0)T(0)R(0)T(0) \\ X = -\i T(\pi/2) R(\pi/2) T(0)\\ H = -\i T(\pi/2) R(\pi/4) T(0) \end{align}

page revision: 0, last edited: 18 Nov 2012 18:36

To show the decomposition of $I$, note that $K(0) = T(0) = R(0) = I$. Thus,

(1)To show the decomposition of $X$, note that

(2)Thus,

(3)To show the decomposition of $H$, note that $R(\pi/4) = \left(\begin{array}{cc} \tfrac{1}{\sqrt{2}} & \tfrac{1}{\sqrt{2}} \\ -\tfrac{1}{\sqrt{2}} & \tfrac{1}{\sqrt{2}} \end{array}\right)$. Thus,

(4)ReplyOptions