[NOTES/QCQI-04001] Single Qubit Quantum Gates

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Single Qubit Quantum Gates}

1Hadamard gate

The Hadamard gate can be represented by a matrix $$H = {1\over\sqrt{2}}\begin{pmatrix} 1 & 1\\ 1 & -1\end{pmatrix}$$ in computational bases the states $\ket{0}$ and $\ket{1}$ transform as $$H\ket{0} = {\ket{0}+\ket{1}\over\sqrt{2}} \qquad H\ket{1}={\ket{0}-\ket{1}\over\sqrt{2}}$$
Example For $\ket{\pm}= {\ket{0}\pm\ket{1}\over\sqrt{2}}$ $$H\ket{+}=\ket{0}~,~~H\ket{-}=\ket{1}$$ The action of Hadamard gate is unitary operation.The action of Hadamard gate is diagrammatically represented as \begin{align*} \ket{0} ~\text{------}\fbox{H}\text{------} &~{\ket{0}+\ket{1}\over\sqrt{2}}\\ \ket{1}~ \text{------}\fbox{H}\text{------} &~{\ket{0}-\ket{1}\over\sqrt{2}}\\ \text{or}~~~\ket{x} ~\text{------}\fbox{H}\text{------} &~ \frac{1}{\sqrt{2}}\{(-1)^x~ \ket{x}+\ket{1-x}\} \end{align*} where $\ket{x}$ can be $\ket{0}$ or $\ket{1}$.\\ Ref. Ekert, A. ``Basic Concepts in Quantum Computation'', arxiv-quant-pn/0011013v1, Nov. 2000.

2.Rotation about $X,Y,Z$ axes

X,Y,Z \longrightarrow~ &\text{Pauli~Matrices}\\ &X=\sigma_x\qquad Y=\sigma_y\qquad Z=\sigma_z\\ =&\begin{pmatrix}0&1\\ 1&0\end{pmatrix} = \begin{pmatrix} 0&-i\\ i & 0\end{pmatrix} = \begin{pmatrix} 1&0\\ 0&-1\end{pmatrix}. \end{align*} Operation of rotation about $X,Y,Z$ axes can be built in terms of Pauli matrices $$ R_x(\theta) = \exp\left(-i\alpha {X\over2}\right) = \begin{pmatrix} \cos(\alpha/2)& -i\sin(\alpha/2)\\ -i\sin(\alpha/2) & \cos(\alpha/2)\end{pmatrix} $$ performs a rotation about $X$ axis. $$ R_y(\theta)=\exp\left(-i\theta{Y\over2}\right) \quad\text{and}\quad R_z(\theta) = \exp\left(-i\theta{Z\over2}\right) $$ rotate the state of a qubit about $Y$ and $Z$ axes. The matrix $U(\phi)=\begin{pmatrix}1&0\\ 0 & e^{i\phi}\end{pmatrix}$ changes phase of $\ket{1}$ and leaves $\ket{0}$ unchanged $$ U(\phi)\ket{0}=\ket{0}\qquad U(\phi)\ket{1} = e^{i\phi}\ket{1}\,. $$ An arbitrary rotation about $\hat{n}=(n_1,n_2,n_3)$ by angle $\alpha$ can be written in terms of $R_x,R_y$ and $R_z$

Question1: Write an arbitrary rotation 
U(\hat{n},\alpha) =&\exp\left[-i\alpha\big( n_1X+n_2Y+n_3Z\big)\right]  in terms of rotations \(R_x, R_y, R_z\) about the coordinate axes. 

The phase gate $\phi$ changes the phase of a qubit.

The usefulness of $R_x,R_y,R_z$ and $ph$ lies in the fact that an arbitrary unitary operator \(V\) with determinant \(\det V=1\) has a representation in terms of rotations about the coordinate axes $$ V\equiv R_x(\alpha) R_y(\beta) R_z(\gamma). $$ An arbitrary unitary matrix can be written as a product of rotations about the coordinate axes and a phase transformation $$ U=R_z(\alpha)~ R_y(\beta)~ R_z(\gamma)~ ph(\delta).

$$ Transformation from $\ket{0}$ to an arbitrary state

A general state, $\theta,\phi$ on Bloch sphere can be obtained from $\ket{0}$ by a sequence of operations \begin{align*} \ket{0}~\text{------}\fbox{H}\text{------}~ &~{\ket{0}+\ket{1}\over\sqrt{2}}\\ {\ket{0}+\ket{1}\over\sqrt{2}}~\text{------}\fbox{$2\theta$}\text{------}~ &~{\ket{0}+e^{2i\theta}\ket{1}\over\sqrt{2}}\\ {e^{-i\theta}\ket{0}+e^{i\theta}\ket{1}\over\sqrt{2}}~\text{------}\fbox{H} \text{------}&~\frac{1}{\sqrt{2}}e^{-i\theta}\left\{\ket{0}+\ket{1}\right\} + \frac{1}{\sqrt{2}} e^{i\theta}\left\{\ket{0}-\ket{1}\right\}\\ =\cos\theta\ket{0}-&i\sin\theta\ket{1} \cos\theta\ket{0}-i\sin\theta\ket{1}~\text{------}\fbox{$\pi/2+\phi$}\text{------}~&~\cos\theta\ket{0}+\sin\theta e^{i\phi}\ket{1} \end{align*} \noindent Thus a single qubit $\ket{0}$ can be transformed into a general state $\ket{\theta,\phi}$ by using Hadamard and phase gates $$ \ket{0}~\text{------}\fbox{H}\overset{2\theta}{\text{------}}\fbox{H}\overset{\pi/2+\phi}{\text{---------}} ~ \cos\theta\ket{0}+\sin\theta e^{i\theta}\ket{0} $$ In general an $n$ qubit state $\ket{\psi_1}\ket{\psi_2}\cdot\ket{\psi_n}$, can be generated from $\ket{0}\ket{0}\cdots\ket{0}$ using Hadamard gates and phase gates.

Question 2
Verify the following identities: ph(\theta)\ket{0}&=e^{i\phi}\ket{0} ~~ph(\phi)\ket{1}=e^{i\phi}\ket{1}.