applying-gates-to-zero-state
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Table of Contents
Applying gates to $\lvert 0\rangle$ (Zero state)
| Gate | Matrix form | Derivation |
|---|---|---|
| I gate | $I = \begin{pmatrix}1&0\\0&1\end{pmatrix}$ | $I\lvert 0\rangle = \begin{pmatrix}1&0\\0&1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}1\\0\end{pmatrix} = \lvert 0\rangle$ |
| X gate | $X = \begin{pmatrix}0&1\\1&0\end{pmatrix}$ | $X\lvert 0\rangle = \begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}0\\1\end{pmatrix} = \lvert 1\rangle$ |
| Y gate | $Y = \begin{pmatrix}0&-i\\i&0\end{pmatrix}$ | $Y\lvert 0\rangle = \begin{pmatrix}0&-i\\i&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}0\\i\end{pmatrix} = i\lvert 1\rangle$ |
| Z gate | $Z = \begin{pmatrix}1&0\\0&-1\end{pmatrix}$ | $Z\lvert 0\rangle = \begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}1\\0\end{pmatrix} = \lvert 0\rangle$ |
| Hadamard gate | $H = \tfrac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}$ | $H\lvert 0\rangle = \tfrac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \tfrac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix} = \lvert +\rangle$ |
| S gate | $S = \begin{pmatrix}1&0\\0&i\end{pmatrix}$ | $S\lvert 0\rangle = \begin{pmatrix}1&0\\0&i\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}1\\0\end{pmatrix} = \lvert 0\rangle$ |
| T gate | $T = \begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix}$ | $T\lvert 0\rangle = \begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}1\\0\end{pmatrix} = \lvert 0\rangle$ |
| $R_x(\theta)$ | $R_x(\theta) = \begin{pmatrix}\cos\tfrac{\theta}{2}&-i\sin\tfrac{\theta}{2}\\-i\sin\tfrac{\theta}{2}&\cos\tfrac{\theta}{2}\end{pmatrix}$ | $R_x(\theta)\lvert 0\rangle = \begin{pmatrix}\cos\tfrac{\theta}{2}&-i\sin\tfrac{\theta}{2}\\-i\sin\tfrac{\theta}{2}&\cos\tfrac{\theta}{2}\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}\cos\tfrac{\theta}{2}\\-i\sin\tfrac{\theta}{2}\end{pmatrix}$ |
| $R_y(\theta)$ | $R_y(\theta) = \begin{pmatrix}\cos\tfrac{\theta}{2}&-\sin\tfrac{\theta}{2}\\\sin\tfrac{\theta}{2}&\cos\tfrac{\theta}{2}\end{pmatrix}$ | $R_y(\theta)\lvert 0\rangle = \begin{pmatrix}\cos\tfrac{\theta}{2}&-\sin\tfrac{\theta}{2}\\\sin\tfrac{\theta}{2}&\cos\tfrac{\theta}{2}\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}\cos\tfrac{\theta}{2}\\\sin\tfrac{\theta}{2}\end{pmatrix}$ |
| $R_z(\theta)$ | $R_z(\theta) = \begin{pmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{pmatrix}$ | $R_z(\theta)\lvert 0\rangle = \begin{pmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = e^{-i\theta/2}\begin{pmatrix}1\\0\end{pmatrix}$ |
| $U(\theta,\phi,\lambda)$ | $U = \begin{pmatrix}\cos\tfrac{\theta}{2}&-e^{i\lambda}\sin\tfrac{\theta}{2}\\e^{i\phi}\sin\tfrac{\theta}{2}&e^{i(\phi+\lambda)}\cos\tfrac{\theta}{2}\end{pmatrix}$ | $U\lvert 0\rangle = \begin{pmatrix}\cos\tfrac{\theta}{2}&-e^{i\lambda}\sin\tfrac{\theta}{2}\\e^{i\phi}\sin\tfrac{\theta}{2}&e^{i(\phi+\lambda)}\cos\tfrac{\theta}{2}\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}\cos\tfrac{\theta}{2}\\e^{i\phi}\sin\tfrac{\theta}{2}\end{pmatrix}$ |
applying-gates-to-zero-state.1781302621.txt.gz ยท Last modified: by Ivan Janevski