Quantum Gate

A quantum gate is a function of a quantum computer. It can be expressed as a unital matrix:

X-gate

Not gate

$${\sigma }_X=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$ $$\begin{array}{rl}|0⟩& \mapsto |1⟩\\ |1⟩& \mapsto |0⟩\\ |+⟩& \mapsto |+⟩\\ |-⟩& \mapsto i|-⟩\\ |i⟩& \mapsto |i⟩\\ |-i⟩& \mapsto i|-i⟩\end{array}$$

Y-gate

$\frac{\pi }{4}$ rotation.

$${\sigma }_Y=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$$ $$\begin{array}{rl}|0⟩& \mapsto i|1⟩\\ |1⟩& \mapsto -i|0⟩\\ |+⟩& \mapsto |-⟩\\ |-⟩& \mapsto |+⟩\\ |i⟩& \mapsto |i⟩\\ |-i⟩& \mapsto i|-i⟩\end{array}$$

Z-gate

Reflection along the $|0⟩$ axis. — check

$${\sigma }_Z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$ $$\begin{array}{rl}|0⟩& \mapsto |0⟩\\ |1⟩& \mapsto -|1⟩\\ |+⟩& \mapsto |-⟩\\ |-⟩& \mapsto |+⟩\\ |i⟩& \mapsto |-i⟩\\ |-i⟩& \mapsto |i⟩\end{array}$$

Hadamard-gate

‘Rotation by $\frac{\pi }{4}$’
Takes superpositions into pure states and vice versa.

$${\sigma }_H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]$$

TODO: Check the superposition for +, shouldn’t it be a pure state?

$$\begin{array}{rl}|0⟩& \mapsto |+⟩\\ |1⟩& \mapsto -|-⟩\\ |+⟩& \mapsto |0⟩+|1⟩\\ |-⟩& \mapsto |0⟩-|1⟩\\ |i⟩& \mapsto |-i⟩\\ |-i⟩& \mapsto |i⟩\end{array}$$

Controlled Not (CNOT)

Negate B only if A is $|1⟩$ .

$${\sigma }_{\mathrm{CNOT}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]$$ \begin{aligned} \ket{ 0+ } \mapsto \ket{ 0+ } \\ \ket{ 0- } \mapsto \ket{ 0- } \\ \ket{ 1+ } \mapsto \ket{ 1+ } \\ \ket{ 1- } \mapsto -\ket{ 1- }\\ \end{aligned} $$ On a [[hadamard basis]], CNOT actually work in reverse, switching the A instead of B. See [[Phase Kickback]]. ### SWAP Swaps A and B

\sigma _{\mathrm{SWAP}}=\begin{bmatrix}
1 & 0 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 0 & 1
\end{bmatrix}

### General controlled gate

\sigma _{\mathrm{SWAP}}=\begin{bmatrix}
I & 0 \
0 & U
\end{bmatrix}

Where $U$ is an arbitrary gate in multiplevariables. Given two gates $A=(a_{i,j}); B=(b_{i,j})$ of sizes m and n, then the [[tensor product]]

A\otimes B=\begin{bmatrix}
a_{i,j} B
\end{bmatrix}

Essentially multiplying $B$ by the scalar $a_{i.j}$ for all $i,j$. Another way to thing about it is that it's creating a multiple dimension matrix, or a [[tensor]].