Quantum Gate

A Quantum gate represents a logic operation that can be applied on Qubits. It is always a Unitary Operator on a Quantum Hilbert Space, and as such is always reversible.

Quantum Gate Summary

The notation for quantum gates has been mostly standardised. Examples include the Pauli Matrixes, the Hadamard gate and the Phase gate.

$$\begin{array}{cccccccc}X& Y& Z& H& S& T& B& \\ \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)& \left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)& \left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)& \frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)& \left(\begin{array}{cc}1& 0\\ 0& i\end{array}\right)& \left(\begin{array}{cc}1& 0\\ 0& e^{\frac{i\pi }{4}}\end{array}\right)& \frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& i\\ i& 1\end{array}\right)& \end{array}$$

In the above we have:

• $X$ is the Bit flip aka NOT Operator
• $H$ is the Hadamard gate
• $S,T$ are Phase Flip operators, by $\frac{\pi }{2}$ and $\frac{\pi }{4}$, respectively
• $B$ is the Beam splitter gate, satisfying $B^2=X$.

Source: wikipedia

Source: IBM Research