Quantum Gates

Hadamard Gate (H)

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

Phase operation Gate (S) → $\theta =\pi /2$

$$S=\left[\begin{array}{cc}1& 0\\ 0& i\end{array}\right]$$

Phase operation Gate (T) → $\theta =\pi /4$

$$T=\left[\begin{array}{cc}1& 0\\ 0& \frac{1+i}{\sqrt{2}}\end{array}\right]$$

Controlled NOT Gate / CNOT

• Y is the control qubit, depicted by the smaller circle
• X is the target qubit, depicted by the larger circle with a + in it
• This is an XOR operation ( X $\oplus$ Y )

$$CNOT=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\end{array}\right]$$

Standard Basis Measurements Represented by a speedometer-like symbol

• A Toffoli gate is a controlled CNOT gate
• A Fredkin gate is a controlled SWAP gate