Quantum Gates

Define how group of qubits are affected.

Controlled-Not Gate (CNOT)

The CNOT gate acts on two qubits. One is the control qubit and the other is the target qubit.

$$ CNOT = \begin{bmatrix} 1 \ 0 \ 0 \ 0 \\ 0 \ 1 \ 0 \ 0 \\ 0 \ 0 \ 0 \ 1 \\ 0 \ 0\ 1\ 0 \\ \end{bmatrix} $$

$$ |a\rang |b\rang \rarr |a\rang |a\oplus b\rang $$

• $a$ is the control bit and $b$ is the target bit
• if the control bit is $0$, do nothing with the target bit
• if the control bit is $1$, flip the target bit.

Table:

• $|00\rang \rarr |00\rang$
• $|01\rang \rarr |01\rang$
• $|10\rang \rarr |11\rang$
• $|11\rang \rarr |10\rang$

Toffoli Gate (CCNOT)

$$ |a\rang |b\rang |c\rang \rarr|a\rang |b\rang |c\oplus ab\rang $$

• $a$ and $b$ are the control bits and $c$ is the target bit.
• if either control bit is 0, do nothing with the target bit
• if both control bits are 1, flip the target bit.