In the vibrant landscape of quantum computing, quantum gates function as the bedrock that powers this revolutionary technology. Much like how classical gates are the building blocks in traditional computing, quantum gates operate on a similar principle but with a twist — they deal with qubits instead of bits.
These gates maneuver qubits into various states of superposition, leveraging the phenomena of quantum interference and entanglement to perform complex calculations at speeds unimaginable in classical systems. As we delve deeper into the intricacies of quantum gates such as Pauli-X, Hadamard, and CNOT, we open doors to a realm where computing meets the fantastical, promising advancements that could redefine the frontier of technology.
Pauli-X Gate
The Pauli-X gate flips a qubit from \(|0\rangle\) to \(|1\rangle\) and vice versa.
$$ X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$
Pauli-Y Gate
The Pauli-Y gate introduces both a bit-flip and a phase-flip to the qubit.
$$ Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$
Pauli-Z Gate
This is a phase-flip gate that changes the phase of the \(|1\rangle\) state by \( \pi \).
$$ Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$
Hadamard Gate
The Hadamard gate creates a superposition.
$$ H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} $$
S Gate (Phase Gate)
The S gate rotates the \(|1\rangle\) state by \( \frac{\pi}{2} \).
$$ S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} $$
T Gate
The T gate rotates the \(|1\rangle\) state by \( \frac{\pi}{4} \).
$$ T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix} $$
Controlled-NOT (CNOT) Gate
The CNOT gate flips the ‘target’ qubit if the ‘control’ qubit is \(|1\rangle\).
$$ \text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix} $$
Toffoli Gate (CCNOT)
The Toffoli gate flips the ‘target’ qubit if both ‘control’ qubits are \(|1\rangle\).
$$ \text{Toffoli} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{pmatrix} $$
SWAP Gate
The SWAP gate swaps two qubits \(|a\rangle\) and \(|b\rangle\).
$$ \text{SWAP} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
