# Quantum gates

- I. Introduction
- II. Elements of quantum mechanics
- III. The electromagnetic field
- IV. The qubit
**V. Quantum gates**- VI. States and operators
- VII. Shor’s algorithm
- VIII. Quantum teleportation
- IX. The Deutsch algorithm
- X. The no-cloning theorem
- XI. About anyons
- XII. About SU(2)
- XIII. Teleportation topology
- XIV. The Simon algorithm
- XV. Superdense coding
- XVI. Chern-Simons theory
- XVII. The toric code
- REF. Some pointers

A unitary operator on a qubit is usually called *a (quantum) gate* and acts on the basis

\(|0\rangle = \begin{pmatrix}1\\0\end{pmatrix}\;|1\rangle = \begin{pmatrix}0\\1\end{pmatrix}.\)

The **NOT** gate \(N\) is defined as

\(N\,|0\rangle = |1\rangle,\; N\,|1\rangle = |0\rangle\)

and has matrix representation

\(N = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}\).

The \(N = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}\). operation \(\oplus\) corresponds to a \(\mathbb{Z}_2\) addition and hence

\(N\,|x\rangle = |x\oplus 1\rangle\).

The identity gate \(\mathbb{1}\) is

\(\mathbb{1} = \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\)

and **the Hadamard operator** is

\(H =\frac{1}{\sqrt{2}} \begin{pmatrix}1 & 1\\ 1 & -1\end{pmatrix}\). The **CNOT operator** and acts on two qubits as follows

\(CNOT\,|00\rangle = |00\rangle,\;CNOT\,|01\rangle = |01\rangle,\\CNOT\,|10\rangle = |11\rangle,\;CNOT\,|11\rangle = |10\rangle\)

meaning that the second qubit is flipped if the first one is 1. The matrix representation is

\(CNOT = \begin{pmatrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{pmatrix}\).

A state which cannot be written as a direct product of two states is called entangled and it means that the state should be considered as one. It effectively means that in reality such a system acts as one despite long distances and does not obbey special relativity. This does not break any fundamental theory because there is no information exchange. A two qubit system has four entangled states called **ebits**:

\(\frac{1}{\sqrt{2}}(\,|00\rangle+|11\rangle\,),\\\frac{1}{\sqrt{2}}(\,|00\rangle-|11\rangle\,),\\\frac{1}{\sqrt{2}}(\,|01\rangle+|10\rangle\,),\\\frac{1}{\sqrt{2}}(\,|01\rangle-|10\rangle\,).\)

The **Toffoli gate** (also known as the CCNOT gate) acts on three qubits and maps \(a, b, c\) to \(c\oplus ab\) with matrix representation

\(CCNOT = \begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\end{pmatrix}\).

It can be shown that the Toffoli gate is a universal reversible logic gate, which means that any reversible circuit can be constructed from Toffoli gates.