# Gate Model¶

## Requirements for the implementation¶

DiVincenzo [32] proposed five criteria necessary to successfully implement a quantum computer 1 3:

A scalable physical system with well characterized qubits

The ability to initialize the state of the qubits to a simple fiducial state, such as \(\ket{000 \ldots}\)

Long relevant decoherence times, much longer than the gate operation time

A “universal” set of quantum gates

A qubit-specific measurement capability

## Gates and circuits¶

Operations on qubits in a quantum computer are realized using so-called **gates**
acting on either only one qubit or on several of them.
A **quantum circuit** describes a sequence of gates applied to a set of qubits.

**Single-qubit gates** can be represented as \(2 \times 2\) matrices acting in
the qubit’s state space spanned by its basis vectors
\(\ket{0}\) and \(\ket{1}\).

Name |
Symbol |
Representation |

Hadamard |
\(\boxed{H}\) |
\(\frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\) |

Pauli-X |
\(\boxed{X}\) |
\(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) |

Pauli-Y |
\(\boxed{Y}\) |
\(\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\) |

Pauli-Z |
\(\boxed{Z}\) |
\(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) |

Phase |
\(\boxed{S}\) |
\(\begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}\) |

\(\pi / 8\) |
\(\boxed{T}\) |
\(\begin{bmatrix} 1 & 0 \\ 0 & e^{i \pi / 4} \end{bmatrix}\) |

The most common and useful **multiple-qubit gates** implement controlled operations,
where the state of one *control* qubit conditions the execution of an operation
on one or several *target* qubits.
The two most common are:

**CNOT**: controlled-NOT gate\[\begin{split}\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}\end{split}\]**Toffoli**gate: a “controlled-controlled-NOT” gate involving two control qubits and one target qubit. “It performs an X on the target only if both controls are in the state \(\ket{1}\). The final state of the target is then equal to either the AND or the NAND of the two controls, depending on whether the initial state of the target was \(\ket{0}\) or \(\ket{1}\).” 4 It is a universal reversible gate, which means that any*classical*reversible circuit can be constructed from Toffoli gates, as for example the AND gate as described previously.

## Universal set of gates¶

2 It can be proven that there exists sets of gates with which any arbitrary unitary operation can be approximated to arbitrary accuracy by a quantum circuit.

Examples are:

CNOT and single qubit unitaries in general

CNOT and Hadamard gate, phase gate, \(\pi / 8\) in particular

Nevertheless, not all unitary operations can be efficiently implemented.

Worth mentioning are also the *Clifford gates*:
quantum circuits that consist only of Clifford gates can be efficiently simulated with a classical computer,
see Clifford operators.