{ "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "cc7d159c", "metadata": {}, "source": [ "# Graph States" ] }, { "attachments": {}, "cell_type": "markdown", "id": "33b229e3", "metadata": {}, "source": [ "This notebook need the python modules qiskit, qutip and seaborn to be able to run. Simply run :\n", "\n", "pip install qiskit qutip seaborn" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a404cf6a", "metadata": {}, "source": [ "## Some definitions and properties of graph states" ] }, { "attachments": {}, "cell_type": "markdown", "id": "81150d0f", "metadata": {}, "source": [ "Graph states are specific entangled states that are represented by a graph. They have interesting properties in many fields of quantum computing [[1]](#References), therefore they are points of interest." ] }, { "attachments": {}, "cell_type": "markdown", "id": "ba744e0b", "metadata": {}, "source": [ "##### Definition:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3673c194", "metadata": {}, "source": [ "Two definitions of a graph state exist. Since they are equivalent, we will only consider the following:\n", "\n", "Given a graph $G=(V,E)$, with the set of vertices $V$ and the set of edges $E$, the corresponding graph state is defined as:\n", "
\n", "$\\left|G\\right\\rangle = \\prod_{(a,b)\\in E} CZ^{\\{a,b\\}} \\left|+\\right\\rangle^{\\otimes V}$\n", "
\n", "\n", "where $|+\\rangle = \\frac{|0\\rangle + |1\\rangle}{\\sqrt2}$ and $CZ^{\\{a,b\\}}$ is the controlled-Z interaction between the two vertices (corresponding to two qubits) $a$ and $b$. The operators order in the product doesn't matter since CZ gates commute between themselves. We can write the CZ gate with the following matrix :\n", "\n", "$CZ = \\begin{bmatrix}\n", "1 & 0 & 0 & 0 \\\\\n", "0 & 1 & 0 & 0 \\\\\n", "0 & 0 & 1 & 0 \\\\\n", "0 & 0 & 0 & -1 \\\\\n", "\\end{bmatrix}$\n", "
\n", "\n", "