QuTiP Majoranas, Mar 6, 2019.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "toc": true
   },
   "source": [
    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Mathjax-custom-macros\" data-toc-modified-id=\"Mathjax-custom-macros-1\" data-vivaldi-spatnav-clickable=\"1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Mathjax custom macros</a></span></li><li><span><a href=\"#Majorana-Fermions-With-QuTiP\" data-toc-modified-id=\"Majorana-Fermions-With-QuTiP-2\" data-vivaldi-spatnav-clickable=\"1\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Majorana Fermions With QuTiP</a></span><ul class=\"toc-item\"><li><span><a href=\"#Fermion-Ladder-Operators\" data-toc-modified-id=\"Fermion-Ladder-Operators-2.1\" data-vivaldi-spatnav-clickable=\"1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>Fermion Ladder Operators</a></span></li><li><span><a href=\"#Construct-Majorana-From-Fermions\" data-toc-modified-id=\"Construct-Majorana-From-Fermions-2.2\" data-vivaldi-spatnav-clickable=\"1\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>Construct Majorana From Fermions</a></span></li><li><span><a href=\"#Reconstruct-Fermion-From-Majorana\" data-toc-modified-id=\"Reconstruct-Fermion-From-Majorana-2.3\" data-vivaldi-spatnav-clickable=\"1\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>Reconstruct Fermion From Majorana</a></span></li><li><span><a href=\"#Collection-of-Majoranas\" data-toc-modified-id=\"Collection-of-Majoranas-2.4\" data-vivaldi-spatnav-clickable=\"1\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>Collection of Majoranas</a></span></li><li><span><a href=\"#Kitaev's-Construction\" data-toc-modified-id=\"Kitaev's-Construction-2.5\" data-vivaldi-spatnav-clickable=\"1\"><span class=\"toc-item-num\">2.5&nbsp;&nbsp;</span>Kitaev's Construction</a></span></li></ul></li></ul></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Mathjax custom macros\n",
    "\n",
    "$ \\newcommand{\\opexpect}[3]{\\langle #1 \\vert #2 \\vert #3 \\rangle} $\n",
    "$ \\newcommand{\\rarrow}{\\rightarrow} $\n",
    "$ \\newcommand{\\bra}{\\langle} $\n",
    "$ \\newcommand{\\ket}{\\rangle} $\n",
    "\n",
    "$ \\newcommand{\\up}{\\uparrow} $\n",
    "$ \\newcommand{\\down}{\\downarrow} $\n",
    "\n",
    "$ \\newcommand{\\mb}[1]{\\mathbf{#1}} $\n",
    "$ \\newcommand{\\mc}[1]{\\mathcal{#1}} $\n",
    "$ \\newcommand{\\mbb}[1]{\\mathbb{#1}} $\n",
    "$ \\newcommand{\\mf}[1]{\\mathfrak{#1}} $\n",
    "\n",
    "$ \\newcommand{\\vect}[1]{\\boldsymbol{\\mathrm{#1}}} $\n",
    "$ \\newcommand{\\expect}[1]{\\langle #1\\rangle} $\n",
    "\n",
    "$ \\newcommand{\\innerp}[2]{\\langle #1 \\vert #2 \\rangle} $\n",
    "$ \\newcommand{\\fullbra}[1]{\\langle #1 \\vert} $\n",
    "$ \\newcommand{\\fullket}[1]{\\vert #1 \\rangle} $\n",
    "$ \\newcommand{\\supersc}[1]{^{\\text{#1}}} $\n",
    "$ \\newcommand{\\subsc}[1]{_{\\text{#1}}} $\n",
    "$ \\newcommand{\\sltwoc}{SL(2,\\mathbb{C})} $\n",
    "$ \\newcommand{\\sltwoz}{SL(2,\\mathbb{Z})} $\n",
    "\n",
    "$ \\newcommand{\\utilde}[1]{\\underset{\\sim}{#1}} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib\n",
    "matplotlib.use('qt4agg')\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 247,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from qutip import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "metadata": {},
   "outputs": [],
   "source": [
    "from hubbard import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Majorana Fermions With QuTiP"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fermion Ladder Operators"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = destroy(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 250,
   "metadata": {},
   "outputs": [],
   "source": [
    "adag = create(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 251,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0\\\\0.0 & -1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 1.  0.]\n",
       " [ 0. -1.]]"
      ]
     },
     "execution_count": 251,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "commutator(a,adag)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 252,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0\\\\0.0 & 1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[1. 0.]\n",
       " [0. 1.]]"
      ]
     },
     "execution_count": 252,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "anticommutator(a,adag)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0\\\\0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0.]\n",
       " [0. 0.]]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "adag**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\n",
       " Qobj data =\n",
       " [[0. 1.]\n",
       "  [0. 0.]],\n",
       " Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False\n",
       " Qobj data =\n",
       " [[0. 0.]\n",
       "  [1. 0.]])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a, adag"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 1.0\\\\1.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 1.]\n",
       " [1. 0.]]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a + adag"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Construct Majorana From Fermions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given a fermionic operator $\\psi$ one can construct two majoranas from it:\n",
    "\\begin{align}\n",
    "    x & = \\psi + \\psi^\\dagger \\nonumber \\\\\n",
    "    y & = -i(\\psi - \\psi^\\dagger)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 1.0\\\\1.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 1.]\n",
       " [1. 0.]]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maj1 = a + adag\n",
    "maj1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can check that the majorana operator satisfies:\n",
    "$$ x^2 = 1 $$\n",
    "i.e. the majorana is its own anti-particle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0\\\\0.0 & 1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[1. 0.]\n",
       " [0. 1.]]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maj1**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "maj2 = (a - adag)/1j"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & -1.0j\\\\1.0j & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0.+0.j 0.-1.j]\n",
       " [0.+1.j 0.+0.j]]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maj2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0\\\\0.0 & 1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[1. 0.]\n",
       " [0. 1.]]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maj2**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check that two Majoranas anticommute:\n",
    "$$ \\{ x, y \\} = 0 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def anticommutator(X,Y):\n",
    "    return commutator(X,Y,kind='anti')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0\\\\0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0.]\n",
       " [0. 0.]]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "anticommutator(maj1,maj2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reconstruct Fermion From Majorana"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given two majoranas $x, y$, one can construct a single fermionic operator $\\psi$ from these:\n",
    "\\begin{align}\n",
    "    \\psi & = (x + iy)/2 \\nonumber \\\\\n",
    "    \\psi^\\dagger & = (x - iy)/2\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "heading_collapsed": true
   },
   "source": [
    "## Collection of Majoranas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "posOperatorN?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "n = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "majoranas = [posOperatorN(maj1, i = k, N = n) for k in range(n)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 0. 1. 0.]\n",
       "  [0. 0. 0. 1.]\n",
       "  [1. 0. 0. 0.]\n",
       "  [0. 1. 0. 0.]],\n",
       " Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 1. 0. 0.]\n",
       "  [1. 0. 0. 0.]\n",
       "  [0. 0. 0. 1.]\n",
       "  [0. 0. 1. 0.]]]"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "majoranas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0. 0. ... 0. 0. 0.]\n",
       " [0. 0. 0. ... 0. 0. 0.]\n",
       " [0. 0. 0. ... 0. 0. 0.]\n",
       " ...\n",
       " [0. 0. 0. ... 0. 0. 0.]\n",
       " [0. 0. 0. ... 0. 0. 0.]\n",
       " [0. 0. 0. ... 0. 0. 0.]]"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "anticommutator(majoranas[0],majoranas[4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 1), type = oper, isherm = False\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0\\\\0.0\\\\0.0\\\\0.0\\\\0.0\\\\\\vdots\\\\0.0\\\\0.0\\\\0.0\\\\0.0\\\\0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 1), type = oper, isherm = False\n",
       "Qobj data =\n",
       "[[0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]\n",
       " [0.]]"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zero_ket(32,dims=[[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "def zero_operatorN(N=2):\n",
    "    zeroket = qutip.zero_ket(2**N,dims=[[2]*N,[2]*N])\n",
    "    return zeroket * zeroket.dag()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0. 0. ... 0. 0. 0.]\n",
       " [0. 0. 0. ... 0. 0. 0.]\n",
       " [0. 0. 0. ... 0. 0. 0.]\n",
       " ...\n",
       " [0. 0. 0. ... 0. 0. 0.]\n",
       " [0. 0. 0. ... 0. 0. 0.]\n",
       " [0. 0. 0. ... 0. 0. 0.]]"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zero_operatorN(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isequal(anticommutator(majoranas[0],majoranas[4]),zero_operatorN(5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isequal(majoranas[0]**2,idOp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "commutator(majoranas[0],majoranas[4]) == zero_operatorN(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "idOp = tensor(identityList(N=n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "braidops = [(idOp + majoranas[k+1]*majoranas[k])/np.sqrt(2) for k in range(n-1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "braidopsinverse = [(idOp - majoranas[k+1]*majoranas[k])/np.sqrt(2) for k in range(n-1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.000 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 1.000 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 1.000 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.000 & 0.0 & \\cdots & 1.000 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 1.000 & \\cdots & 0.0 & 1.000 & 0.0 & 0.0 & 0.0\\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots\\\\0.0 & 0.0 & 0.0 & 1.000 & 0.0 & \\cdots & 1.000 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 1.000 & \\cdots & 0.0 & 1.000 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 1.000 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 1.000 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 1.000\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[1. 0. 0. ... 0. 0. 0.]\n",
       " [0. 1. 0. ... 0. 0. 0.]\n",
       " [0. 0. 1. ... 0. 0. 0.]\n",
       " ...\n",
       " [0. 0. 0. ... 1. 0. 0.]\n",
       " [0. 0. 0. ... 0. 1. 0.]\n",
       " [0. 0. 0. ... 0. 0. 1.]]"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "braidops[0]*braidops[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 1.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 1.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 1.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 1.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 1.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 1.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 1.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[1. 0. 0. ... 0. 0. 0.]\n",
       " [0. 1. 0. ... 0. 0. 0.]\n",
       " [0. 0. 1. ... 0. 0. 0.]\n",
       " ...\n",
       " [0. 0. 0. ... 1. 0. 0.]\n",
       " [0. 0. 0. ... 0. 1. 0.]\n",
       " [0. 0. 0. ... 0. 0. 1.]]"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "idOp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "qutip.isequal?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "hidden": true
   },
   "outputs": [],
   "source": [
    "majoranas2 = [tensor([*[identity(2)]*k,maj1,*[identity(2)]*(5-k-1)]) for k in range(5)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  ...\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]],\n",
       " Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  ...\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]],\n",
       " Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  ...\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]],\n",
       " Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 0. 1. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [1. 0. 0. ... 0. 0. 0.]\n",
       "  ...\n",
       "  [0. 0. 0. ... 0. 0. 1.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 1. 0. 0.]],\n",
       " Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 1. 0. ... 0. 0. 0.]\n",
       "  [1. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  ...\n",
       "  [0. 0. 0. ... 0. 0. 0.]\n",
       "  [0. 0. 0. ... 0. 0. 1.]\n",
       "  [0. 0. 0. ... 0. 1. 0.]]]"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "majoranas2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "isequal(anticommutator(majoranas2[0],majoranas2[4]), zero_operatorN(5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "hidden": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 1.]\n",
       "  [1. 0.]],\n",
       " Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 1.]\n",
       "  [1. 0.]],\n",
       " Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 1.]\n",
       "  [1. 0.]],\n",
       " Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[1. 0.]\n",
       "  [0. 1.]]]"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[*[maj1]*3,identity(2)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hidden": true
   },
   "source": [
    "**Conclusion:** Implement Majoranas using formalism in Kitaev2008Anyons paper, Sec 4.1 onwards."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kitaev's Construction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a system with $n$ fermionic operators: $ a_k, a^\\dagger_k; k \\in \\{1,\\ldots,n\\} $. From these fermionic operators we can construct $2n$ Majorana spinors:\n",
    "$$ c_{2k} = -i ( a_k - a^\\dagger_k); \\quad c_{2k-1} = a_k + a^\\dagger_k $$\n",
    "One can check that these operators satisfy the following relations:\n",
    "\\begin{align}\n",
    "    c_j = c^\\dagger_j \\quad & \\text{hermitian} \\nonumber \\\\\n",
    "    c^2_j = 1 \\quad & \\text{root of identity} \\nonumber \\\\\n",
    "    \\{c_j, c_l \\} = 0; \\text{if} j\\ne l ; \\quad & \\text{anticommuting}\n",
    "\\end{align}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 253,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Specify number of fermionic operators:\n",
    "\n",
    "N = 5\n",
    "\n",
    "# Generate array of fermionic creation and annihilation operators for N site system\n",
    "\n",
    "createOps = []\n",
    "destroyOps = []\n",
    "\n",
    "createOps = [posOperatorN(create(2), k, N) for k in range(N)]\n",
    "destroyOps = [posOperatorN(destroy(2), k, N) for k in range(N)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 254,
   "metadata": {},
   "outputs": [],
   "source": [
    "def majoranaOps(N = 1, dims = 2):\n",
    "    '''Creates 2N Majorana operators using Kitaev's construction\n",
    "    '''\n",
    "    \n",
    "    # Generate array of fermionic creation and annihilation operators for N site system\n",
    "\n",
    "    createOps = []\n",
    "    destroyOps = []\n",
    "\n",
    "    createOps = [posOperatorN(create(2), k, N) for k in range(N)]\n",
    "    destroyOps = [posOperatorN(destroy(2), k, N) for k in range(N)]    \n",
    "\n",
    "    majoranas = []\n",
    "    \n",
    "    for k in range(N):\n",
    "        majoranas.append((1j)*(destroyOps[k] - createOps[k]))\n",
    "        majoranas.append(destroyOps[k] + createOps[k])\n",
    "        \n",
    "    return majoranas\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 255,
   "metadata": {},
   "outputs": [],
   "source": [
    "majoranas = majoranaOps()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 256,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[1. 0.]\n",
      " [0. 1.]]\n",
      "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[1. 0.]\n",
      " [0. 1.]]\n"
     ]
    }
   ],
   "source": [
    "for i in majoranas:\n",
    "    print(i**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 257,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0\\\\0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0.]\n",
       " [0. 0.]]"
      ]
     },
     "execution_count": 257,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qtAntiCommutator(majoranas[0], majoranas[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 258,
   "metadata": {},
   "outputs": [],
   "source": [
    "majoranas4 = majoranaOps(N = 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 265,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0 & 1.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.0\\\\1.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 1.0 & 0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0. 1. 0.]\n",
       " [0. 0. 0. 1.]\n",
       " [1. 0. 0. 0.]\n",
       " [0. 1. 0. 0.]]"
      ]
     },
     "execution_count": 265,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "majoranas4[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[1. 0. 0. 0.]\n",
      " [0. 1. 0. 0.]\n",
      " [0. 0. 1. 0.]\n",
      " [0. 0. 0. 1.]]\n",
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[1. 0. 0. 0.]\n",
      " [0. 1. 0. 0.]\n",
      " [0. 0. 1. 0.]\n",
      " [0. 0. 0. 1.]]\n",
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[1. 0. 0. 0.]\n",
      " [0. 1. 0. 0.]\n",
      " [0. 0. 1. 0.]\n",
      " [0. 0. 0. 1.]]\n",
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[1. 0. 0. 0.]\n",
      " [0. 1. 0. 0.]\n",
      " [0. 0. 1. 0.]\n",
      " [0. 0. 0. 1.]]\n"
     ]
    }
   ],
   "source": [
    "for i in majoranas4:\n",
    "    print(i**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 266,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "0 1\n",
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[0. 0. 0. 0.]\n",
      " [0. 0. 0. 0.]\n",
      " [0. 0. 0. 0.]\n",
      " [0. 0. 0. 0.]]\n",
      "0 2\n",
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[ 0.  0.  0. -2.]\n",
      " [ 0.  0.  2.  0.]\n",
      " [ 0.  2.  0.  0.]\n",
      " [-2.  0.  0.  0.]]\n",
      "0 3\n",
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[0.+0.j 0.+0.j 0.+0.j 0.+2.j]\n",
      " [0.+0.j 0.+0.j 0.+2.j 0.+0.j]\n",
      " [0.+0.j 0.-2.j 0.+0.j 0.+0.j]\n",
      " [0.-2.j 0.+0.j 0.+0.j 0.+0.j]]\n",
      "True\n",
      "1 2\n",
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[0.+0.j 0.+0.j 0.+0.j 0.+2.j]\n",
      " [0.+0.j 0.+0.j 0.-2.j 0.+0.j]\n",
      " [0.+0.j 0.+2.j 0.+0.j 0.+0.j]\n",
      " [0.-2.j 0.+0.j 0.+0.j 0.+0.j]]\n",
      "1 3\n",
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[0. 0. 0. 2.]\n",
      " [0. 0. 2. 0.]\n",
      " [0. 2. 0. 0.]\n",
      " [2. 0. 0. 0.]]\n",
      "True\n",
      "2 3\n",
      "Quantum object: dims = [[2, 2], [2, 2]], shape = (4, 4), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[0. 0. 0. 0.]\n",
      " [0. 0. 0. 0.]\n",
      " [0. 0. 0. 0.]\n",
      " [0. 0. 0. 0.]]\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "for i in range(len(majoranas4)):\n",
    "    \n",
    "    print(isherm(majoranas4[i]))\n",
    "    \n",
    "    for j in range(i+1, len(majoranas4)):\n",
    "        print(i,j)\n",
    "        \n",
    "        print(qtAntiCommutator(majoranas4[i],majoranas4[j]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [],
   "source": [
    "for k in range(N):\n",
    "    majoranaOps.append((1j)*(destroyOps[k] - createOps[k]))\n",
    "    majoranaOps.append(destroyOps[k] + createOps[k])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(majoranaOps)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {},
   "outputs": [],
   "source": [
    "identityOp = identity([2]*N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "majoranaOps[5]**2 == identityOp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 1\n",
      "True\n",
      "0 2\n",
      "False\n",
      "0 3\n",
      "False\n",
      "0 4\n",
      "False\n",
      "0 5\n",
      "False\n",
      "0 6\n",
      "False\n",
      "0 7\n",
      "False\n",
      "0 8\n",
      "False\n",
      "0 9\n",
      "False\n",
      "1 2\n",
      "False\n",
      "1 3\n",
      "False\n",
      "1 4\n",
      "False\n",
      "1 5\n",
      "False\n",
      "1 6\n",
      "False\n",
      "1 7\n",
      "False\n",
      "1 8\n",
      "False\n",
      "1 9\n",
      "False\n",
      "2 3\n",
      "True\n",
      "2 4\n",
      "False\n",
      "2 5\n",
      "False\n",
      "2 6\n",
      "False\n",
      "2 7\n",
      "False\n",
      "2 8\n",
      "False\n",
      "2 9\n",
      "False\n",
      "3 4\n",
      "False\n",
      "3 5\n",
      "False\n",
      "3 6\n",
      "False\n",
      "3 7\n",
      "False\n",
      "3 8\n",
      "False\n",
      "3 9\n",
      "False\n",
      "4 5\n",
      "True\n",
      "4 6\n",
      "False\n",
      "4 7\n",
      "False\n",
      "4 8\n",
      "False\n",
      "4 9\n",
      "False\n",
      "5 6\n",
      "False\n",
      "5 7\n",
      "False\n",
      "5 8\n",
      "False\n",
      "5 9\n",
      "False\n",
      "6 7\n",
      "True\n",
      "6 8\n",
      "False\n",
      "6 9\n",
      "False\n",
      "7 8\n",
      "False\n",
      "7 9\n",
      "False\n",
      "8 9\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "for j in range(2*N):\n",
    "    for k in range(j+1,2*N):\n",
    "        print(j,k)\n",
    "        print(anticommutator(majoranaOps[j],majoranaOps[k]) == zero_operatorN(5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 1.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 1.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 1.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 1.0 & 0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 1.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 1.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 1.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \\cdots & 0.0 & 0.0 & 0.0 & 0.0 & 1.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2, 2, 2, 2, 2], [2, 2, 2, 2, 2]], shape = (32, 32), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[1. 0. 0. ... 0. 0. 0.]\n",
       " [0. 1. 0. ... 0. 0. 0.]\n",
       " [0. 0. 1. ... 0. 0. 0.]\n",
       " ...\n",
       " [0. 0. 0. ... 1. 0. 0.]\n",
       " [0. 0. 0. ... 0. 1. 0.]\n",
       " [0. 0. 0. ... 0. 0. 1.]]"
      ]
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "identity([2]*5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {},
   "outputs": [
    {
     "ename": "AttributeError",
     "evalue": "'numpy.ndarray' object has no attribute 'full'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-198-f957e706d0cf>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mallclose\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0manticommutator\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmajoranaOps\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfull\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmajoranaOps\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfull\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfull\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mzeros\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m32\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m32\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAttributeError\u001b[0m: 'numpy.ndarray' object has no attribute 'full'"
     ]
    }
   ],
   "source": [
    "np.allclose(anticommutator(majoranaOps[0].full(), majoranaOps[1].full()).full(), np.zeros([32,32]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.+0.j, 0.+0.j, 0.+0.j, ..., 0.+0.j, 0.+0.j, 0.+0.j],\n",
       "       [0.+0.j, 0.+0.j, 0.+0.j, ..., 0.+0.j, 0.+0.j, 0.+0.j],\n",
       "       [0.+0.j, 0.+0.j, 0.+0.j, ..., 0.+0.j, 0.+0.j, 0.+0.j],\n",
       "       ...,\n",
       "       [0.+0.j, 0.+0.j, 0.+0.j, ..., 0.+0.j, 0.+0.j, 0.+0.j],\n",
       "       [0.+0.j, 0.+0.j, 0.+0.j, ..., 0.+0.j, 0.+0.j, 0.+0.j],\n",
       "       [0.+0.j, 0.+0.j, 0.+0.j, ..., 0.+0.j, 0.+0.j, 0.+0.j]])"
      ]
     },
     "execution_count": 197,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "anticommutator(majoranaOps[0].full(), majoranaOps[1].full())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {},
   "outputs": [],
   "source": [
    "commutator?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [],
   "source": [
    "def qtAntiCommutator(X,Y):\n",
    "    if type(X) == Qobj and type(Y) == Qobj:\n",
    "        if len(X.shape) == 2 and len(Y.shape) == 2:\n",
    "            if X.shape[1] == Y.shape[0]:\n",
    "                return qutip.commutator(X,Y,kind='anti')\n",
    "            else:\n",
    "                raise ValueError(\"Number of columns of first argument must be equal to number of rows of second argument\")\n",
    "        else:\n",
    "            raise ValueError(\"Arguements must be QuTiP arrays of shape M x N\")\n",
    "    else:\n",
    "        raise ValueError(\"Arguments must be QuTiP arrays of compatible dimensions\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 2)"
      ]
     },
     "execution_count": 217,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maj1.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0\\\\0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0.]\n",
       " [0. 0.]]"
      ]
     },
     "execution_count": 220,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qtAntiCommutator(maj1,maj2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "metadata": {},
   "outputs": [],
   "source": [
    "def npAntiCommutator(X,Y):\n",
    "    if type(X) == np.ndarray and type(Y) == np.ndarray:\n",
    "        if len(X.shape) == 2 and len(Y.shape) == 2:\n",
    "            if X.shape[1] == Y.shape[0]:\n",
    "                return X*Y + Y*X\n",
    "            else:\n",
    "                raise ValueError(\"Number of columns of first argument must be equal to number of rows of second argument\")\n",
    "        else:\n",
    "            raise ValueError(\"Arguements must be numpy arrays of shape M x N\")\n",
    "    else:\n",
    "        raise ValueError(\"Arguments must be numpy arrays of compatible dimensions\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.+0.j, 0.-2.j],\n",
       "       [0.+2.j, 0.+0.j]])"
      ]
     },
     "execution_count": 227,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "npAntiCommutator(maj1.full(),maj2.full())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.ndarray"
      ]
     },
     "execution_count": 210,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(maj2.full())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.+0.j, 0.-1.j],\n",
       "       [0.+1.j, 0.+0.j]])"
      ]
     },
     "execution_count": 221,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr = maj2.full()\n",
    "arr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 2)"
      ]
     },
     "execution_count": 222,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "qutip.qobj.Qobj"
      ]
     },
     "execution_count": 204,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(maj1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "qutip.qobj.Qobj"
      ]
     },
     "execution_count": 207,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maj1.__class__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "metadata": {},
   "outputs": [],
   "source": [
    "majoranaOps = []"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "metadata": {},
   "outputs": [],
   "source": [
    "majoranaOps.append((-1j)*(destroy(2) - create(2)))\n",
    "majoranaOps.append((destroy(2) + create(2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0.+0.j 0.-1.j]\n",
       "  [0.+1.j 0.+0.j]],\n",
       " Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       " Qobj data =\n",
       " [[0. 1.]\n",
       "  [1. 0.]]]"
      ]
     },
     "execution_count": 232,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "majoranaOps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[1. 0.]\n",
      " [0. 1.]]\n",
      "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
      "Qobj data =\n",
      "[[1. 0.]\n",
      " [0. 1.]]\n"
     ]
    }
   ],
   "source": [
    "for i in majoranaOps:\n",
    "    print(i**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}0.0 & 0.0\\\\0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[0. 0.]\n",
       " [0. 0.]]"
      ]
     },
     "execution_count": 235,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qtAntiCommutator(majoranaOps[0],majoranaOps[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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