ToolBox.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "codetobasis (generic function with 2 methods)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function codetobasis(a::Int64,L::Int64,base=4) #convert a state code a to basis, e.g. 1=0000001\n",
    "    basis=zeros(Int64,L);\n",
    "    temp=a;\n",
    "    for i=1:L\n",
    "        basis[i]=rem(temp,base);\n",
    "        temp=div(temp,base);\n",
    "    end\n",
    "    return basis;\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "num2basis (generic function with 2 methods)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using SparseArrays\n",
    "num2basis(a::Int64,L::Int64,base=4)=codetobasis(a,L,base)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "basistocode (generic function with 2 methods)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#basis to number converter\n",
    "function basistocode(basis::Array{Int64},base=4)\n",
    "    temp=BigInt(0);\n",
    "    for i=1:length(basis)\n",
    "        temp+=basis[i]*big(base)^(i-1);\n",
    "    end\n",
    "    return temp;\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "basis2num (generic function with 2 methods)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "basis2num(basis::Array{Int64},base=4)=basistocode(basis,base)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Pauli_Algebra (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Pauli_Algebra(x::Int64, y::Int64)\n",
    "           if x == 0 && y == 0 || x == 1 && y == 1 || x == 2 && y == 2 || x == 3 && y == 3\n",
    "               c, w = 1, 0\n",
    "           elseif x == 2 && y == 3\n",
    "               c, w = 1im, 1\n",
    "           elseif x == 3 && y == 1\n",
    "               c, w = 1im, 2\n",
    "           elseif x == 1 && y == 2\n",
    "               c, w = 1im, 3\n",
    "           elseif x == 3 && y == 2 \n",
    "               c, w = -1im, 1\n",
    "           elseif x == 1 && y == 3 \n",
    "               c, w = -1im, 2\n",
    "           elseif x == 2 && y == 1 \n",
    "               c, w = -1im, 3\n",
    "           elseif x == 0 && y == 1 || x == 1 && y == 0 \n",
    "               c, w = 1, 1\n",
    "           elseif x == 0 && y == 2 || x == 2 && y == 0 \n",
    "               c, w = 1, 2\n",
    "           elseif x == 0 && y == 3 || x == 3 && y == 0 \n",
    "               c, w = 1, 3\n",
    "           end\n",
    "            return c, w\n",
    "       end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PauliAlg (generic function with 1 method)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PauliAlg(x::Int64, y::Int64) = Pauli_Algebra(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Initial_Operator (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Initial_Operator(c::Complex{Float64},a::Int64,L::Int64)\n",
    "    z = convert(Array{Int64},zeros(L))\n",
    "    z[convert(Int64,ceil(L/2))] = a\n",
    "     return (c,z)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "IoP (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function IoP(a,L)\n",
    "    z = convert(Array{Int64},zeros(L))\n",
    "    z[convert(Int64,ceil(L/2))] = a\n",
    "     return z\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "InitialOp (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "InitialOp(c,a::Int64,L::Int64) = Initial_Operator(c,a,L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Pauli_Commutator (generic function with 1 method)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Pauli_Commutator(x::Int64, y::Int64)\n",
    "           if  x == 1 && y == 1 || x == 2 && y == 2 || x == 3 && y == 3\n",
    "               return 0 ,0\n",
    "           elseif x == 0 || y == 0\n",
    "               return 0 ,0\n",
    "           elseif x == 1 && y == 2 \n",
    "               return  2im, 3\n",
    "           elseif x == 1 && y == 3 \n",
    "               return  -2im, 2\n",
    "           elseif x == 2 && y == 1 \n",
    "               return  -2im, 3\n",
    "           elseif x == 2 && y == 3 \n",
    "               return  2im, 1\n",
    "           elseif x == 3 && y == 1 \n",
    "               return  2im, 2\n",
    "           elseif x == 3 && y == 2 \n",
    "               return  -2im, 1\n",
    "           end\n",
    "            return c, w\n",
    "       end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PauliComm (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PauliComm(x::Int64, y::Int64) = Pauli_Commutator(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Pauli_Algebra2 (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Pauli_Algebra2(x::Int64, y::Int64) #This product is blind to coefficients and signs\n",
    "           if  x == 1 && y == 1 || x == 2 && y == 2 || x == 3 && y == 3\n",
    "               return 0\n",
    "           elseif x == 0 && y == 0\n",
    "               return 0\n",
    "           elseif x == 0 && y != 0\n",
    "                return y\n",
    "           elseif x != 0 && y == 0\n",
    "                return x\n",
    "           elseif x == 1 && y == 2 \n",
    "               return  3\n",
    "           elseif x == 1 && y == 3 \n",
    "               return  2\n",
    "           elseif x == 2 && y == 1 \n",
    "               return  3\n",
    "           elseif x == 2 && y == 3 \n",
    "               return  1\n",
    "           elseif x == 3 && y == 1 \n",
    "               return  2\n",
    "           elseif x == 3 && y == 2 \n",
    "               return  1\n",
    "           end\n",
    "            return w\n",
    "       end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PC2 (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PC2(x::Int64, y::Int64) = Pauli_Algebra2(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Commutator (generic function with 1 method)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Commutator(A::Array{Int64,1},B::Array{Int64,1},L::Int64) #Commutation of Pauli (Basis)\n",
    "    cm = convert(Array{Int64},zeros(L))\n",
    "    zl = 1\n",
    "    zr = 1\n",
    "    for i = 1:L\n",
    "        cm[i] += PauliAlg(A[i],B[i])[2] \n",
    "    end\n",
    "    for i = 1:L\n",
    "        zl = zl * PauliAlg(A[i],B[i])[1] \n",
    "        zr = zr * PauliAlg(B[i],A[i])[1]\n",
    "    end \n",
    "    z = zl - zr \n",
    "    if z == 0\n",
    "        return (convert(Complex{Float64},0),cm)\n",
    "    else\n",
    "        return (convert(Complex{Float64},z), cm)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Comm (generic function with 1 method)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Comm(A::Array{Int64,1},B::Array{Int64,1},L::Int64)= Commutator(A,B,L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Interaction_Hamiltonian (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Interaction_Hamiltonian(j::Int64,A::Int64,B::Int64,L::Int64)\n",
    "    h = convert(Array{Int64},zeros(L))\n",
    "    h[j] = A\n",
    "    h[j+1] = B\n",
    "    return h\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LocalField_Hamiltonian (generic function with 1 method)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function LocalField_Hamiltonian(j::Int64,A::Int64,L::Int64)\n",
    "    h = convert(Array{Int64},zeros(L))\n",
    "    h[j] = A\n",
    "    return h\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "IntHam (generic function with 1 method)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IntHam(j::Int64,A::Int64,B::Int64,L::Int64) = Interaction_Hamiltonian(j,A,B,L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "#I will define as code the map from paulis to a number by means of the power of 4 sum. \n",
    "#I will define as index the corresponding Julia entry for the array In this very trivial case that means\n",
    "# Index = code + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index_to_Pauli (generic function with 1 method)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Index_to_Pauli(a::Int64,L::Int64) #Thisfunctionsendscoeffentrytotheassocpaulistring a>=1\n",
    "    if a >= 1\n",
    "        z = a-1\n",
    "        p = codetobasis(z,L)\n",
    "    else\n",
    "        println(\"Error!, index starts at 1 \")\n",
    "    end\n",
    "    \n",
    "    return p\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index2Pauli (generic function with 1 method)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Index2Pauli(a::Int64,L::Int64) = Index_to_Pauli(a,L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Pauli_to_Index (generic function with 1 method)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Pauli_to_Index(A::Array{Int64})\n",
    "    z = BigInt(basis2num(A)) + 1\n",
    "    return z\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Pauli2Index (generic function with 1 method)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pauli2Index(A::Array{Int64}) = Pauli_to_Index(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Now I want to make a function that I give it a coefficient and a Pauli and in returns\n",
    "#a coefficient and a index"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Coef_and_Pauli_to_Index (generic function with 1 method)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Coef_and_Pauli_to_Index(V::Tuple{Complex{Float64},Array{Int64,1}},L::Int64)\n",
    "    cf = V[1]\n",
    "    index = Pauli2Index(V[2])\n",
    "    return (cf,index)\n",
    "end "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CnP2I (generic function with 1 method)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CnP2I(V::Tuple{Complex{Float64},Array{Int64,1}},L::Int64) = Coef_and_Pauli_to_Index(V,L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Reminder: Index to pauli needs L.  Pauli to Index doesnt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Coef_and_Index_to_Pauli (generic function with 1 method)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Coef_and_Index_to_Pauli(a::Complex{Float64},b::Int64,L::Int64)\n",
    "    c = a\n",
    "    z = Index2Pauli(b,L)\n",
    "    \n",
    "    return (c,z)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CnI2P (generic function with 1 method)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CnI2P(a::Complex{Int64},b::Int64,L::Int64) = Coef_and_Index_to_Pauli(a,b,L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Hamiltonian (generic function with 1 method)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Hamiltonian(i::Int64,c,A::Int64,B::Int64,L::Int64)\n",
    "    coef = c\n",
    "    h = convert(Array{Int64},zeros(L))\n",
    "    h[i] = A\n",
    "    h[i+1] = B\n",
    "    \n",
    "    return (coef, h)\n",
    "end  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "HamP (generic function with 1 method)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function HamP(c::Int64,A::Int64,B::Int64)\n",
    "    \n",
    "    return (c,A,B)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Liouvillian (generic function with 1 method)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Liouvillian(H::Tuple{Int64,Int64,Int64},Op::Tuple{Complex{Float64},Array{Int64,1}},L::Int64)\n",
    "    coefvec = convert(Array{Complex{Float64}},ones(L-1))\n",
    "    v = [Vector{Int64}() for i in 1:L-1]\n",
    "    for i=1:L-1\n",
    "        v[i] = Comm(Hamiltonian(i,H[1],H[2],H[3],L)[2],Op[2],L)[2]\n",
    "    end\n",
    "    for i=1:L-1\n",
    "        coefvec[i] = Hamiltonian(i,H[1],H[2],H[3],L)[1] * Op[1] * Comm(Hamiltonian(i,H[1],H[2],H[3],L)[2],Op[2],L)[1]\n",
    "    end\n",
    "    \n",
    "    return (coefvec, v)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Important:Some Pauli strings will show up that are associated to a 0 coefficient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LiouvillianIndexForm (generic function with 1 method)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function LiouvillianIndexForm(H::Tuple{Int64,Int64,Int64},Op::Tuple{Complex{Float64},Array{Int64,1}},L::Int64)\n",
    "    coefvec = convert(Array{Complex{Float64}},ones(L-1))\n",
    "    v = [Vector{Int64}() for i in 1:L-1]\n",
    "    VI = convert(Array{Int64},zeros(L-1))\n",
    "    for i=1:L-1\n",
    "        v[i] = Comm(Hamiltonian(i,H[1],H[2],H[3],L)[2],Op[2],L)[2]\n",
    "    end\n",
    "    for i=1:L-1\n",
    "        coefvec[i] = Hamiltonian(i,H[1],H[2],H[3],L)[1] * Op[1] * Comm(Hamiltonian(i,H[1],H[2],H[3],L)[2],Op[2],L)[1]\n",
    "    end\n",
    "    for i=1:L-1\n",
    "        VI[i] = Pauli2Index(v[i])\n",
    "    end\n",
    "        \n",
    "    return (coefvec,VI)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LiouvillianIndexForm4Lvector (generic function with 1 method)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function LiouvillianIndexForm4Lvector(H::Tuple{Int64,Int64,Int64},Op::Tuple{Complex{Float64},Array{Int64,1}},L::Int64)\n",
    "    coefvec = convert(Array{Complex{Float64}},ones(L-1))\n",
    "    v = [Vector{Int64}() for i in 1:L-1]\n",
    "    VI = convert(Array{Int64},zeros(L-1))\n",
    "    FV = convert(Array{Complex{Float64}},zeros(4^L))\n",
    "    for i=1:L-1\n",
    "        v[i] = Comm(Hamiltonian(i,H[1],H[2],H[3],L)[2],Op[2],L)[2]\n",
    "    end\n",
    "    for i=1:L-1\n",
    "        coefvec[i] = Hamiltonian(i,H[1],H[2],H[3],L)[1] * Op[1] * Comm(Hamiltonian(i,H[1],H[2],H[3],L)[2],Op[2],L)[1]\n",
    "    end\n",
    "    for i=1:L-1\n",
    "        VI[i] = Pauli2Index(v[i])\n",
    "    end\n",
    "    for i=1:L-1\n",
    "        FV[VI[i]] = coefvec[i]\n",
    "    end\n",
    " \n",
    "    \n",
    "    return sparse(FV)\n",
    "            \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CoefandPauli_to_IndexVector (generic function with 1 method)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function CoefandPauli_to_IndexVector(A::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},L::Int64)\n",
    "    k = convert(Array{BigInt,1},zeros(length(A[1]))) \n",
    "    for i = 1:length(A[1])\n",
    "        k[i] = Pauli2Index(A[2][i])\n",
    "    end\n",
    "    return sparsevec(k,A[1],4^L,+)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "IndexVector_to_CoefandPauli (generic function with 1 method)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Here we aim to write a function that takes a 4^L sized vector which is actually our index vector\n",
    "#And spits out coeficients and associated Pauli string IndexVector_to_CoefandPauli\n",
    "function IndexVector_to_CoefandPauli(V::SparseVector{Complex{Float64},Int64},L::Int64)\n",
    "    k = findnz(V)[2]\n",
    "    w = [Vector{Int64}() for i in 1:nnz(V)]\n",
    "\n",
    "    for i =1:nnz(V)\n",
    "        w[i] = Index2Pauli(findnz(V)[1][i],L)\n",
    "    end\n",
    "    return (k,w)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LiouvillianE (generic function with 1 method)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function LiouvillianE(H::Tuple{Int64,Int64,Int64},Op::Tuple{Complex{Float64},Array{Int64,1}},L::Int64)\n",
    "    coefvec = convert(Array{Complex{Float64}},ones(L-1))\n",
    "    v = [Vector{Int64}() for i in 1:L-1]\n",
    "    VI = convert(Array{Int64},zeros(L-1))\n",
    "    FV = convert(Array{Complex{Float64}},zeros(4^L))\n",
    "    for i=1:L-1\n",
    "        v[i] = Comm(Hamiltonian(i,H[1],H[2],H[3],L)[2],Op[2],L)[2]\n",
    "    end\n",
    "    for i=1:L-1\n",
    "        coefvec[i] = Hamiltonian(i,H[1],H[2],H[3],L)[1] * Op[1] * Comm(Hamiltonian(i,H[1],H[2],H[3],L)[2],Op[2],L)[1]\n",
    "    end\n",
    "    for i=1:L-1\n",
    "        VI[i] = Pauli2Index(v[i])\n",
    "    end\n",
    "    for i=1:L-1\n",
    "        FV[VI[i]] = coefvec[i]\n",
    "    end\n",
    " \n",
    "    return IndexVector_to_CoefandPauli(sparse(FV),L)\n",
    "            \n",
    "end\n",
    "#This function is the same as Liouvillian but doesn't give ugly 0 coefficient terms!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TupleCommu (generic function with 1 method)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function TupleCommu(A::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},B::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},L::Int64)\n",
    "    k = Complex{Float64}[]\n",
    "    w = [Vector{Int64}() for i in 1:length(A[1])*length(B[1])]\n",
    "    x = 0\n",
    "    for i =1:length(A[1])\n",
    "        for j =1:length(B[1])\n",
    "            if A[1][i] * B[1][j] * Comm(A[2][i],B[2][j],L)[1] != 0\n",
    "                x+=1\n",
    "                append!( k, A[1][i] * B[1][j] * Comm(A[2][i],B[2][j],L)[1] )\n",
    "                w[x] = Comm(A[2][i],B[2][j],L)[2]\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    return CoefandPauli_to_IndexVector((k,w[1:length(k)]),TT[2])\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "#function TupleCommu(A::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},B::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},L::Int64)\n",
    " #   k = convert(Array{Complex{Float64}},ones(length(A[1])*length(B[1])))\n",
    "  #  w = [Vector{Int64}() for i in 1:length(A[1])*length(B[1])]\n",
    "   # x = 0\n",
    "    #for i =1:length(A[1])\n",
    "     #   for j =1:length(B[1])\n",
    "      #      x+=1\n",
    "       #     k[x] = A[1][i] * B[1][j] * Comm(A[2][i],B[2][j],L)[1]\n",
    "        #    w[x] = Comm(A[2][i],B[2][j],L)[2]\n",
    "        #end\n",
    "    #end\n",
    "    #return (k,w)\n",
    "    #thisone original return: return CoefandPauli_to_IndexVector((k,w),L)\n",
    "#end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Now before writing the Lanczos we need the Liouvillian for the XXX Hamiltonian\n",
    "#Notice this Liouvillian is different from previous since it must me able \n",
    "#to receive a N-TuplePauli \n",
    "#Once we have this we can easily write the Lanczos algorithm for n-iterations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Function to create a disorder vector of length L with random variables in the [-h,h] box"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "disorderField (generic function with 1 method)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function disorderField(h,L::Int64)\n",
    "    V = zeros(L)\n",
    "    for i=1:L\n",
    "        V[i] = rand(Uniform(-h, h))\n",
    "    end\n",
    "    return convert(Array{Complex{Float64}},V)\n",
    "end"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "HamiltonInt (generic function with 1 method)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function HamiltonInt(C::Array{Complex{Float64}},P::Int64,L::Int64)\n",
    "    w = [Vector{Int64}() for i in 1:L-1]\n",
    "    for i=1:L-1\n",
    "        w[i] = Interaction_Hamiltonian(i,P,P,L)\n",
    "    end\n",
    "    return (C,w)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "HamiltonFloat (generic function with 2 methods)"
      ]
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function HamiltonFloat(C::Array{Complex{Float64},1},P::Int64,L::Int64)\n",
    "    w = [Vector{Int64}() for i in 1:L]\n",
    "    for i=1:L\n",
    "        w[i] = LocalField_Hamiltonian(i,P,L)\n",
    "    end\n",
    "    return (C,w)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Liouville2 (generic function with 1 method)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Liouville2(H::Tuple{Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}}},O::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},L::Int64)\n",
    "    W = TupleCommu(H[1],O,L) + TupleCommu(H[2],O,L) + TupleCommu(H[3],O,L) \n",
    "\n",
    "    return W\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Liouville (generic function with 1 method)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Liouville(H::NTuple{4,Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}}},O::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},L::Int64)\n",
    "    W = TupleCommu(H[1],O,L) + TupleCommu(H[2],O,L) + TupleCommu(H[3],O,L) - TupleCommu(H[4],O,L)\n",
    "\n",
    "    return W\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "DisorderMatrix (generic function with 2 methods)"
      ]
     },
     "execution_count": 201,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function DisorderMatrix(h_dis,L,m)\n",
    "    A = Complex.(zeros(L,m))\n",
    "    for i = 1:m\n",
    "        A[:,i] = disorderField(h_dis,L)\n",
    "    end\n",
    "    return A\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10×5 Array{Complex{Float64},2}:\n",
       " -6.16381+0.0im  -2.34731+0.0im  …    2.6376+0.0im  -4.88753+0.0im\n",
       " -5.38882+0.0im   1.74386+0.0im      4.99162+0.0im  -2.14892+0.0im\n",
       "  7.69218+0.0im    2.8811+0.0im     -7.35007+0.0im  -7.38761+0.0im\n",
       " -9.58959+0.0im  -5.24401+0.0im     -8.37444+0.0im   6.36394+0.0im\n",
       "  11.8692+0.0im   4.94122+0.0im     -1.20038+0.0im   -4.0335+0.0im\n",
       " -5.02774+0.0im  -4.73679+0.0im  …  0.960661+0.0im  -3.26596+0.0im\n",
       " -5.85263+0.0im   4.91254+0.0im      1.65719+0.0im  -4.00009+0.0im\n",
       " -5.18658+0.0im    11.273+0.0im     -5.34343+0.0im  -1.72527+0.0im\n",
       "  11.8706+0.0im  -5.09477+0.0im     -7.51711+0.0im  -1.47073+0.0im\n",
       "  -9.4534+0.0im  -11.8412+0.0im     -1.41343+0.0im   -11.915+0.0im"
      ]
     },
     "execution_count": 202,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "DisorderMatrix(12,10,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = 12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "19"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LT = 19"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_iter = convert(Int64,(LT - 1)/2) + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [],
   "source": [
    "Oj = (Complex{Float64}[1 + 0im], [IoP(1,LT)]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [],
   "source": [
    "H_X = HamiltonInt(convert(Array{Complex{Float64}},ones(LT-1)),1,LT);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [],
   "source": [
    "H_Y = HamiltonInt(convert(Array{Complex{Float64}},ones(LT-1)),2,LT);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [],
   "source": [
    "H_Z = HamiltonInt(convert(Array{Complex{Float64}},ones(LT-1)),3,LT);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "TT = (N_iter,LT);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [],
   "source": [
    "AH = (H_X,H_Y,H_Z);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [],
   "source": [
    "H_F = HamiltonFloat(disorderField(h,LT),3,LT);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [],
   "source": [
    "FH = (H_X,H_Y,H_Z,H_F);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [],
   "source": [
    "HE=HamiltonInt(convert(Array{Complex{Float64}},zeros(LT-1)),3,LT);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Lanczos3 (generic function with 1 method)"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Lanczos3(H::Tuple{Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}}},O::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},V::Tuple{Int64,Int64})\n",
    "    bn  = convert(Array{Float64},zeros(V[1]))\n",
    "    On = [(Complex{Float64}[], Array{Int64,1}[]) for i in 1:V[1]]\n",
    "    An = [sparse(convert(Array{Complex{Float64}},zeros(4^5))) for i in 1:V[1]]\n",
    "    \n",
    "    \n",
    "    On[1] = O\n",
    "    \n",
    "    An[1] = Liouville2(H,On[1],V[2])\n",
    "    bn[2] = sqrt(dot(An[1],An[1]))\n",
    "    On[2] = IndexVector_to_CoefandPauli((1/(bn[2])) * An[1],V[2])\n",
    "    \n",
    "    for i=2:V[1]-1\n",
    "        An[i] = Liouville2(H,On[i],V[2]) - bn[i] * CoefandPauli_to_IndexVector(On[i-1],V[2])\n",
    "        bn[i+1] = sqrt(dot(An[i],An[i]))\n",
    "        \n",
    "        On[i+1] = IndexVector_to_CoefandPauli(1/bn[i+1] * An[i],V[2]) \n",
    "    \n",
    "    end\n",
    "    \n",
    "    return bn\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Lanczos (generic function with 1 method)"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Lanczos(H::NTuple{4,Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}}},O::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},V::Tuple{Int64,Int64})\n",
    "    bn  = convert(Array{Float64},zeros(V[1]))\n",
    "    On = [(Complex{Float64}[], Array{Int64,1}[]) for i in 1:V[1]]\n",
    "    An = [sparse(convert(Array{Complex{Float64}},zeros(4^5))) for i in 1:V[1]]\n",
    "    \n",
    "    \n",
    "    On[1] = O\n",
    "    \n",
    "    An[1] = Liouville(H,On[1],V[2])\n",
    "    bn[2] = sqrt(dot(An[1],An[1]))\n",
    "    On[2] = IndexVector_to_CoefandPauli((1/(bn[2])) * An[1],V[2])\n",
    "    \n",
    "    for i=2:V[1]-1\n",
    "        An[i] = Liouville(H,On[i],V[2]) - bn[i] * CoefandPauli_to_IndexVector(On[i-1],V[2])\n",
    "        bn[i+1] = sqrt(dot(An[i],An[i]))\n",
    "        \n",
    "        On[i+1] = IndexVector_to_CoefandPauli(1/bn[i+1] * An[i],V[2]) \n",
    "        \n",
    "    end\n",
    "    \n",
    "    return bn\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 33.382921 seconds (76.56 M allocations: 38.053 GiB, 20.68% gc time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "10-element Array{Float64,1}:\n",
       " 0.0\n",
       " 4.0\n",
       " 5.291502622129181\n",
       " 4.898979485566356\n",
       " 6.187545093630777\n",
       " 6.677567491596992\n",
       " 7.797744991997943\n",
       " 8.096633251217776\n",
       " 9.232217972128058\n",
       " 9.84220586403025"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time Lanczos3(AH,Oj,TT)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11-element Array{Float64,1}:\n",
       "  4.0\n",
       "  5.291502622129181\n",
       "  4.898979485566356\n",
       "  6.187545093630777\n",
       "  6.677567491596992\n",
       "  7.797744991997943\n",
       "  8.096633251217776\n",
       "  9.232217972128058\n",
       "  9.84220586403025\n",
       " 10.555151445890598\n",
       " 10.89773312688247"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "VXXX = [4.0,5.291502622129181,4.898979485566356,6.187545093630777,6.677567491596992,7.797744991997943,8.096633251217776,9.232217972128058,9.84220586403025,10.555151445890598,10.89773312688247]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "147.972073 seconds (176.67 M allocations: 143.419 GiB, 21.61% gc time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "10-element Array{Float64,1}:\n",
       "  0.0\n",
       "  7.348380770554922\n",
       "  7.4660349914248565\n",
       " 11.149133054770367\n",
       " 12.723254991651443\n",
       " 20.3428421504593\n",
       " 19.491952500329774\n",
       " 16.580809422101\n",
       " 20.95005420479415\n",
       " 26.092591634643387"
      ]
     },
     "execution_count": 180,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time Lanczos(FH,Oj,TT)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [],
   "source": [
    "using CSV, DataFrames"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"DisorderMatrix.csv\""
      ]
     },
     "execution_count": 203,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 10 #number of reps\n",
    "H_Matrix = DisorderMatrix(h,LT,m);\n",
    "CSV.write(\"DisorderMatrix.csv\", DataFrame(H_Matrix),  \n",
    "                         header = false)   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"LanczosMatrix.csv\""
      ]
     },
     "execution_count": 204,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = zeros(10,10)\n",
    "for i=1:10\n",
    "    A[:,i] = Lanczos((H_X,H_Y,H_Z,HamiltonFloat(H_Matrix[:,i],3,LT)),Oj,TT)\n",
    "end\n",
    "CSV.write(\"LanczosMatrix.csv\", DataFrame(A),  \n",
    "                         header = false)   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {},
   "outputs": [],
   "source": [
    "meanA = zeros(10)\n",
    "for i = 1:10\n",
    "    meanA[i] = sum(A[i,:])/10\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Float64,1}:\n",
       "  0.0\n",
       " 16.38711035991349\n",
       "  7.2115094667915915\n",
       " 18.834703960175084\n",
       " 13.79248623456991\n",
       " 34.06682780708835\n",
       " 29.326891141650616\n",
       " 26.42130422323878\n",
       " 27.38979573582435\n",
       " 38.272969375689044"
      ]
     },
     "execution_count": 208,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "meanA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LCLHAM (generic function with 1 method)"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function LCLHAM(H::Tuple{Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}}},i::Int64)\n",
    "    \n",
    "    return ((H[1][1][convert(Int64,floor(LT/2))-(i-1):convert(Int64,ceil(LT/2))+(i-1)],H[1][2][convert(Int64,floor(LT/2))-(i-1):convert(Int64,ceil(LT/2))+(i-1)]),(H[2][1][convert(Int64,floor(LT/2))-(i-1):convert(Int64,ceil(LT/2))+(i-1)],H[2][2][convert(Int64,floor(LT/2))-(i-1):convert(Int64,ceil(LT/2))+(i-1)]),(H[3][1][convert(Int64,floor(LT/2))-(i-1):convert(Int64,ceil(LT/2))+(i-1)],H[3][2][convert(Int64,floor(LT/2))-(i-1):convert(Int64,ceil(LT/2))+(i-1)]))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Lanczos3LOCAL (generic function with 1 method)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Lanczos3LOCAL(H::Tuple{Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}}},O::Tuple{Array{Complex{Float64},1},Array{Array{Int64,1},1}},V::Tuple{Int64,Int64})\n",
    "    bn  = convert(Array{Float64},zeros(V[1]))\n",
    "    On = [(Complex{Float64}[], Array{Int64,1}[]) for i in 1:V[1]]\n",
    "    An = [sparse(convert(Array{Complex{Float64}},zeros(4^5))) for i in 1:V[1]]\n",
    "    \n",
    "    \n",
    "    On[1] = O\n",
    "    \n",
    "    An[1] = Liouville2(LCLHAM(AH,1),On[1],V[2])\n",
    "    bn[2] = sqrt(dot(An[1],An[1]))\n",
    "    On[2] = IndexVector_to_CoefandPauli((1/(bn[2])) * An[1],V[2])\n",
    "    \n",
    "    for i=2:V[1]-1\n",
    "        An[i] = Liouville2(LCLHAM(AH,i),On[i],V[2]) - bn[i] * CoefandPauli_to_IndexVector(On[i-1],V[2])\n",
    "        bn[i+1] = sqrt(dot(An[i],An[i]))\n",
    "        \n",
    "        On[i+1] = IndexVector_to_CoefandPauli(1/bn[i+1] * An[i],V[2]) \n",
    "    \n",
    "    end\n",
    "    \n",
    "    return bn\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  6.263867 seconds (22.02 M allocations: 5.238 GiB, 21.60% gc time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9-element Array{Float64,1}:\n",
       " 0.0\n",
       " 4.0\n",
       " 5.291502622129181\n",
       " 4.898979485566356\n",
       " 6.187545093630777\n",
       " 6.677567491596992\n",
       " 7.797744991997943\n",
       " 8.096633251217776\n",
       " 9.232217972128058"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time Lanczos3LOCAL(AH,Oj,TT)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "#The speed up due to locality is very small. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = 1:length(VXXX)-1; y = VXXX[2:end]; # Lanczos Coefficients XXX model\n",
    "plot(sqrt.(x), y,title = \"Lanczos coefficients XXX model\",seriestype = :scatter,label = false)\n",
    "ylabel!(\"bn\")\n",
    "xlabel!(\"sqrt(n)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
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     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "x = 1:length(VXXX); y = VXXX; # Lanczos Coefficients XXX model\n",
    "plot(x, y,title = \"Lanczos coefficients XXX model\",seriestype = :scatter,label = false)\n",
    "ylabel!(\"bn\")\n",
    "xlabel!(\"n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = 1:length(VXXX); y = VXXX; # Lanczos Coefficients XXX model\n",
    "plot(log.(x), log.(y),title = \"Lanczos coefficients XXX model\",seriestype = :scatter,label = false)\n",
    "ylabel!(\"log(bn)\")\n",
    "xlabel!(\"log(n)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11-element Array{Float64,1}:\n",
       " 0.0\n",
       " 0.6931471805599453\n",
       " 1.0986122886681098\n",
       " 1.3862943611198906\n",
       " 1.6094379124341003\n",
       " 1.791759469228055\n",
       " 1.9459101490553132\n",
       " 2.0794415416798357\n",
       " 2.1972245773362196\n",
       " 2.302585092994046\n",
       " 2.3978952727983707"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log.(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11-element Array{Float64,1}:\n",
       " 1.3862943611198906\n",
       " 1.666102255087602\n",
       " 1.5890269151739727\n",
       " 1.822538415727772\n",
       " 1.898753773266062\n",
       " 2.0538345883070397\n",
       " 2.091448327275015\n",
       " 2.22269931998616\n",
       " 2.2866798591149164\n",
       " 2.3566140294579454\n",
       " 2.388554797595554"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log.(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5208149925794373"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(log.(y)[11] - log.(y)[8])/(log.(x)[11] - log.(x)[8])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [
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       "<circle clip-path=\"url(#clip652)\" cx=\"1784.11\" cy=\"628.096\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"/>\n",
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       "<circle clip-path=\"url(#clip652)\" cx=\"2291.83\" cy=\"160.256\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 221,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = 1:length(meanA)-1; y = meanA[2:10]; # Lanczos Coefficients XXX model\n",
    "plot(x, y,title = \"Average Lanczos coefficients MBL, h = 12\",seriestype = :scatter,label = false)\n",
    "ylabel!(\"bn\")\n",
    "xlabel!(\"n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Float64,1}:\n",
       "  0.0\n",
       " 16.38711035991349\n",
       "  7.2115094667915915\n",
       " 18.834703960175084\n",
       " 13.79248623456991\n",
       " 34.06682780708835\n",
       " 29.326891141650616\n",
       " 26.42130422323878\n",
       " 27.38979573582435\n",
       " 38.272969375689044"
      ]
     },
     "execution_count": 212,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "meanA\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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