philzook58/zx_catlab.ipynb

## gistfile1.txt
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# ZX calculus in Catlab"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Catlab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Theory and symbolic expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define a  *generalized algebraic theory (GAT)* for the ZX calculus."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Catlab.Doctrines\n",
    "import Catlab.Doctrines: Ob, Hom, dom, codom, compose, ⋅, id, otimes, ⊗, munit, braid,\n",
    "  hom, ev, curry, dual, dunit, dcounit, dagger, mate\n",
    "\n",
    "\"\"\" ZX calculus, presented in biased style.\n",
    "\"\"\"\n",
    "@theory DaggerCompactCategory(Ob,Hom) => ZXCategory(Ob,Hom) begin\n",
    "    # Argument α is the phase, usually <: Real\n",
    "    zphase(A::Ob, α)::(A → A)\n",
    "    zcopy(A::Ob, α)::(A → (A⊗A))\n",
    "    zdelete(A::Ob, α)::(A → munit())\n",
    "    zmerge(A::Ob, α)::((A⊗A) → A)\n",
    "    zcreate(A::Ob, α)::(munit() → A)\n",
    "    \n",
    "    xphase(A::Ob, α)::(A → A)\n",
    "    xcopy(A::Ob, α)::(A → (A⊗A))\n",
    "    xdelete(A::Ob, α)::(A → munit())\n",
    "    xmerge(A::Ob, α)::((A⊗A) → A)\n",
    "    xcreate(A::Ob, α)::(munit() → A)\n",
    "    \n",
    "    hadamard(A::Ob)::(A → A)\n",
    "    \n",
    "    # Example axioms.\n",
    "    zphase(A,0) == id(A) ⊣ (A::Ob)\n",
    "    zcopy(A,α)⋅(zdelete(A,β) ⊗ id(A)) == zphase(A, α+β) ⊣ (A::Ob, α, β)\n",
    "    (zcreate(A,α) ⊗ id(A))⋅zmerge(A,β) == zphase(A, α+β) ⊣ (A::Ob, α, β)\n",
    "end\n",
    "\n",
    "# Convenience methods for phaseless spiders.\n",
    "zcopy(A) = zcopy(A,0)\n",
    "zdelete(A) = zdelete(A,0)\n",
    "zmerge(A) = zmerge(A,0)\n",
    "zcreate(A) = zcreate(A,0)\n",
    "\n",
    "xcopy(A) = xcopy(A,0)\n",
    "xdelete(A) = xdelete(A,0)\n",
    "xmerge(A) = xmerge(A,0)\n",
    "xcreate(A) = xcreate(A,0);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Symbolic expressions, with simplification for associativity and unitality laws of monoidal categeory."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Main.ZXCalculus"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@syntax ZXCalculus(ObExpr,HomExpr) ZXCategory begin\n",
    "    otimes(A::Ob, B::Ob) = associate_unit(new(A,B), munit)\n",
    "    otimes(f::Hom, g::Hom) = associate(new(f,g))\n",
    "    compose(f::Hom, g::Hom) = associate_unit(new(f,g; strict=true), identity)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pretty-printing symbolic expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import Catlab: show_latex\n",
    "\n",
    "latex_spider(io, expr, color, head) = Syntax.show_latex_script(io, expr, \"{\\\\color{$color} $head}\")\n",
    "\n",
    "const zcolor = \"green\"\n",
    "const xcolor = \"red\"\n",
    "const hcolor = \"yellow\"\n",
    "\n",
    "show_latex(io::IO, expr::HomExpr{:zphase}; kw...) = latex_spider(io, expr, zcolor, \"\\\\bullet\")\n",
    "show_latex(io::IO, expr::HomExpr{:zcopy}; kw...) = latex_spider(io, expr, zcolor, \"\\\\blacktriangle\")\n",
    "show_latex(io::IO, expr::HomExpr{:zdelete}; kw...) = latex_spider(io, expr, zcolor, \"\\\\blacklozenge\")\n",
    "show_latex(io::IO, expr::HomExpr{:zmerge}; kw...) = latex_spider(io, expr, zcolor, \"\\\\blacktriangledown\")\n",
    "show_latex(io::IO, expr::HomExpr{:zcreate}; kw...) = latex_spider(io, expr, zcolor, \"\\\\blacksquare\")\n",
    "\n",
    "show_latex(io::IO, expr::HomExpr{:xphase}; kw...) = latex_spider(io, expr, xcolor, \"\\\\bullet\")\n",
    "show_latex(io::IO, expr::HomExpr{:xcopy}; kw...) = latex_spider(io, expr, xcolor, \"\\\\blacktriangle\")\n",
    "show_latex(io::IO, expr::HomExpr{:xdelete}; kw...) = latex_spider(io, expr, xcolor, \"\\\\blacklozenge\")\n",
    "show_latex(io::IO, expr::HomExpr{:xmerge}; kw...) = latex_spider(io, expr, xcolor, \"\\\\blacktriangledown\")\n",
    "show_latex(io::IO, expr::HomExpr{:xcreate}; kw...) = latex_spider(io, expr, xcolor, \"\\\\blacksquare\")\n",
    "\n",
    "show_latex(io::IO, expr::HomExpr{:hadamard}; kw...) = Syntax.show_latex_script(io, expr, \"H\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hadamard gate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$H_{A} : A \\to A$"
      ],
      "text/plain": [
       "hadamard(A)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Ob(ZXCalculus, :A)\n",
    "hadamard(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "CNOT (controlled NOT) gate in ZX calculus:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left({\\color{green} \\blacktriangle}_{A,0} \\otimes \\mathrm{id}_{A}\\right) \\cdot \\left(\\mathrm{id}_{A} \\otimes {\\color{red} \\blacktriangledown}_{A,0}\\right) : A \\otimes A \\to A \\otimes A$"
      ],
      "text/plain": [
       "compose(otimes(zcopy(A,0),id(A)),otimes(id(A),xmerge(A,0)))"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cnot = (zcopy(A) ⊗ id(A))⋅(id(A) ⊗ xmerge(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another, equivalent CNOT gate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left(\\mathrm{id}_{A} \\otimes {\\color{red} \\blacktriangle}_{A,0}\\right) \\cdot \\left({\\color{green} \\blacktriangledown}_{A,0} \\otimes \\mathrm{id}_{A}\\right) : A \\otimes A \\to A \\otimes A$"
      ],
      "text/plain": [
       "compose(otimes(id(A),xcopy(A,0)),otimes(zmerge(A,0),id(A)))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cnot_alt = (id(A) ⊗ xcopy(A))⋅(zmerge(A) ⊗ id(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Wiring diagrams"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Catlab.WiringDiagrams\n",
    "\n",
    "# Dagger compact structure.\n",
    "dunit(A::Ports{ZXCategory.Hom}) = junction_caps(A)\n",
    "dcounit(A::Ports{ZXCategory.Hom}) = junction_cups(A)\n",
    "dagger(f::WiringDiagram{ZXCategory.Hom}) = functor(f, identity, dagger, contravariant=true)\n",
    "\n",
    "struct PhasedValue\n",
    "    value::Any\n",
    "    phase::Any\n",
    "end\n",
    "\n",
    "function phased_junction_diagram(A::Ports, phase, nin::Int, nout::Int; op=nothing)\n",
    "    junction_diagram(A, nin, nout) do port, inputs, outputs\n",
    "        Junction{op}(PhasedValue(port, phase), inputs, outputs)\n",
    "    end\n",
    "end\n",
    "\n",
    "zphase(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 1, 1, op=:Z)\n",
    "zcopy(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 1, 2, op=:Z)\n",
    "zdelete(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 1, 0, op=:Z)\n",
    "zmerge(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 2, 1, op=:Z)\n",
    "zcreate(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 0, 1, op=:Z)\n",
    "\n",
    "xphase(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 1, 1, op=:X)\n",
    "xcopy(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 1, 2, op=:X)\n",
    "xdelete(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 1, 0, op=:X)\n",
    "xmerge(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 2, 1, op=:X)\n",
    "xcreate(A::Ports{ZXCategory.Hom}, α) = phased_junction_diagram(A, α, 0, 1, op=:X)\n",
    "\n",
    "hadamard(A::Ports{ZXCategory.Hom}) = junction_diagram(A, 1, 1, op=:H);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Converting expressions to wiring diagrams"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "WiringDiagram{Main.ZXCategory.Hom}([:A], [:A], \n",
       "[ 1 => {inputs},\n",
       "  2 => {outputs},\n",
       "  3 => Junction{:H,Symbol}(:A, [:A], [:A]) ],\n",
       "[ Wire((1,1) => (3,1)),\n",
       "  Wire((3,1) => (2,1)) ])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "to_wiring_diagram(hadamard(A))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "WiringDiagram{Main.ZXCategory.Hom}([:A,:A], [:A,:A], \n",
       "[ 1 => {inputs},\n",
       "  2 => {outputs},\n",
       "  3 => Junction{:Z,PhasedValue}(PhasedValue(:A, 0), [:A], [:A, :A]),\n",
       "  4 => Junction{:X,PhasedValue}(PhasedValue(:A, 0), [:A, :A], [:A]) ],\n",
       "[ Wire((1,1) => (3,1)),\n",
       "  Wire((1,2) => (4,2)),\n",
       "  Wire((3,1) => (2,1)),\n",
       "  Wire((3,2) => (4,1)),\n",
       "  Wire((4,1) => (2,2)) ])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "to_wiring_diagram(cnot)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graphics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "graphviz_box (generic function with 5 methods)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Catlab.Graphics\n",
    "import Catlab.Graphics.GraphvizWiringDiagrams: graphviz_box, graphviz_junction\n",
    "\n",
    "node_label(phased::PhasedValue) = phased.phase == 0 ? \"\" : string(phased.phase)\n",
    "\n",
    "graphviz_box(junction::Junction{:Z}, node_id; kw...) = graphviz_junction(junction, node_id;\n",
    "    label=node_label(junction.value), shape=\"circle\", style=\"filled\", fillcolor=zcolor,\n",
    "    width=\"0.15\", height=\"0.15\")\n",
    "graphviz_box(junction::Junction{:X}, node_id; kw...) = graphviz_junction(junction, node_id;\n",
    "    label=node_label(junction.value), shape=\"circle\", style=\"filled\", fillcolor=xcolor,\n",
    "    width=\"0.15\", height=\"0.15\")\n",
    "graphviz_box(junction::Junction{:H}, node_id; kw...) = graphviz_junction(junction, node_id;\n",
    "    label=\"\", shape=\"rect\", style=\"filled\", fillcolor=hcolor, width=\"0.15\", height=\"0.15\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Visualizing wiring diagrams using Graphviz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "to_graphviz(hadamard(A), orientation=LeftToRight)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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       "Catlab.Graphics.Graphviz.Graph(\"G\", true, Catlab.Graphics.Graphviz.Statement[Catlab.Graphics.Graphviz.Subgraph(\"\", Catlab.Graphics.Graphviz.Statement[Catlab.Graphics.Graphviz.Node(\"n1p1\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:id => \"in1\")), Catlab.Graphics.Graphviz.Node(\"n1p2\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:id => \"in2\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n1p1\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n1p2\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}())], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:rank => \"source\",:rankdir => \"TB\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:style => \"invis\",:shape => \"none\",:label => \"\",:width => \"0\",:height => \"0.333\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:style => \"invis\")), Catlab.Graphics.Graphviz.Subgraph(\"\", Catlab.Graphics.Graphviz.Statement[Catlab.Graphics.Graphviz.Node(\"n2p1\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:id => \"out1\")), Catlab.Graphics.Graphviz.Node(\"n2p2\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:id => \"out2\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n2p1\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n2p2\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}())], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:rank => \"sink\",:rankdir => \"TB\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:style => \"invis\",:shape => \"none\",:label => \"\",:width => \"0\",:height => \"0.333\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:style => \"invis\")), Catlab.Graphics.Graphviz.Node(\"n3\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fillcolor => \"green\",:height => \"0.15\",:id => \"n3\",:label => \"\",:shape => \"circle\",:style => \"filled\",:width => \"0.15\")), Catlab.Graphics.Graphviz.Node(\"n4\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fillcolor => \"red\",:height => \"0.15\",:id => \"n4\",:label => \"\",:shape => \"circle\",:style => \"filled\",:width => \"0.15\")), Catlab.Graphics.Graphviz.Node(\"n5\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fillcolor => \"red\",:height => \"0.15\",:id => \"n5\",:label => \"\",:shape => \"circle\",:style => \"filled\",:width => \"0.15\")), Catlab.Graphics.Graphviz.Node(\"n6\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fillcolor => \"green\",:height => \"0.15\",:id => \"n6\",:label => \"\",:shape => \"circle\",:style => \"filled\",:width => \"0.15\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n1p1\", \"e\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n3\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e1\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n1p2\", \"e\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n4\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e2\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n3\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n4\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e3\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n3\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n6\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e4\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n4\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n5\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e5\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n5\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n2p2\", \"w\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e6\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n5\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n6\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e7\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n6\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n2p1\", \"w\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e8\"))], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fontname => \"Serif\",:rankdir => \"LR\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fontname => \"Serif\",:shape => \"none\",:width => \"0\",:height => \"0\",:margin => \"0\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:arrowsize => \"0.5\",:fontname => \"Serif\"))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "to_graphviz(to_wiring_diagram(cnot⋅cnot_alt), orientation=LeftToRight)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "${\\color{green} \\bullet}_{A,\"π/2\"} \\cdot {\\color{red} \\bullet}_{A,\"π/2\"} \\cdot {\\color{green} \\bullet}_{A,\"π/2\"} : A \\to A$"
      ],
      "text/plain": [
       "compose(zphase(A,π/2),xphase(A,π/2),zphase(A,π/2))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "euler_decomp = zphase(A, \"π/2\") ⋅ xphase(A, \"π/2\") ⋅ zphase(A, \"π/2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
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       " -->\n",
       "<!-- Title: G Pages: 1 -->\n",
       "<svg width=\"235pt\" height=\"35pt\"\n",
       " viewBox=\"0.00 0.00 234.61 34.87\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
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       "<text text-anchor=\"middle\" x=\"50.435\" y=\"-9.735\" font-family=\"Serif\" font-size=\"14.00\" fill=\"#000000\">π/2</text>\n",
       "</g>\n",
       "<!-- n1p1&#45;&gt;n3 -->\n",
       "<!-- A -->\n",
       "<g id=\"e1\" class=\"edge\">\n",
       "<title>n1p1:e&#45;&gt;n3</title>\n",
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       "</g>\n",
       "<!-- n2p1 -->\n",
       "<!-- n4 -->\n",
       "<g id=\"n4\" class=\"node\">\n",
       "<title>n4</title>\n",
       "<ellipse fill=\"#ff0000\" stroke=\"#000000\" cx=\"113.3051\" cy=\"-13.435\" rx=\"13.3704\" ry=\"13.3704\"/>\n",
       "<text text-anchor=\"middle\" x=\"113.3051\" y=\"-9.735\" font-family=\"Serif\" font-size=\"14.00\" fill=\"#000000\">π/2</text>\n",
       "</g>\n",
       "<!-- n3&#45;&gt;n4 -->\n",
       "<!-- A -->\n",
       "<g id=\"e2\" class=\"edge\">\n",
       "<title>n3&#45;&gt;n4</title>\n",
       "<path fill=\"none\" stroke=\"#000000\" d=\"M64.0769,-13.435C72.9746,-13.435 84.7081,-13.435 94.488,-13.435\"/>\n",
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       "</g>\n",
       "<!-- n5 -->\n",
       "<g id=\"n5\" class=\"node\">\n",
       "<title>n5</title>\n",
       "<ellipse fill=\"#00ff00\" stroke=\"#000000\" cx=\"176.1751\" cy=\"-13.435\" rx=\"13.3704\" ry=\"13.3704\"/>\n",
       "<text text-anchor=\"middle\" x=\"176.1751\" y=\"-9.735\" font-family=\"Serif\" font-size=\"14.00\" fill=\"#000000\">π/2</text>\n",
       "</g>\n",
       "<!-- n4&#45;&gt;n5 -->\n",
       "<!-- A -->\n",
       "<g id=\"e3\" class=\"edge\">\n",
       "<title>n4&#45;&gt;n5</title>\n",
       "<path fill=\"none\" stroke=\"#000000\" d=\"M126.947,-13.435C135.8446,-13.435 147.5782,-13.435 157.3581,-13.435\"/>\n",
       "<polygon fill=\"#000000\" stroke=\"#000000\" points=\"157.5599,-15.1851 162.5599,-13.435 157.5599,-11.6851 157.5599,-15.1851\"/>\n",
       "</g>\n",
       "<!-- n5&#45;&gt;n2p1 -->\n",
       "<!-- A -->\n",
       "<g id=\"e4\" class=\"edge\">\n",
       "<title>n5&#45;&gt;n2p1:w</title>\n",
       "<path fill=\"none\" stroke=\"#000000\" d=\"M190.0258,-13.435C198.5428,-13.435 209.8275,-13.435 220.4587,-13.435\"/>\n",
       "<polygon fill=\"#000000\" stroke=\"#000000\" points=\"220.6102,-15.1851 225.6102,-13.435 220.6101,-11.6851 220.6102,-15.1851\"/>\n",
       "</g>\n",
       "</g>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "Catlab.Graphics.Graphviz.Graph(\"G\", true, Catlab.Graphics.Graphviz.Statement[Catlab.Graphics.Graphviz.Subgraph(\"\", Catlab.Graphics.Graphviz.Statement[Catlab.Graphics.Graphviz.Node(\"n1p1\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:id => \"in1\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n1p1\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}())], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:rank => \"source\",:rankdir => \"TB\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:style => \"invis\",:shape => \"none\",:label => \"\",:width => \"0\",:height => \"0.333\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:style => \"invis\")), Catlab.Graphics.Graphviz.Subgraph(\"\", Catlab.Graphics.Graphviz.Statement[Catlab.Graphics.Graphviz.Node(\"n2p1\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:id => \"out1\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n2p1\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}())], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:rank => \"sink\",:rankdir => \"TB\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:style => \"invis\",:shape => \"none\",:label => \"\",:width => \"0\",:height => \"0.333\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:style => \"invis\")), Catlab.Graphics.Graphviz.Node(\"n3\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fillcolor => \"green\",:height => \"0.15\",:id => \"n3\",:label => \"π/2\",:shape => \"circle\",:style => \"filled\",:width => \"0.15\")), Catlab.Graphics.Graphviz.Node(\"n4\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fillcolor => \"red\",:height => \"0.15\",:id => \"n4\",:label => \"π/2\",:shape => \"circle\",:style => \"filled\",:width => \"0.15\")), Catlab.Graphics.Graphviz.Node(\"n5\", OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fillcolor => \"green\",:height => \"0.15\",:id => \"n5\",:label => \"π/2\",:shape => \"circle\",:style => \"filled\",:width => \"0.15\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n1p1\", \"e\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n3\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e1\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n3\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n4\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e2\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n4\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n5\", \"\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e3\")), Catlab.Graphics.Graphviz.Edge(Catlab.Graphics.Graphviz.NodeID[Catlab.Graphics.Graphviz.NodeID(\"n5\", \"\", \"\"), Catlab.Graphics.Graphviz.NodeID(\"n2p1\", \"w\", \"\")], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:comment => \"A\",:id => \"e4\"))], OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fontname => \"Serif\",:rankdir => \"LR\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:fontname => \"Serif\",:shape => \"none\",:width => \"0\",:height => \"0\",:margin => \"0\"), OrderedCollections.OrderedDict{Symbol,Union{String, Catlab.Graphics.Graphviz.Html}}(:arrowsize => \"0.5\",:fontname => \"Serif\"))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "to_graphviz(to_wiring_diagram(euler_decomp), orientation=LeftToRight)"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Julia 1.4.0",
   "language": "julia",
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   "file_extension": ".jl",
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## zx_catlab.ipynb

      
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