Routing in directed graph.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Routing in a directed graph\n",
    "Consider a directed graph with edges $i=0, ..., N$ and edge costs $C_i$. We want to find a path from a start vertex to a target vertex. Let $X_i\\in\\{0,1\\}$ be binary variables indicating if edge $i$ is chosen along the path.\n",
    "\n",
    "The cost function is based on the following requirements:\n",
    "- The start vertex has exactly one outgoing edge.\n",
    "- The start vertex has no incoming edge.\n",
    "- The target vertex has exactly one incoming edge.\n",
    "- The target vertex has no outgoing edge.\n",
    "- Every other vertex has\n",
    "    - either one or zero incoming edges.\n",
    "    - either one or zero outgoing edges.\n",
    "    - an equal number of incoming and outgoing edges.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem definition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle P \\left(\\sum_{i=0}^{LenVsO - 1} {X}_{{VsO}_{i}} - 1\\right)^{2} + P \\left(\\sum_{i=0}^{LenVtI - 1} {X}_{{VtI}_{i}} - 1\\right)^{2} + P \\sum_{v=0}^{LenV - 1} \\left(\\sum_{i=0}^{{LenVI}_{v} - 1} {X}_{{VI}_{v,i}} - 1\\right) \\sum_{i=0}^{{LenVI}_{v} - 1} {X}_{{VI}_{v,i}} + P \\sum_{v=0}^{LenV - 1} \\left(\\sum_{i=0}^{{LenVO}_{v} - 1} {X}_{{VO}_{v,i}} - 1\\right) \\sum_{i=0}^{{LenVO}_{v} - 1} {X}_{{VO}_{v,i}} + P \\sum_{v=0}^{LenV - 1} \\left(\\sum_{i=0}^{{LenVI}_{v} - 1} {X}_{{VI}_{v,i}} - \\sum_{i=0}^{{LenVO}_{v} - 1} {X}_{{VO}_{v,i}}\\right)^{2} + P \\sum_{i=0}^{LenVsI - 1} {X}_{{VsI}_{i}} + P \\sum_{i=0}^{LenVtO - 1} {X}_{{Vt O}_{i}} + \\sum_{i=0}^{N - 1} {C}_{i} {X}_{i}$"
      ],
      "text/plain": [
       "P*(Sum(X[VsO[i]], (i, 0, LenVsO - 1)) - 1)**2 + P*(Sum(X[VtI[i]], (i, 0, LenVtI - 1)) - 1)**2 + P*Sum((Sum(X[VI[v, i]], (i, 0, LenVI[v] - 1)) - 1)*Sum(X[VI[v, i]], (i, 0, LenVI[v] - 1)), (v, 0, LenV - 1)) + P*Sum((Sum(X[VO[v, i]], (i, 0, LenVO[v] - 1)) - 1)*Sum(X[VO[v, i]], (i, 0, LenVO[v] - 1)), (v, 0, LenV - 1)) + P*Sum((Sum(X[VI[v, i]], (i, 0, LenVI[v] - 1)) - Sum(X[VO[v, i]], (i, 0, LenVO[v] - 1)))**2, (v, 0, LenV - 1)) + P*Sum(X[VsI[i]], (i, 0, LenVsI - 1)) + P*Sum(X[Vt O[i]], (i, 0, LenVtO - 1)) + Sum(C[i]*X[i], (i, 0, N - 1))"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## Cost function specification in sympy\n",
    "\n",
    "import sympy as sym\n",
    "\n",
    "# number of edges (Qubits)\n",
    "N = sym.symbols('N')\n",
    "\n",
    "# possible edges X_0, ..., X_{N-1} with values in {0,1}\n",
    "X = sym.IndexedBase('X')\n",
    "\n",
    "# costs for edges\n",
    "C = sym.IndexedBase('C')\n",
    "\n",
    "# penalty\n",
    "P = sym.symbols('P')\n",
    "\n",
    "# indices\n",
    "i, j, k, v = sym.symbols('i j k v')\n",
    "\n",
    "# start vertex: incoming and outgoing edges\n",
    "VsI = sym.IndexedBase('VsI')    # incoming edges\n",
    "LenVsI = sym.symbols('LenVsI')  # number of incoming edges\n",
    "VsO = sym.IndexedBase('VsO')    # outgoing edges\n",
    "LenVsO = sym.symbols('LenVsO')  # number of incoming edges\n",
    "\n",
    "# target vertex: incoming and outgoing edges\n",
    "VtI = sym.IndexedBase('VtI')    # incoming edges\n",
    "LenVtI = sym.symbols('LenVtI')  # number of incoming edges\n",
    "VtO = sym.IndexedBase('Vt O')   # outgoing edges\n",
    "LenVtO = sym.symbols('LenVtO')  # number of incoming edges\n",
    "\n",
    "# middle vertices: incoming and outgoing edges\n",
    "# two indices: VI_(i,k) is the kth incoming edge at vertex i\n",
    "# Similarly, LenVI_i is the number of incoming edges at vertex i\n",
    "# the number of lists in LenVI and LenVO is the number of central vertices and must equal\n",
    "LenV = sym.IndexedBase('LenV')   # number central vertices\n",
    "\n",
    "VI = sym.IndexedBase('VI')        # incoming edges\n",
    "LenVI = sym.IndexedBase('LenVI')  # number of incoming edges\n",
    "VO = sym.IndexedBase('VO')        # outgoing edges\n",
    "LenVO = sym.IndexedBase('LenVO')  # number of outgoing edges\n",
    "\n",
    "cost_function = (\n",
    "    sym.Sum( C[i] * X[i] , (i,0,N-1) ) +                    # costs for edge i\n",
    "    P * (sym.Sum( X[VsO[i]], (i, 0, LenVsO-1) ) - 1 )**2 +  # exactly one outgoing edge at start vertex\n",
    "    P * sym.Sum( X[VsI[i]], (i, 0, LenVsI-1) ) +            # no incoming edge at start vertex\n",
    "    P * (sym.Sum( X[VtI[i]], (i, 0, LenVtI-1) ) - 1 )**2 +  # exactly one incoming edge at target vertex\n",
    "    P * sym.Sum( X[VtO[i]], (i, 0, LenVtO-1) ) +            # no incoming edge at target vertex\n",
    "    P * sym.Sum(                                            # one or zero incoming edges at each central point\n",
    "        sym.Sum(  X[VI[v,i]] , (i, 0, LenVI[v]-1) ) *\n",
    "        (sym.Sum(  X[VI[v,i]] , (i, 0, LenVI[v]-1) ) -1) ,\n",
    "        (v, 0, LenV-1) ) +\n",
    "    P * sym.Sum(                                            # one or zero outgoing edges at each central point\n",
    "        sym.Sum(  X[VO[v,i]] , (i, 0, LenVO[v]-1) ) *\n",
    "        (sym.Sum(  X[VO[v,i]] , (i, 0, LenVO[v]-1) ) -1) ,\n",
    "        (v, 0, LenV-1) ) +\n",
    "    P * sym.Sum(                                            # equal number of incoming and outgoing edges at each central point\n",
    "        (sym.Sum(  X[VI[v,i]] , (i, 0, LenVI[v]-1) ) -\n",
    "        sym.Sum(  X[VO[v,i]] , (i, 0, LenVO[v]-1) ))**2 ,\n",
    "        (v, 0, LenV-1) ) )\n",
    "cost_function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data input"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 2,
     "metadata": {
      "image/jpeg": {
       "width": 400
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## graph structure\n",
    "## external input\n",
    "\n",
    "TestN = 5                  # number of edges\n",
    "TestVsI = []            # indices of incoming edges at start vertex\n",
    "TestVsO = [0,1]            # indices of outgoing edges at start vertex\n",
    "TestVtI = [3,4]          # indices of incoming edges at target vertex\n",
    "TestVtO = []            # indices of outgoing edges at target vertex\n",
    "TestVI = [[0],[1,2]]     # indices of incoming edges at central vertices\n",
    "TestVO = [[2,3],[4]]   # indices of outgoing edges at central vertices\n",
    "\n",
    "## edge costs\n",
    "## external input\n",
    "\n",
    "TestC = [2,4,1,7,3]\n",
    "TestP = 50\n",
    "\n",
    "from IPython.display import Image\n",
    "Image(\"graph.jpg\", width = 400)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data processing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "## translation of graph structure into dictionaries for sympy\n",
    "\n",
    "single_valued_dict = {\n",
    "    N: TestN,\n",
    "    P: TestP,\n",
    "    LenVsI: len(TestVsI),\n",
    "    LenVsO: len(TestVsO),\n",
    "    LenVtI: len(TestVtI),\n",
    "    LenVtO: len(TestVtO),\n",
    "    LenV:  len(TestVI)\n",
    "}\n",
    "\n",
    "# multi-valued dictionaries\n",
    "dict_VsI = { VsI[k]: TestVsI[k] for k in range(len(TestVsI)) }\n",
    "\n",
    "dict_VsO = { VsO[k]: TestVsO[k] for k in range(len(TestVsO)) }\n",
    "\n",
    "dict_VtI = { VtI[k]: TestVtI[k] for k in range(len(TestVtI)) }\n",
    "\n",
    "dict_VtO = { VtO[k]: TestVtO[k] for k in range(len(TestVtO)) }\n",
    "\n",
    "dict_LenVI = { LenVI[k]: len(TestVI[k]) for k in range(len(TestVI)) }\n",
    "\n",
    "dict_LenVO = { LenVO[k]: len(TestVO[k]) for k in range(len(TestVO)) }\n",
    "\n",
    "dict_VI = { VI[k, i]: TestVI[k][i] for k in range(len(TestVI)) for i in range(len(TestVI[k])) }\n",
    "\n",
    "dict_VO = { VO[k, i]: TestVO[k][i] for k in range(len(TestVO)) for i in range(len(TestVO[k])) }\n",
    "\n",
    "num_qubits = TestN\n",
    "\n",
    "dict_costs = { C[k]: TestC[k] for k in range(len(TestC)) }\n",
    "\n",
    "# definition of cost polynomial\n",
    "cost_poly = sym.Poly(cost_function\n",
    "                     .subs(single_valued_dict)\n",
    "                     .doit()\n",
    "                     .subs(dict_VsI)\n",
    "                     .subs(dict_VsO)\n",
    "                     .subs(dict_VtI)\n",
    "                     .subs(dict_VtO)\n",
    "                     .subs(dict_LenVI)\n",
    "                     .subs(dict_LenVO)\n",
    "                     .doit()\n",
    "                     .subs(dict_VI)\n",
    "                     .subs(dict_VO)\n",
    "                     .subs(dict_costs),\n",
    "                     [X[i] for i in range(num_qubits)])\n",
    "# in order to visualize the QUBO-matrix\n",
    "# do not set P in single_valued_dict\n",
    "# and do not substitute the cost in cost_poly\n",
    "# then procede as in the minimal optimization example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## QAOA and VQE in Qiskit\n",
    "### Cost function transfer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import qiskit\n",
    "from qiskit.aqua import QuantumInstance\n",
    "from qiskit.aqua.algorithms import QAOA, VQE\n",
    "from qiskit.optimization.algorithms import MinimumEigenOptimizer\n",
    "\n",
    "# generate qiskit's cost function\n",
    "qiskit_cost_function = qiskit.optimization.QuadraticProgram()\n",
    "\n",
    "# define qiskit variables\n",
    "for i in range(num_qubits):\n",
    "    qiskit_cost_function.binary_var('X_' + str(i))\n",
    "\n",
    "# specify qiskit cost function\n",
    "qiskit_cost_function.minimize(\n",
    "    linear = [int(cost_poly.coeff_monomial(X[i]**1)) for i in range(num_qubits)],\n",
    "    quadratic = {\n",
    "        ('X_'+str(i), 'X_'+str(j)): cost_poly.coeff_monomial(X[i]**1 * X[j]**1)\n",
    "        for i in range(num_qubits)\n",
    "        for j in range(i,num_qubits)\n",
    "    })"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Backend selection"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit import IBMQ\n",
    "provider = IBMQ.load_account()\n",
    "\n",
    "# 5 qubit quantum computer\n",
    "backend = provider.get_backend('ibmq_london')\n",
    "\n",
    "# ignore warnings\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### QAOA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "QAOA: x=[1.0,0.0,1.0,0.0,1.0], fval=-94.0\n"
     ]
    }
   ],
   "source": [
    "qaoa = QAOA(quantum_instance =\n",
    "             QuantumInstance(backend, skip_qobj_validation=False))\n",
    "optimizer_qaoa = MinimumEigenOptimizer(qaoa)\n",
    "result_qaoa = optimizer_qaoa.solve(qiskit_cost_function)\n",
    "print(\"QAOA:\", result_qaoa)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### VQE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "VQE: x=[1.0,0.0,1.0,0.0,1.0], fval=-94.0\n"
     ]
    }
   ],
   "source": [
    "vqe = VQE(quantum_instance =\n",
    "             QuantumInstance(backend, skip_qobj_validation=False))\n",
    "optimizer_vqe = MinimumEigenOptimizer(vqe)\n",
    "result_vqe = optimizer_vqe.solve(qiskit_cost_function)\n",
    "\n",
    "print(\"VQE:\", result_vqe)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## D-Wave\n",
    "### Cost function transfer and execution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{0: 1, 1: 0, 2: 1, 3: 0, 4: 1}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# generate QUBO-matrix for the given number of Qubits\n",
    "Qubo = { (i,j) : (cost_poly.coeff_monomial(X[i]**1 * X[j]**1)\n",
    "                  + ( cost_poly.coeff_monomial(X[i]**1) if (i==j) else 0 ))\n",
    "        for i in range(num_qubits)\n",
    "        for j in range(i,num_qubits) }\n",
    "# execution on D-Wave simulator\n",
    "from dwave_qbsolv import QBSolv\n",
    "result_dwave = QBSolv().sample_qubo(Qubo)\n",
    "\n",
    "# print result\n",
    "list(result_dwave.samples())[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Result visualization\n",
    "All methods give the correct result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 9,
     "metadata": {
      "image/jpeg": {
       "width": 400
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image(\"graph_solution.jpg\", width = 400)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "quantum",
   "language": "python",
   "name": "quantum"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}