examples/quantum_annealing/Dwave_Factoring/Dwave_factoring.ipynb

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 "cells": [
  {
   "cell_type": "markdown",
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   "source": [
    "# Demo for D-Wave on Braket: Factoring"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this tutorial we show how to solve a [constraint satisfaction problem](https://docs.ocean.dwavesys.com/en/stable/concepts/csp.html)(CSP) on a quantum computer with the example of factoring, which is realized by running a multiplication circuit in reverse using the D-Wave device on Amazon Braket.\n",
    "\n",
    "Disclaimer: The code shown in this example has been taken from D-Wave tutorial available online [here](https://github.com/dwave-examples/factoring-notebook), with copyright to D-Wave Systems, Inc., licensed under the Apache License. The purpose of this example is to show how existing code using D-Wave's Ocean tool suite can easily be run on Amazon Braket, with minimal code changes, using the ```BraketDWaveSampler```."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Background\n",
    "\n",
    "This notebook solves factoring on a D-Wave quantum computer by formulating it as a *[constraint satisfaction problem](https://docs.ocean.dwavesys.com/en/stable/concepts/csp.html)*. CSPs require that all a problem's variables be assigned values that result in the satisfying of all constraints. For factoring, the problem's constraints are that the two variables representing factors, $a$ and $b$, be assigned only natural numbers and that their multiplication be equal to the factored number, $P$. \n",
    "\n",
    "D-Wave systems solve [binary quadratic models](https://docs.ocean.dwavesys.com/en/stable/concepts/bqm.html) (BQM) problems. Given $N$ variables $x_1,...,x_N$, where each variable $x_i$ can have binary values $0$ or $1$, the system finds assignments of values that minimize the expression\n",
    "\n",
    "$\\sum_i^N q_ix_i + \\sum_{i<j}^N q_{i,j}x_i  x_j$,\n",
    "\n",
    "where $q_i$ and $q_{i,j}$ are configurable (linear and quadratic) coefficients.\n",
    "\n",
    "To implement the factoring problem, we express the constraints of the problem with Boolean logic and follow these steps:\n",
    "\n",
    "1. Express $P=ab$ as a CSP with a binary multiplication circuit.\n",
    "2. Convert to a BQM.\n",
    "3. Program the quantum computer with the BQM's coefficients.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Imports and setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
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      "Requirement already satisfied: pyparsing>=2.0.2 in /home/ec2-user/anaconda3/envs/Braket/lib/python3.7/site-packages (from packaging>=16.8->bokeh==0.12.15) (2.4.7)\n",
      "Building wheels for collected packages: bokeh\n",
      "  Building wheel for bokeh (setup.py) ... \u001b[?25ldone\n",
      "\u001b[?25h  Created wheel for bokeh: filename=bokeh-0.12.15-py3-none-any.whl size=4939343 sha256=8625923e839a0073dcbe9579c4e1d961e033cac7133f120d9996928bc5bf3b4b\n",
      "  Stored in directory: /home/ec2-user/.cache/pip/wheels/5d/cb/6b/ffc8cf076dc7666fd6f184842ca5949505fdce6d06bec2bc8f\n",
      "Successfully built bokeh\n",
      "Installing collected packages: bokeh\n",
      "  Attempting uninstall: bokeh\n",
      "    Found existing installation: bokeh 2.2.0\n",
      "    Uninstalling bokeh-2.2.0:\n",
      "      Successfully uninstalled bokeh-2.2.0\n",
      "Successfully installed bokeh-0.12.15\n"
     ]
    }
   ],
   "source": [
    "!pip install autopep8\n",
    "!pip install jupyter_contrib_nbextensions==0.5.1\n",
    "!pip install bokeh==0.12.15"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import json\n",
    "from braket.aws import AwsDevice\n",
    "from braket.ocean_plugin import BraketSampler, BraketDWaveSampler\n",
    "\n",
    "from dwave.system.composites import EmbeddingComposite"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Device: Device('name': DW_2000Q_6, 'arn': arn:aws:braket:::device/qpu/d-wave/DW_2000Q_6)\n"
     ]
    }
   ],
   "source": [
    "# session and device\n",
    "device = AwsDevice(\"arn:aws:braket:::device/qpu/d-wave/DW_2000Q_6\")\n",
    "print('Device:', device)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A Simple Example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In alignment with the D-Wave example [here](https://github.com/dwave-examples/factoring-notebook). We firstly show a simple example: a circuit with two switches that turns on a light. \n",
    "\n",
    "<div align=\"center\">\n",
    "    <img src=\"images/example_light_and_switches.png\" width=300x/>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 1: Express as a CSP with Boolean Logic\n",
    "\n",
    "You can express the problem of how to turn on the light as a CSP with a single constraint: for light $L$ to shine, both switches $SW_1$ and $SW_2$ must be on. In logic, you express this constraint as the [conjunction](https://en.wikipedia.org/wiki/Logical_conjunction) $L = SW_1 \\wedge SW_2$. \n",
    "\n",
    "First, express the CSP with binary variables:\n",
    "\n",
    "* Switches $SW_1$ and $SW_2$ are represented by binary variables $x_1, x_2$ with values 1 when on and 0 when off.\n",
    "* Light $L$ is represented by binary variable $x_3$ with value 1 if it's shining and 0 if not.\n",
    "* Logical conjunction, $L = SW_1 \\wedge SW_2$, is expressed in binary format as $x_3 = x_1 \\wedge x_2$. \n",
    "\n",
    "Ocean's [dwavebinarycsp](https://docs.ocean.dwavesys.com/en/stable/docs_binarycsp/sdk_index.html) binary CSP tool provides factories for useful constraints such as logic gates. The first code cell adds an AND gate as a constraint to CSP `and_csp`, defined for binary variables, and tests the CSP for configuration $x1, x2, x3 = 1, 1, 1$ (the bottom row of the AND truth table). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import dwavebinarycsp as dbc\n",
    "\n",
    "and_gate = dbc.factories.and_gate([\"x1\", \"x2\", \"x3\"])\n",
    "and_csp = dbc.ConstraintSatisfactionProblem('BINARY')\n",
    "and_csp.add_constraint(and_gate)\n",
    "\n",
    "# Test the CSP\n",
    "and_csp.check({\"x1\": 1, \"x2\": 1, \"x3\": 1})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 2: Convert to a BQM \n",
    "\n",
    "The D-wave quantum device solves binary quadratic models. Express the light-circuit CSP as a BQM. More than one BQM can represent an AND gate: it's just a polynomial of binary variables, with only linear and quadratic terms, that has lowest value for variables that match rows of the AND truth table; for example, \n",
    "\n",
    "$3x_3 + x_1x_2 - 2x_1x_3 - 2x_2x_3$.\n",
    "\n",
    "To see that this BQM represents the AND gate, you can set its variables to the values of the AND truth table (e.g., $x_1, x_2, x_3=0,0,0$) and to non-valid values (e.g., $ x_1, x_2, x_3=0,0,1$). All the former should produce lower values than any of the latter. \n",
    "\n",
    "The code cell below uses `itertools` to produce all possible configurations of $ x_1, x_2, x_3$, and then prints these (under \"x1, x2, x3\") to the right of the BQM's value (under \"E\"), sorted from low to high.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "E, x1, x2, x3\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[(0, 0, 0, 0),\n",
       " (0, 0, 1, 0),\n",
       " (0, 1, 0, 0),\n",
       " (0, 1, 1, 1),\n",
       " (1, 0, 1, 1),\n",
       " (1, 1, 0, 1),\n",
       " (1, 1, 1, 0),\n",
       " (3, 0, 0, 1)]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import itertools\n",
    "\n",
    "configurations = []\n",
    "for (x1, x2, x3) in  list(itertools.product([0, 1], repeat=3)):\n",
    "     E = 3*x3+x1*x2-2*x1*x3-2*x2*x3\n",
    "     configurations.append((E, x1, x2, x3))\n",
    "\n",
    "configurations.sort()\n",
    "\n",
    "print(\"E, x1, x2, x3\")\n",
    "configurations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we use Ocean's `stitch` function to convert the binary CSP to a BQM. \n",
    "\n",
    "The code cell below does so and prints out the BQM's coefficients, the inputs used to program the D-Wave system. As noted, more than one BQM can represent an AND gate, so the BQM generated here does not have to match the one written above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Linear coefficients:\n",
      "\t{'x1': -0.0, 'x2': -0.0, 'x3': 6.0}\n",
      "Quadratic coefficients:\n",
      "\t{('x1', 'x2'): 2.0, ('x1', 'x3'): -4.0, ('x2', 'x3'): -4.0}\n"
     ]
    }
   ],
   "source": [
    "and_bqm = dbc.stitch(and_csp)\n",
    "and_bqm.remove_offset()\n",
    "\n",
    "print(\"Linear coefficients:\\n\\t{}\".format(\n",
    "      {key: round(val, 2) for key, val in and_bqm.linear.items()}))\n",
    "print(\"Quadratic coefficients:\\n\\t{}\".format(\n",
    "      {key: round(val, 2) for key, val in and_bqm.quadratic.items()}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 3: Solve By Minimization "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lastly, solve the problem by finding variable values that produce the BQM's lowest values. For this trivial example, we use the exact solver, which simply returns the BQM's value for every possible assignment of variable values. For real-world problems, with large numbers of variables and constraints, minimizing a BQM is hard: this is where a quantum computer comes in handy. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  x1 x2 x3 energy num_oc.\n",
      "5  1  1  1   -0.0       1\n",
      "1  1  0  0   -0.0       1\n",
      "3  0  1  0   -0.0       1\n",
      "0  0  0  0    0.0       1\n",
      "6  1  0  1    2.0       1\n",
      "4  0  1  1    2.0       1\n",
      "2  1  1  0    2.0       1\n",
      "7  0  0  1    6.0       1\n",
      "['BINARY', 8 rows, 8 samples, 3 variables]\n"
     ]
    }
   ],
   "source": [
    "from dimod import ExactSolver\n",
    "\n",
    "sampler = ExactSolver()\n",
    "solution = sampler.sample(and_bqm)\n",
    "\n",
    "print(solution)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Factoring on the Quantum Computer\n",
    "\n",
    "This section solves a factoring problem as a CSP, following the same steps used for the simple problem of the light circuit:\n",
    "\n",
    "1. Express factoring as a CSP using Boolean logic operations.\n",
    "2. Convert to a BQM.\n",
    "3. Minimize the BQM."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 1: Express Factoring as Multiplication Circuit\n",
    "\n",
    "Again start by expressing the problem with Boolean logic gates, in this case a multiplication circuit.\n",
    "\n",
    "This example factors integer 21 which is represented as a 6-bit binary number. To express the factoring equation, $P = ab$, in Boolean logic, use a simple 3-bit multiplier (a circuit that takes two 3-bit binary numbers and outputs their 6-bit binary product).  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "010101\n"
     ]
    }
   ],
   "source": [
    "P = 21  \n",
    "\n",
    "bP = \"{:06b}\".format(P)    # \"{:06b}\" formats for 6-bit binary\n",
    "print(bP)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, obtain a multiplication circuit using `multiplication_circuit()` as done before for the AND gate. The next cell creates a CSP for a 3-bit multiplication circuit and prints one of its constraints."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Constraint.from_configurations(frozenset({(1, 0, 0), (1, 1, 1), (0, 1, 0), (0, 0, 0)}), ('a0', 'b0', 'p0'), Vartype.BINARY, name='AND(a0, b0) = p0')\n"
     ]
    }
   ],
   "source": [
    "csp = dbc.factories.multiplication_circuit(3)\n",
    "\n",
    "print(next(iter(csp.constraints)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 2: Convert to a BQM\n",
    "\n",
    "Convert the multiplication circuit as a BQM that provides the coefficients used to program the problem on a D-Wave system."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "BQM has 27 variables: \n",
      "\t['a0', 'b0', 'p0', 'b1', 'and0,1', 'b2', 'and0,2', 'a1', 'and1,0', 'p1', 'carry1,0', 'and1,1', 'sum1,1', 'carry1,1', 'and1,2', 'a2', 'and2,0', 'p2', 'carry2,0', 'and2,1', 'sum2,1', 'carry2,1', 'and2,2', 'p3', 'carry3,0', 'p4', 'p5']\n"
     ]
    }
   ],
   "source": [
    "bqm = dbc.stitch(csp, min_classical_gap=.1)\n",
    "\n",
    "print(\"BQM has {} variables: \\n\\t{}\".format(len(bqm.variables), list(bqm.variables)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The next cell just creates a nice visualization of the BQM. Each node of the graph represents a variable; these include P and its factors as binary numbers, and some internal variables of the multiplication circuit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    var el = document.getElementById(\"f2291a5d-44bc-4a32-9fd8-037a2aceb4bf\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) { callback() });\n    }\n    finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.info(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(js_urls, callback) {\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.log(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.log(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = js_urls.length;\n    for (var i = 0; i < js_urls.length; i++) {\n      var url = js_urls[i];\n      var s = document.createElement('script');\n      s.src = url;\n      s.async = false;\n      s.onreadystatechange = s.onload = function() {\n        root._bokeh_is_loading--;\n        if (root._bokeh_is_loading === 0) {\n          console.log(\"Bokeh: all BokehJS libraries loaded\");\n          run_callbacks()\n        }\n      };\n      s.onerror = function() {\n        console.warn(\"failed to load library \" + url);\n      };\n      console.log(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.getElementsByTagName(\"head\")[0].appendChild(s);\n    }\n  };var element = document.getElementById(\"f2291a5d-44bc-4a32-9fd8-037a2aceb4bf\");\n  if (element == null) {\n    console.log(\"Bokeh: ERROR: autoload.js configured with elementid 'f2291a5d-44bc-4a32-9fd8-037a2aceb4bf' but no matching script tag was found. \")\n    return false;\n  }\n\n  var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-0.12.15.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.12.15.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-0.12.15.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-0.12.15.min.js\"];\n\n  var inline_js = [\n    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\n    \n    function(Bokeh) {\n      \n    },\n    function(Bokeh) {\n      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-0.12.15.min.css\");\n      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-0.12.15.min.css\");\n      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.12.15.min.css\");\n      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-widgets-0.12.15.min.css\");\n      console.log(\"Bokeh: injecting CSS: https://cdn.pydata.org/bokeh/release/bokeh-tables-0.12.15.min.css\");\n      Bokeh.embed.inject_css(\"https://cdn.pydata.org/bokeh/release/bokeh-tables-0.12.15.min.css\");\n    }\n  ];\n\n  function run_inline_js() {\n    \n    if ((root.Bokeh !== undefined) || (force === true)) {\n      for (var i = 0; i < inline_js.length; i++) {\n        inline_js[i].call(root, root.Bokeh);\n      }if (force === true) {\n        display_loaded();\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      var cell = $(document.getElementById(\"f2291a5d-44bc-4a32-9fd8-037a2aceb4bf\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.log(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(js_urls, function() {\n      console.log(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));",
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      "application/vnd.bokehjs_exec.v0+json": ""
     },
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       "id": "31612af7-3b9c-4c5f-bdc7-30d89b16caff"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from helpers import draw\n",
    "\n",
    "draw.circuit_from(bqm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The D-Wave system factors this integer by running a multiplication circuit in reverse. The code below fixes the variables of the multiplication circuit's BQM to the binary digits of the factored number P (represented by binary variables P0, P1...P6). This modifies the BQM by removing the known variables and updating neighboring values accordingly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "BQM has 21 non-fixed variables: \n",
      "\t['a0', 'b0', 'b1', 'and0,1', 'b2', 'and0,2', 'a1', 'and1,0', 'carry1,0', 'and1,1', 'sum1,1', 'carry1,1', 'and1,2', 'a2', 'and2,0', 'carry2,0', 'and2,1', 'sum2,1', 'carry2,1', 'and2,2', 'carry3,0']\n"
     ]
    }
   ],
   "source": [
    "p_vars = ['p0', 'p1', 'p2', 'p3', 'p4', 'p5']\n",
    "\n",
    "# Convert P from decimal to binary\n",
    "fixed_variables = dict(zip(reversed(p_vars), \"{:06b}\".format(P)))\n",
    "fixed_variables = {var: int(x) for(var, x) in fixed_variables.items()}\n",
    "\n",
    "# Fix product variables\n",
    "for var, value in fixed_variables.items():\n",
    "    bqm.fix_variable(var, value)\n",
    "    \n",
    "print(\"BQM has {} non-fixed variables: \\n\\t{}\".format(len(bqm.variables), list(bqm.variables)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 3: Submit to the Quantum Computer\n",
    "\n",
    "Lastly, solve the BQM with the D-wave device by finding variable assignments that produce its lowest values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setting Up a Solver\n",
    "\n",
    "The code below sets up a D-wave solver as a QPU supported with Braket. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "sampler = BraketDWaveSampler(device_arn='arn:aws:braket:::device/qpu/d-wave/DW_2000Q_6')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A problem can be minor embedded onto the QPU in a variety of ways and this affects solution quality and performance. The cell below uses `FixedEmbeddingComposite()` with a pre-calculated minor-embedding."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mapping from BQM variables to qubits for the Dwave system:\n",
      "\n",
      "\t{} {'and2,1': [590, 582, 450, 578], 'sum1,1': [469, 464, 592, 720], 'carry1,0': [727, 851, 723], 'and2,2': [577, 449, 455, 448], 'a1': [707, 711, 719], 'and2,0': [709, 717, 725], 'and0,1': [854, 843, 846], 'a0': [585, 841, 713], 'and0,2': [456, 597, 584, 589], 'carry2,1': [457, 454, 462], 'carry2,0': [722, 594, 466], 'sum2,1': [452, 467, 468, 460], 'a2': [710, 704, 576], 'b0': [840, 712, 718], 'b1': [706, 834, 842, 839, 847], 'b2': [579, 583, 591], 'and1,2': [715, 587, 459], 'carry3,0': [471, 463], 'and1,0': [835, 837, 853, 845], 'and1,1': [586, 724, 716, 714], 'carry1,1': [458, 453, 461]}\n"
     ]
    }
   ],
   "source": [
    "from helpers.embedding import embeddings\n",
    "from dwave.system import FixedEmbeddingComposite\n",
    "\n",
    "sampler_fixed_embedding = FixedEmbeddingComposite(sampler, embeddings['DW_2000Q_6'])\n",
    "\n",
    "print(\"Mapping from BQM variables to qubits for the Dwave system:\\n\\n\\t{}\", sampler_fixed_embedding.embedding)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Submit the sample to the D-wave device to solve the problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best solution found: \n",
      " {'a0': 1, 'a1': 1, 'a2': 0, 'and0,1': 1, 'and0,2': 1, 'and1,0': 1, 'and1,1': 1, 'and1,2': 1, 'and2,0': 0, 'and2,1': 0, 'and2,2': 0, 'b0': 1, 'b1': 1, 'b2': 1, 'carry1,0': 1, 'carry1,1': 1, 'carry2,0': 0, 'carry2,1': 1, 'carry3,0': 0, 'sum1,1': 1, 'sum2,1': 0}\n"
     ]
    }
   ],
   "source": [
    "sampleset = sampler_fixed_embedding.sample(bqm, num_reads=100)\n",
    "print(\"Best solution found: \\n\",sampleset.first.sample)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Viewing the Solution\n",
    "\n",
    "Convert back from binary numbers to integers. Because quantum computing is probabilistic, there is a slight chance that in many executions of this example, your execution might return an incorrect solution. Rerunning the previous cell will most likely produce a correct answer. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Given integer P=21, found factors a=3 and b=7\n"
     ]
    }
   ],
   "source": [
    "from helpers.convert import to_base_ten\n",
    " \n",
    "a, b = to_base_ten(sampleset.first.sample)\n",
    "\n",
    "print(\"Given integer P={}, found factors a={} and b={}\".format(P, a, b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The next cell views the energy (rounded to two decimal places) of the samples, using a `dict` mapping pairs `(a, b)` to information about them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OrderedDict([((3, 7), 2.0),\n",
       "             ((7, 3), 3.0),\n",
       "             ((5, 3), 3.0),\n",
       "             ((7, 1), 3.0),\n",
       "             ((3, 6), 4.0),\n",
       "             ((7, 5), 5.0),\n",
       "             ((3, 1), 6.0),\n",
       "             ((5, 5), 7.0)])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from collections import OrderedDict\n",
    "\n",
    "def response_to_dict(sampleset):\n",
    "    results_dict = OrderedDict()\n",
    "    for sample, energy in sampleset.data(['sample', 'energy']):\n",
    "        a, b = to_base_ten(sample)\n",
    "        if (a, b) not in results_dict:\n",
    "            results_dict[(a, b)] = round(energy, 2)\n",
    "            \n",
    "    return results_dict\n",
    "\n",
    "results = response_to_dict(sampleset)\n",
    "results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a scatter plot of the samples and their energies, showing that the lowest energy states correspond to correct answers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
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    "## Alternative minor-embedding\n",
    "\n",
    "The pre-calculated minor-embedding used above was selected for its good performance from many candidate embeddings generated by a heuristic algorithm called [minorminer](https://docs.ocean.dwavesys.com/en/stable/docs_minorminer/source/sdk_index.html). Alternatively, you can use the function `EmbeddingComposite()` to generate a new embedding that may or may not have as good performance. Here, ask for more samples from the QPU by raising the `num_reads` parameter to 1000 to increase your chances of getting a correct answer. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Best solution found: \n",
      " {'a0': 1, 'a1': 1, 'a2': 1, 'and0,1': 1, 'and0,2': 0, 'and1,0': 1, 'and1,1': 1, 'and1,2': 0, 'and2,0': 1, 'and2,1': 1, 'and2,2': 0, 'b0': 1, 'b1': 1, 'b2': 0, 'carry1,0': 1, 'carry1,1': 0, 'carry2,0': 1, 'carry2,1': 0, 'carry3,0': 1, 'sum1,1': 1, 'sum2,1': 1}\n"
     ]
    }
   ],
   "source": [
    "from dwave.system import EmbeddingComposite\n",
    "\n",
    "sampleset = EmbeddingComposite(sampler).sample(bqm, num_reads=1000)\n",
    "print(\"Best solution found: \\n\",sampleset.first.sample)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "OrderedDict([((7, 3), -0.0),\n",
       "             ((5, 5), 1.0),\n",
       "             ((3, 3), 2.0),\n",
       "             ((1, 7), 2.0),\n",
       "             ((3, 5), 3.0),\n",
       "             ((5, 7), 4.0),\n",
       "             ((4, 7), 4.0),\n",
       "             ((7, 2), 4.0),\n",
       "             ((1, 5), 4.0),\n",
       "             ((0, 7), 5.0),\n",
       "             ((0, 3), 5.0),\n",
       "             ((7, 4), 9.0)])"
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     "execution_count": 20,
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   ],
   "source": [
    "results = response_to_dict(sampleset)\n",
    "results"
   ]
  },
  {
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   "execution_count": 21,
   "metadata": {},
   "outputs": [
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