zero_sum_subset_main.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "##### >> If you have never used a Jupyter notebook before, check out the documentation before running the code: https://jupyter-notebook-beginner-guide.readthedocs.io/ and install Anaconda <<"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Zero-sum subset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are given a set of integer numbers, for example $\\left\\{2,5,4,-3,-11\\right\\}$. Right now, we have chosen a very small set of only $5$ numbers, but in principal our set could be very large. \n",
    "\n",
    "The goal is now to find a subset of these numbers that add up to zero. In our case, one can check that the solution is the subset $\\left\\{2,5,4,-11\\right\\}$, since $2+5+4-11=0$. Another possible solution is always $\\{\\}$, the empty set. But this is not very interesting and we will call it the trivial solution.\n",
    "\n",
    "\n",
    "To make everything more general, we can replace these numbers by $N$ variables $C=\\left\\{ c_1, c_2, c_3, \\dots, c_N\\right\\}$, where the dots ($\\dots$) imply that you have to fill in the remaining numbers.\n",
    "\n",
    "To specify a subset, we will introduce another set of variables $X=\\left\\{ x_1, x_2, x_3, \\dots, x_N\\right\\}$. We will require that any of these variables $x_i$ are either $0$ or $1$. Sometimes these are called binary variables. If $x_i=1$, it means that $c_i$ is included in the subset. If $x_i=0$, the number $c_i$ is not included.\n",
    "\n",
    "In this notation, we can compute the sum $S$ of all numbers in a subset as\n",
    "\n",
    "$S=x_1 \\cdot c_1 + x_2 \\cdot c_2 + x_3 \\cdot c_3 + \\dots + x_N \\cdot c_N$.\n",
    "\n",
    "You can see that the notation using the dots ($\\dots$) becomes a little cumbersome. For this reason, people have come up with the following notation:\n",
    "\n",
    "$S =  \\sum_{i=1}^{N} x_i \\cdot c_i$\n",
    "\n",
    "The $\\sum$ symbol (greek S) is an abbreviation for the following instructions: Replace all $i$ to the right of it by $1$ (you get $x_1 \\cdot c_1$), then $2$ (you get $x_2 \\cdot c_2$), then $3$, and all the way up to $N$ and then sum all of these terms. Voilà, you can see that you get the original sum $S$.\n",
    "\n",
    "***\n",
    "### Exercise\n",
    "\n",
    "1. *For the set $\\left\\{2,5,4,-3,-11\\right\\}$, what is the set $X=\\{x_1, x_2, x_3, x_4, x_5\\}$ that specifies the zero-sum subset?*\n",
    "\n",
    "\n",
    "2. *What is the subset corresponding to $X=\\{1,0,1,1,0\\}$ and what is its sum?*\n",
    "\n",
    "\n",
    "3. *How about $X=\\{0,0,0,0,0\\}$?*\n",
    "\n",
    "***\n",
    "\n",
    "\n",
    "\n",
    "### Let's code\n",
    "\n",
    "Below is a set of numbers $C$. Change it to something else if you want. \n",
    "\n",
    "To proceed you need to run the code inside the cell. You do this by putting the cursor inside the cell and pressing Shift + Enter. After succesful execution, the text \"In [  ]\" to the left of the cell will be replaced with \"In[1]\" where \"1\" indicates that this is the first cell you have executed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "C = [2,5,4,-3,-11]"
   ]
  },
  {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now want to use our computer to find a zero-sum subset. The simplest approach is to build all possible subsets and check if they sum to zero. \n",
    "\n",
    "For this, we first need to teach our computer how to sum a subset, in the following exercise.\n",
    "\n",
    "***\n",
    "### Exercise\n",
    "\n",
    "*Can you write an implementation of the function $\\text{sum_set}(C,X)$? Given a set of numbers $C=\\{c_1,\\dots,c_N\\}$ and the binary set $X=\\{x_1,\\dots,x_N\\}$, it is supposed to return the sum $S =  \\sum_{i=1}^{N} x_i \\cdot c_i$ as discussed above.*\n",
    "\n",
    "*In the python code, the elements of the set $C$ and $X$ are retrieved using $C[i]$ and $X[i]$, respectively. Note that the first element is $C[0]$ and the last element is $C[N-1]$, i.e. python starts counting at 0! The following table shows an example for $C=\\{2,5,4,-3,-11\\}$ and $X=\\{0,1,1,0,1\\}$.* \n",
    "\n",
    "![table1.jpg](attachment:table1.jpg)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sum_set(C,X):\n",
    "    # takes sets C={c_1,...,c_N} and X={x_1,...,x_N} and returns x_1*c_1 + x_2*c_2 + ... + x_N*c_N\n",
    "    s = 0 # this is the variable we want to store the sum in, initially it is 0.\n",
    "    for i in np.arange(len(X)): # i is a number from 0 to N-1\n",
    "        s += C[i] * X[i]\n",
    "    return s"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "The code below builds all possible subsets and checks if any of them sum to zero. Run it and check that it gives the expected output!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2 + 5 + 4 - 11 = 0\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import time\n",
    "\n",
    "def solve_exactly(C):\n",
    "    \"\"\"\n",
    "    Given a set of numbers C={c_1,...,c_N} this function \n",
    "    returns: exact_result, a list of all sets X that correspond to zero-sum subset\n",
    "             duration, the time it took your computer to find all zero-sum subsets\n",
    "    \"\"\"\n",
    "    exact_result = []\n",
    "    start = time.time()\n",
    "    N = len(C) # N numbers in set S={c_1,...,c_N}\n",
    "    for i in np.arange(2**N):\n",
    "        X = np.array([int(x_i) for x_i in (list(format(i, 'b').zfill(N)))]) # this is the set X={x_1,...,x_N}.\n",
    "        # In every iteration of the for-loop X will be a different combination of 1 and 0 corresponding to a \n",
    "        # different subset\n",
    "        S = sum_set(C,X) # compute sum of subset\n",
    "        if S == 0:\n",
    "            # if the sum of the subset is zero, add it to the results\n",
    "            exact_result.append(X)\n",
    "    end = time.time()\n",
    "    duration = end - start\n",
    "    return exact_result, duration\n",
    "\n",
    "def print_result(C, results):\n",
    "    \"\"\"\n",
    "    prints all subsets\n",
    "    \"\"\"\n",
    "    for sample in results:\n",
    "        answer = ''\n",
    "        s = 0\n",
    "        for i in np.arange(len(sample)):\n",
    "            if sample[i] == 1:\n",
    "                if C[i] > 0:\n",
    "                    answer += \" + \"\n",
    "                else:\n",
    "                    answer += \" - \"\n",
    "                answer += str(abs(C[i]))\n",
    "                s += C[i]\n",
    "\n",
    "        if len(answer) > 0: # don't print anything if it's the trivial set\n",
    "            print(answer[3:],'=', s)\n",
    "\n",
    "exact_result, duration = solve_exactly(np.array(C))\n",
    "print_result(C, exact_result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How fast is your computer?\n",
    "\n",
    "A very important property of our code is how long it takes to run for a specific size of the set of numbers, $N$.\n",
    "\n",
    "We can figure this out by asking the following question:\n",
    "***\n",
    "### Exercise\n",
    "\n",
    "*How many different subsets are there?*\n",
    "***\n",
    "\n",
    "Well, each number can either be included or not included in a subset. If I just have one number, there is only two options. If I have two numbers, there are $2\\cdot 2=2^2=4$ options: include none, include only the first number, include only the second number, include both. \n",
    "For three numbers, we have $2\\cdot 2\\cdot 2 =2^3= 8$ subsets. And in general, for $N$ numbers, we have $2^N$ subsets.\n",
    "\n",
    "If it takes some time $a$ to build a subset, and we are building $2^N$ subsets, the total time is $a \\cdot 2^N$.\n",
    "\n",
    "But now we also need to also need to sum all subsets. This takes the time $b\\cdot N\\cdot 2^N$. \n",
    "\n",
    "The total computation time, will be $a \\cdot 2^N+ b \\cdot N \\cdot 2^N$. Since the second term gets larger much faster than the first one, the first term is not that important and it is a good approximation to neglect it.\n",
    "\n",
    "So the total time needed is approximately $b \\cdot N\\cdot 2^N$. We call this *exponential scaling* of order $\\mathcal{O}\\left(N \\cdot 2^N\\right)$. The notation with the $\\mathcal{O}$ (\"order\") simply means that we don't really care about the prefactor $b$, but only about the $N$-dependence.\n",
    "\n",
    "Below is a code script that builds all possible subsets of sets of different size $N$ and measures the time needed for that."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.optimize import curve_fit\n",
    "import random\n",
    "\n",
    "def fit_function(N,a):\n",
    "    return a*N*2**N\n",
    "\n",
    "set_sizes = np.arange(5,18) # perform simulations for sets of size 5 to 19\n",
    "sets = []\n",
    "durations = []\n",
    "for size in set_sizes:\n",
    "    C_random = ((np.random.rand(size)-0.5)*10).astype(int) # get a random set of integer numbers \n",
    "    sets.append(C_random)\n",
    "    exact_result, duration = solve_exactly(C_random)\n",
    "    durations.append(duration)\n",
    "\n",
    "popt, pcov = curve_fit(fit_function, set_sizes[7:],durations[7:])\n",
    "x = np.linspace(min(set_sizes),max(set_sizes),1000)\n",
    "y = fit_function(x, *popt)\n",
    "plt.plot(x, y, 'k--', label='$b \\cdot N \\cdot 2^N$');\n",
    "plt.plot(set_sizes, durations,'r.', label='measured time');\n",
    "plt.xlabel('Size of Set $N$');\n",
    "plt.ylabel('Computation Time in Seconds');\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The red data points show the time your computer needed to build all subsets of a set of $N$ randomly generated numbers. The black dashed curve is the function $b \\cdot N \\cdot 2^N$ where $b$ has been chosen to fit the red dots. \n",
    "\n",
    "A different way of plotting is to use a log scale. For this, the y-axis is streched in a way that makes an exponential function look linear. This way of visualizing the data makes it easier to anticipate how long the computation time will take for large $N$.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(min(set_sizes),60,1000)\n",
    "y = fit_function(x, *popt)\n",
    "seconds_per_week = 604800\n",
    "plt.semilogy(x, y/seconds_per_week, 'k--');\n",
    "plt.semilogy(set_sizes, np.array(durations)/seconds_per_week,'r.');\n",
    "plt.xlabel('Size of Set $N$');\n",
    "plt.ylabel('Computation Time in Days');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "### Exercise\n",
    "\n",
    "\n",
    "*How long would it supposedly take our code to check all subsets of a set of $N=50$ numbers?*\n",
    "***"
   ]
  },
  {
   "attachments": {
    "table2.jpg": {
     "image/jpeg": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Can a Quantum Computer Help?\n",
    "\n",
    "We have seen that it becomes nearly impossible to solve the zero sum subset problem when $N$ gets too large. \n",
    "\n",
    "So we need a different approach: We will use a quantum computer. There are three different types of quantum computation. Here, we will focus on an approach called *adiabatic quantum computation*. The leading company who develops this type of quantum computer id D-Wave. It is located in Burnaby and we will use their cloud service to run programs on the quantum computer in real time.\n",
    "\n",
    "Remember the soap bubble experiment, where a physical system minimzed the total length of a Steiner tree and thereby solved a computationally hard math problem. \n",
    "\n",
    "The D-Wave quantum computer also minimizes an object. Instead of a Steiner tree this object is a mathematatical function $E(x_1,x_2,\\dots,x_N)$ that depends on a bunch of binary variables $x_i$. This is more abstract, but it also turns out to be more useful in that it allows to solve a variety of different problems. The specific way of the minimization relies on quantum effects and we won't have time to explain it in this lecture.\n",
    "\n",
    "For now, all we need to know is:\n",
    "*The D-Wave quantum computer minimizes the function $ E(x_1,x_2,\\dots,x_N) $ which is given by*\n",
    "\n",
    "$E(x_1,x_2,\\dots,x_N) = \\sum_{i=1}^{N}\\sum_{j=1}^{N} a_{i,j} \\cdot x_i \\cdot x_j$\n",
    "\n",
    "Here, the $x_i$ are binary variables that can only have the value $0$ or $1$ and the $a_{i,j}$ are coefficients that are real numbers and can be adjusted to suit the math problem we want to solve.\n",
    "\n",
    "Notice that we have used the sum-notation again. This time, there are two $\\sum$-symbols. The strategy is the same as previously: Replace $i$ and $j$ by all combinations of numbers between $1$ and $N$ and sum up all terms. It looks like this: $E=a_{1,1} \\cdot x_1 \\cdot x_1+ a_{2,1} \\cdot x_2 \\cdot x_1 + a_{1,2} \\cdot x_1 \\cdot x_2 + \\dots$\n",
    "\n",
    "To be a bit more clear, we can also list all terms of the sum up to $i=4$ and $j=4$ in a table:\n",
    "\n",
    "![table2.jpg](attachment:table2.jpg)\n",
    "\n",
    "***\n",
    "### Exercise\n",
    "1. *Write out a few terms of the sum explicitly, by replacing $i$ and $j$ by numbers.*\n",
    "\n",
    "\n",
    "2. *How many terms in the sum are there in terms of $N$?*\n",
    "***\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# How is this helpful for our problem?\n",
    "\n",
    "Remeber that we want to find a subset whose sum is $S =  \\sum_{i=1}^{N} x_i \\cdot c_i = 0$.\n",
    "\n",
    "Now consider the square of the sum $S^2$. The square of a number is always positive. And the smallest positive number is $0$. But if $S^2=0$ it follows that $S=0$ which is precisely the condition that we have a zero-sum subset. Therefore, minimizing $S^2$ will give us the answer.\n",
    "\n",
    "Let us do the math:\n",
    "\n",
    "$S^2 = \\left(\\sum_{i=1}^{N} x_i \\cdot c_i \\right)^2 = \\left(\\sum_{i=1}^{N} x_i \\cdot c_i \\right)\\left(\\sum_{j=1}^{N} x_j \\cdot c_j \\right) = \\sum_{i=1}^{N}\\sum_{j=1}^{N} c_i \\cdot c_j \\cdot x_i \\cdot x_j$\n",
    "\n",
    "\n",
    "***\n",
    "\n",
    "### Exercise\n",
    "\n",
    "1. *Can you get an idea that above calculation is correct by writing out some terms of the double-sum? If all the different terms confuse you, you could also arrange them in a table.*\n",
    "\n",
    "\n",
    "2. *Calculate $S^2$ for the set $C=\\{3,1,-4\\}$. (This means that $c_1=3, c_2=1, c_3=-4$ and $N=3$)*\n",
    "***\n",
    "\n",
    "We see that $S^2$ exactly looks like $E(x_1,x_2,\\dots,x_N)$ if we identify\n",
    "\n",
    "$a_{i,j} = c_i \\cdot c_j$ .\n",
    "\n",
    "And as we discussed before, the D-Wave quantum computer's purpose is to minimize $E$. \n",
    "\n",
    "The upshot is that we only need to hand the coefficients $a_{i,j}$ to the D-Wave quantum computer. The results of the computation will then be the set $X=\\{x_1,x_2,\\dots,x_N\\}$ that specifies the zero-sum subset."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "### Exercise\n",
    "\n",
    "*Implement the function $\\text{coefficient_a}(i,j,C)$ that takes indices $i,j$ and the set of numbers $C$ as input and returns the coefficient $a_{i,j} = c_i \\cdot c_j$.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def coefficient_a(i,j, C):\n",
    "    return C[i] * C[j]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now implement the function $\\text{coefficient_matrix_a}(C)$ that returns a table with entries $a_{i,j}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def coefficient_matrix_a(C):\n",
    "    a = np.zeros((len(C),len(C)))\n",
    "    for i in np.arange(len(C)):\n",
    "        for j in np.arange(len(C)):\n",
    "            a[i,j] = coefficient_a(i,j, C)\n",
    "    return a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*If you want you can choose a new set of numbers:*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "C = np.array([1,3,2,-3,-4,-4,3,2,6,3,-2,3,-5,4,-2,1-5,-4,2,-5,-1,4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "Now we will solve the problem on the quantum computer."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### >> In order to run the following code, you first need to create an account at D-Wave Leap (https://www.dwavesys.com/take-leap) and follow the instructions to install dwave-ocean-sdk. It's free! <<"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3 - 4 + 2 + 3 - 5 + 4 - 2 - 4 - 1 + 4 = 0\n",
      "1 - 3 - 4 + 2 + 6 + 3 - 5 + 4 - 2 - 4 - 1 + 4 = 1\n",
      "1 + 2 - 3 - 4 + 3 + 6 - 2 + 3 - 4 + 2 - 5 = -1\n",
      "1 + 2 - 3 - 4 + 3 + 3 + 2 - 5 = -1\n",
      "1 - 3 - 4 + 2 + 6 + 3 - 5 + 4 - 2 - 4 + 4 = 2\n",
      "1 + 3 - 4 + 2 + 6 + 3 - 2 - 5 + 4 - 2 - 4 - 4 + 4 = 2\n",
      "3 + 2 - 4 - 4 + 3 - 2 - 5 - 1 + 4 = -4\n",
      "3 + 2 - 3 - 4 - 4 + 3 - 2 + 3 + 2 - 5 = -5\n",
      "2 - 4 + 3 + 2 + 6 + 3 - 2 + 4 - 4 - 5 - 1 + 4 = 8\n",
      "1 - 3 - 4 + 6 - 2 + 3 - 4 - 4 + 2 - 5 - 1 = -11\n",
      "It took the quantum computer 12.545 milliseconds to perform the minimization 10 times\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import dwavebinarycsp\n",
    "from dimod import BinaryQuadraticModel\n",
    "from dwave.system.samplers import DWaveSampler\n",
    "from dwave.system.composites import EmbeddingComposite\n",
    "\n",
    "def solve_on_quantum_computer(C, num_reads=1):\n",
    "    QUBO = coefficient_matrix_a(C)\n",
    "    bqm = BinaryQuadraticModel.from_numpy_matrix(QUBO)\n",
    "    sampler = EmbeddingComposite(DWaveSampler())\n",
    "    response = sampler.sample(bqm, num_reads=num_reads, return_embedding=True)\n",
    "    duration_in_ms = response.info['timing']['qpu_access_time']/1000\n",
    "    return response, duration_in_ms\n",
    "\n",
    "num_reads = 10\n",
    "response, duration = solve_on_quantum_computer(C, num_reads=num_reads)\n",
    "print_result(C, [x_i.sample for x_i in response.data(['sample'])])\n",
    "print('It took the quantum computer', response.info['timing']['qpu_access_time']/1000, \\\n",
    "      'milliseconds to perform the minimization', num_reads, 'times')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You will see that the D-Wave quantum computer gives you a bunch of different subsets as an answer and many, in some cases even all of them, don't sum to zero. However, you will notice that the absolute values of all of the sums are quite small.\n",
    "\n",
    "In an ideal world, the D-Wave quantum computer would always return the correct answer. Real-world imperfections, however, lead to wrong results. But this isn't so bad. Given an answer to our problem, it is very easy to check if that answer is correct: We just have to sum up the terms. And summing up numbers has complexity $O(N)$, as we learned, so it's very easy!\n",
    "\n",
    "So, the D-Wave computer has a tendency to get answers right, but can be wrong many times. We repeat the computation a bunch of times and check the results. As long as at least one of the answers is correct, we are good!\n",
    "\n",
    "***\n",
    "### Exercise\n",
    "\n",
    "1. *In practice, one may have to run a problem on the D-Wave machine many times. In the code above, it is run $10$ times. Adjust the parameter $\\text{num_reads}$ to somewhat larger numbers. Can you get a zero-sum subset?*\n",
    "\n",
    "\n",
    "2. *Change the set $C$ and see what happends. What happends if you make the set very large?*\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Is it Faster?\n",
    "\n",
    "Congratulations for having it made so far! You are now one of the few people in the world who can say to have programmed a quantum computer.\n",
    "\n",
    "But here is the most important question: Is all of this useful? \n",
    "\n",
    "Previously, we saw that the brute-force solution on your computer became infeasible for large set sizes $N$ because of *exponential scaling*. How does the computation time for the D-Wave quantum computer change with $N$? Run the following code to find out."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "duration_quantum_computer = []\n",
    "for i in np.arange(len(sets)):\n",
    "    quantum_result, quantum_duration = solve_on_quantum_computer(sets[i], num_reads=1)\n",
    "    duration_quantum_computer.append(quantum_duration)\n",
    "\n",
    "plt.plot(set_sizes, durations,'r.-', label=\"Your computer\");\n",
    "plt.plot(set_sizes, np.array(duration_quantum_computer)/1000*100,'b.-', label=\"D-Wave (100 runs each)\");\n",
    "plt.xlabel('Size of Set');\n",
    "plt.ylabel('Computation Time in Seconds');\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The answer is: the computation time does not depend on $N$! \n",
    "\n",
    "We conclude with the main take-away from this lecture: *Quantum Computers may allow us to tackle problems that would be impossible to solve otherwise.*"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}