Notebooks/Find_Edge.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finding the edge of one pinture using Quantum Paralelism\n",
    "\n",
    "This notebook is based on the paper:\n",
    "\n",
    "[Yao, X.-W.](https://arxiv.org/abs/1801.01465), et al. (2017). Quantum Image Processing and Its Application to Edge Detection: Theory and Experiment. Physical Review X, 7(3), 031041. http://doi.org/10.1103/PhysRevX.7.031041\n",
    "\n",
    "The main objective of this algorithm is to find the edge of one image. This means, given the image:\n",
    "\n",
    "<img src=\"./Images/C.png\">\n",
    "\n",
    "the objective is to find its edges to get this other image:\n",
    "\n",
    "<img src=\"./Images/C_EDGES.png\">\n",
    "\n",
    "The algorithm has the next steps:\n",
    "\n",
    "1. Load the image as a matrix os (n,m) elements $c_{ij}$\n",
    "$$D=\\begin{bmatrix} \n",
    "d_{11} & d_{12} & \\cdots&d_{1j}&\\cdots&d_{1m}  \\\\\n",
    "d_{21} & d_{22} & \\cdots&d_{2j}&\\cdots &d_{2m}  \\\\\n",
    "\\vdots&&\\ddots&\\vdots&\\ddots&\\vdots\\\\\n",
    "d_{i1} & d_{i2} & \\cdots&d_{ij}&\\cdots &d_{im}  \\\\\n",
    "\\vdots&&\\ddots&\\vdots&\\ddots&\\vdots\\\\\n",
    "d_{n1} & d_{n2} & \\cdots&d_{nj}&\\cdots &d_{nm}  \\\\\n",
    "\\end{bmatrix}$$\n",
    "2. Convert the array in a vector of N=n\\*m elements\n",
    "$$C=\\begin{bmatrix} \n",
    "d_{11}   \\\\\n",
    "d_{21}   \\\\\n",
    "\\vdots\\\\\n",
    "d_{ij}   \\\\\n",
    "d_{(i+1) j}   \\\\\n",
    "\\vdots\\\\\n",
    "d_{nm} \\\\\n",
    "\\end{bmatrix}=\n",
    "\\begin{bmatrix} \n",
    "c_{0}   \\\\\n",
    "c_{1}   \\\\\n",
    "\\vdots\\\\\n",
    "c_{N-2}   \\\\\n",
    "c_{N-1}   \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "3. Map this vector to one qureg of $q=log_2(N)$ qubits as\n",
    "$${\\lvert}\\phi(C){\\rangle}=\\frac{1}{{\\left\\lVert C \\right\\rVert}^2}\\sum_{k=0}^{N-1}c_{k}{\\lvert}k{\\rangle}$$\n",
    "4. Apply a Hadamard gate to the last qubit\n",
    "$$(I^{\\otimes q-1}\\otimes H)\\lvert\\phi(C)\\rangle=\\begin{bmatrix} \n",
    "c_{0}+c_{1}   \\\\\n",
    "c_{0}-c_{1}   \\\\\n",
    "\\vdots\\\\\n",
    "c_{N-2}+c_{N-1}\\\\\n",
    "c_{N-2}-c_{N-1}\\\\\n",
    "\\end{bmatrix}$$\n",
    "So, even elements will be close to 0 is two pixels have close values -in the case of black and white images, they must be 0-. Measuring the last qubit, if the result is one, the final states $\\lvert k \\rangle$, encode the pixels of the boundaries.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Start importing the packages to use. To load and convert the image, PIL package is needed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "from projectq.cengines import MainEngine\n",
    "from projectq.ops import H,_state_prep\n",
    "from PIL import Image\n",
    "import numpy as np\n",
    "from numpy import linalg as LA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load the image. Because it is too large, rescale it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7fb708e18c50>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "A=Image.open(\"Images/cesga_unha_tinta.jpg\").convert(\"1\")\n",
    "maxsize=(A.size[0]//4,A.size[1]//4)\n",
    "A.thumbnail(maxsize, Image.ANTIALIAS)\n",
    "im_arr = (np.array(A)==False).astype(np.int8)\n",
    "plt.imshow(im_arr, cmap='Greys' )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because even after rescaling the image is to large to simulate, crop it around **C**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7fb708d87da0>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "im_crop=im_arr[46:120,265:320]\n",
    "plt.imshow(im_crop, cmap='Greys' )\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate the number of qubits needed to store all the image. Remember that n qubits can store $N=2^ n$ complex numbers. As consequence, to store m\\*n pixels, a qureg of $q=\\log_2(n*m)$ qubits is used"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "12\n"
     ]
    }
   ],
   "source": [
    "number_of_qubits=int(np.ceil(np.log2(im_crop.shape[0]*im_crop.shape[1])))\n",
    "print(number_of_qubits)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a vector with the number of elements that number_of_qubits can store"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Print 12 qubits can store 4096 elements.\n"
     ]
    }
   ],
   "source": [
    "Vector=np.zeros((2**number_of_qubits))\n",
    "print(\"Print %d qubits can store %d elements.\"%(number_of_qubits,Vector.shape[0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, map the image to the first elements of this vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "shape=im_crop.shape[0]*im_crop.shape[1]\n",
    "Vector[0:shape]=im_crop.reshape((shape))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because the norm of the amplitudes of one quantum state must be one, we must convert this vector to a normalized one, dividing each element by the norm of the vector. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import linalg as LA\n",
    "norma=LA.norm(Vector)\n",
    "Vnorma=Vector/norma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Allocate one qureg with the number of needed qubits."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "eng=MainEngine()\n",
    "q=eng.allocate_qureg(number_of_qubits)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, use the operation [StatePreparation](https://projectq.readthedocs.io/en/latest/projectq.ops.html#projectq.ops.StatePreparation) to generate the quantum circuit to map the Vnorma vector to one quantum state. This operation will generate the set of gates to transform the initial state $\\lvert 0 \\rangle^{\\otimes q}$ to the desired final state $\\lvert \\phi \\rangle$ using the algorithm proposed by [Mikko Mottonen et al.](https://arxiv.org/abs/quant-ph/0407010). There are other methods implemented on the literature that deserve to be explored because this is a critical point of many modern algorithms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "CreateState=_state_prep.StatePreparation(Vnorma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK. Apply this circuit to the allocated qureg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "CreateState | q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Do not to forget that any operation must be executed before getting the results. Because this circuit is simulated and not executed on a quantum processor, it will consume some minutes to finish. Be patient!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "eng.flush()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the simulator, it is possible to obtain the current state and probabilities without measuring it. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "Vout=np.zeros(shape)\n",
    "for i in range(shape):\n",
    "    bits=np.binary_repr(i,width=len(q))\n",
    "    Vout[i]=abs(eng.backend.get_probability(bits[-1::-1],q))\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remember that the amplitude of one state is the square root of the probability. Plot the results and compare the initial normalized vector with the amplitudes of the final state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fb7084cdef0>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 720x216 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax=plt.subplots(1,2,figsize=(10,3))\n",
    "ax[0].plot(Vout**0.5,\"r*\",label=\"State\")\n",
    "ax[0].legend()\n",
    "ax[0].set_ylabel(\"Amplitude\")\n",
    "ax[1].plot(Vnorma,\"*\",label=\"Vnorma\")\n",
    "ax[1].legend()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7fb7084a2828>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "IMOutput=np.round((Vout**0.5)*norma)\n",
    "plt.imshow(IMOutput.reshape((im_crop.shape[0],im_crop.shape[1])), cmap='Greys')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, apply the Hadamard gate to the last qubit. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "H|q[0]\n",
    "eng.flush()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the real case, a large number of measurements of the final state have to be done to obtain the probabilities of the final state. In fact, the algorithm needs to measure the last qubit and, when the results is one, the result state is the solution.However, in the simulator, you can recover them directly, and take only those that are due to last state equal to 1, this means, the natural odd numbers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "eng.backend.collapse_wavefunction([q[0]], [1])\n",
    "Vout=np.zeros(shape)\n",
    "IMOutputOdd=np.zeros(shape)\n",
    "for i in range(shape):\n",
    "    bits=np.binary_repr(i,width=len(q))\n",
    "    Vout[i]=abs(eng.backend.get_probability(bits[-1::-1],q))\n",
    "    if (Vout[i]>0.0001):\n",
    "        IMOutputOdd[i]=1.0\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'New transformed image')"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 864x216 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig,ax=plt.subplots(1,3,figsize=(12,3))\n",
    "ax[0].plot(Vout**0.5,\"r*\",label=\"Final state\")\n",
    "ax[0].set_title(\"Final state\")\n",
    "ax[0].set_ylabel(\"Amplitude\")\n",
    "ax[1].plot(Vnorma,\"*\",label=\"Vnorma\")\n",
    "ax[1].set_title(\"Vnorma\")\n",
    "ax[2].imshow(IMOutputOdd.reshape((im_crop.shape[0],im_crop.shape[1])), cmap='Greys')\n",
    "ax[2].set_title(\"New transformed image\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK. Look the final result after applying a single gate. Remember that only even elements are needed"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Only half of the borders have been calculated. To calculate the even borders, the algorithm must be executed permuting the input vector by one position."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "eng2=MainEngine()\n",
    "q2=eng2.allocate_qureg(number_of_qubits)\n",
    "CreateState2=_state_prep.StatePreparation(np.take(Vnorma,range(1,len(Vnorma)+1),mode=\"warp\"))\n",
    "CreateState2 | q2\n",
    "H |q2[0]\n",
    "eng2.flush()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Get the final amplitudes of this case from the simulator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "eng2.backend.collapse_wavefunction([q[0]], [1])\n",
    "Vout=np.zeros(shape)\n",
    "IMOutputEven=np.zeros(shape)\n",
    "for i in range(shape):\n",
    "    bits=np.binary_repr(i,width=len(q))\n",
    "    Vout[i]=abs(eng2.backend.get_probability(bits[-1::-1],q))\n",
    "    if (Vout[i]>0.0001):\n",
    "        IMOutputEven[i+1]=1.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the result of this second execution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7fb70839e2b0>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.imshow(IMOutputEven.reshape((im_crop.shape[0],im_crop.shape[1])), cmap='Greys')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And combine both outputs to produce the final"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7fb7083534e0>"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.imshow((IMOutputOdd+IMOutputEven).reshape((im_crop.shape[0],im_crop.shape[1])), cmap='Greys')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.5.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}