Merge pull request #37 from aryashah2k/Quantum-Random-Walks

Add Quantum Time Random Walks | Qiskit
QPower-Research · Feb 16, 2021 · 7120a94 · 7120a94
2 parents f65b608 + 6cadd76
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diff --git a/qiskit/Quantum Random Walks/ContinuousTimeQuantumWalks.ipynb b/qiskit/Quantum Random Walks/ContinuousTimeQuantumWalks.ipynb
@@ -0,0 +1,160 @@
+{
+  "cells": [
+    {
+      "cell_type": "code",
+      "source": [
+        "import numpy as np\n",
+        "from numpy import linalg as LA\n",
+        "from scipy.linalg import expm, sinm, cosm\n",
+        "import matplotlib.pyplot as plt\n",
+        "import pandas as pd\n",
+        "import seaborn as sns\n",
+        "import math\n",
+        "from scipy import stats\n",
+        "%matplotlib inline\n",
+        "\n",
+        "from IPython.display import Image, display, Math, Latex\n",
+        "sns.set(color_codes=True)"
+      ],
+      "outputs": [],
+      "execution_count": 1,
+      "metadata": {}
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "# Continuous Time Quantum Walks"
+      ],
+      "metadata": {}
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "## Continuous-Time Quantum Walks"
+      ],
+      "metadata": {}
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "Now let's see an example of how it work on cycle graph $C_n$ for some numbers of vertices."
+      ],
+      "metadata": {}
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "#number of vertices\n",
+        "n = 4\n",
+        "\n",
+        "#Define adjacency matrix A_Cn\n",
+        "A = np.zeros((n, n))\n",
+        "for i in range(n):\n",
+        "    j1 = (i - 1)%n\n",
+        "    j2 = (i + 1)%n\n",
+        "    A[i][j1] = 1\n",
+        "    A[i][j2] = 1\n",
+        "\n",
+        "#Define our initial state Psi_a\n",
+        "psi_a = np.zeros(n)\n",
+        "psi_a[3] = 1\n",
+        "\n",
+        "#Define the time t >= 0\n",
+        "t = math.pi/2\n",
+        "\n",
+        "#Exponentiate or hamiltonian\n",
+        "U_t = expm(1j*t*A)\n",
+        "U_mt = expm(1j*(-t)*A)\n",
+        "\n",
+        "#Compute Psi_t\n",
+        "psi_t = U_t @ psi_a\n",
+        "\n",
+        "#Compute the probabilities\n",
+        "prob_t = abs(psi_t)**2"
+      ],
+      "outputs": [],
+      "execution_count": 2,
+      "metadata": {}
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "M_t = U_t*U_mt \n",
+        "M_t = np.around(M_t, decimals = 3)\n",
+        "M_t"
+      ],
+      "outputs": [
+        {
+          "output_type": "execute_result",
+          "execution_count": 3,
+          "data": {
+            "text/plain": "array([[0.+0.j, 0.-0.j, 1.+0.j, 0.+0.j],\n       [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j],\n       [1.-0.j, 0.+0.j, 0.+0.j, 0.+0.j],\n       [0.+0.j, 1.-0.j, 0.+0.j, 0.+0.j]])"
+          },
+          "metadata": {}
+        }
+      ],
+      "execution_count": 3,
+      "metadata": {}
+    },
+    {
+      "cell_type": "code",
+      "source": [
+        "x = M_t[:, 0].real\n",
+        "plt.bar(range(len(x)), x, tick_label=[0, 1, 2, 3])\n",
+        "plt.xlabel('Vertices')\n",
+        "plt.ylabel('Probability')"
+      ],
+      "outputs": [
+        {
+          "output_type": "execute_result",
+          "execution_count": 7,
+          "data": {
+            "text/plain": "Text(0, 0.5, 'Probability')"
+          },
+          "metadata": {}
+        },
+        {
+          "output_type": "display_data",
+          "data": {
+            "image/png": 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\n",
+            "text/plain": "<Figure size 432x288 with 1 Axes>"
+          },
+          "metadata": {}
+        }
+      ],
+      "execution_count": 7,
+      "metadata": {}
+    },
+    {
+      "cell_type": "markdown",
+      "source": [
+        "Then we can easily visualize how our quantum-walker behaves in graph $C_n$ given an initial state and a time $t = \\pi/2$, and we can see the Perfect State Transfer phenomena from vertice 0 to 2, that will be explained in the detail [here](https://github.com/matheusmtta/Quantum-Computing/blob/master/Quantum%20Information%20Theory/State_Transfer.ipynb)."
+      ],
+      "metadata": {}
+    }
+  ],
+  "metadata": {
+    "kernelspec": {
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