Add State Transfer Notebook

Miscellaneous Quantum Computing Notebooks/StateTransfer.ipynb

@@ -0,0 +1,85 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\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)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Perfect State Transfer and Uniform Mixing"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "As we have seen in the introduction article about __[CTQW (Continuous Time Quantum Walks)](https://github.com/matheusmtta/Quantum-Computing/blob/master/CTQW.ipynb)__\n",
+    "\n",
+    "If A is the adjacency matrix of a graph G, we will define a Quantum Walk as a trasition matrix $U(t)$,\n",
+    "\n",
+    "$$\n",
+    "    U(t) = exp(itA)\n",
+    "$$\n",
+    "\n",
+    "In this article we are concerned about how these transition matrices works and some of its special properties in quantum states.\n",
+    "\n",
+    "Since we are working with quantum information we are interested in the squared absolute value of $U(t)$ entries, in order to guarantee that its entries can be betermined by measurements.\n",
+    "\n",
+    "These matrices are called unitary, i.e, $U^*U = UU^* = I$. Hence we have that $\\overline{U(t)} = U(-t)$, and we can define a measurement matrix by the Schur Product of its entries, which is defined as the entrywise matrix product \n",
+    "\n",
+    "$$\n",
+    "    M(t) = U(t)\\circ U(-t)\n",
+    "$$\n",
+    "\n",
+    "Hence, giving us a doubly sthocastic measurement matrix $M(t)$, where each column give us the probability density of $V(X)$, i.e, information about the quantum walker into our vertex set, given a initial state $|\\psi\\rangle$ and a time $t$.\n",
+    "\n",
+    "Thus, we are interested in evaluate some special properties of CTQW over specfic graphs at a certain time, and we are mostly interested and the ones that are known as Perfect State Transfer (PST) and Uniform Mixing (UM)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Still in progress..."
+   ]
+  }
+ ],
+ "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": "3.8.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}