Fix qcvv/xeb_theory.ipynb as part of docs cleanup for Cirq 1.0 (quantumlib#5202)

* Moved import to top
* Fixed broken link to quantum ai website
* Minor improvements in text formatting

Author: tanujkhattar

docs/qcvv/xeb_theory.ipynb

@@ -1,14 +1,5 @@
 {
  "cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "id": "SzKwuqYESWwm"
-   },
-   "source": [
-    "##### Copyright 2021 The Cirq Developers"
-   ]
-  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -18,7 +9,8 @@
    },
    "outputs": [],
    "source": [
-    "#@title Licensed under the Apache License, Version 2.0 (the \"License\");\n",
+    "#@title Copyright 2021 The Cirq Developers\n",
+    "# Licensed under the Apache License, Version 2.0 (the \"License\");\n",
     "# you may not use this file except in compliance with the License.\n",
     "# You may obtain a copy of the License at\n",
     "#\n",
@@ -39,7 +31,7 @@
    "source": [
     "<table class=\"tfo-notebook-buttons\" align=\"left\">\n",
     "  <td>\n",
-    "    <a target=\"_blank\" href=\"https://quantumai.google/cirq/qcvv/xeb_theory>\"><img src=\"https://quantumai.google/site-assets/images/buttons/quantumai_logo_1x.png\" />View on QuantumAI</a>\n",
+    "    <a target=\"_blank\" href=\"https://quantumai.google/cirq/qcvv/xeb_theory\"><img src=\"https://quantumai.google/site-assets/images/buttons/quantumai_logo_1x.png\" />View on QuantumAI</a>\n",
     "  </td>\n",
     "  <td>\n",
     "    <a target=\"_blank\" href=\"https://colab.research.google.com/github/quantumlib/Cirq/blob/master/docs/qcvv/xeb_theory.ipynb\"><img src=\"https://quantumai.google/site-assets/images/buttons/colab_logo_1x.png\" />Run in Google Colab</a>\n",
@@ -66,6 +58,7 @@
     "except ImportError:\n",
     "    print(\"installing cirq...\")\n",
     "    !pip install --quiet cirq\n",
+    "    import cirq\n",
     "    print(\"installed cirq.\")"
    ]
   },
@@ -90,7 +83,6 @@
    "source": [
     "# Standard imports\n",
     "import numpy as np\n",
-    "import cirq\n",
     "\n",
     "from cirq.contrib.svg import SVGCircuit"
    ]
@@ -125,9 +117,9 @@
    },
    "source": [
     "### Possible single-qubit rotations\n",
-    "These 8*8 possible rotations are chosen randomly when constructing the circuit.\n",
+    "Geometrically, we choose 8 axes in the XY plane to perform a quarter-turn (pi/2 rotation) around. This is followed by a rotation around the Z axis of 8 different magnitudes.\n",
     "\n",
-    "Geometrically, we choose 8 axes in the XY plane to perform a quarter-turn (pi/2 rotation) around. This is followed by a rotation around the Z axis of 8 different magnitudes."
+    "These 8*8 possible rotations are chosen randomly when constructing the circuit."
    ]
   },
   {
@@ -166,7 +158,7 @@
    "source": [
     "### Random circuit\n",
     "\n",
-    "We use `random_rotations_between_two_qubit_circuit` to generate a random two-qubit circuit. Note that we provide the possible single-qubit rotations from above and declare that our two-qubit operation is the $\\sqrt{i\\mathrm{SWAP}}$ gate."
+    "We use `cirq.experiments.random_quantum_circuit_generation.random_rotations_between_two_qubit_circuit` to generate a random two-qubit circuit. Note that we provide the possible single-qubit rotations from above and declare that our two-qubit operation is the $\\sqrt{i\\mathrm{SWAP}}$ gate."
    ]
   },
   {
@@ -291,8 +283,7 @@
     "        trunc_circuit = circuit[:circuit_depth]\n",
     "\n",
     "        # Pure-state simulation\n",
-    "        psi = pure_sim.simulate(trunc_circuit)\n",
-    "        psi = psi.final_state_vector\n",
+    "        psi = pure_sim.simulate(trunc_circuit).final_state_vector\n",
     "        pure_probs = np.abs(psi)**2\n",
     "\n",
     "        # Noisy execution\n",