notebooks/textbook/Quantum_Walk.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "c5524682",
   "metadata": {},
   "source": [
    "# Quantum Walk\n",
    "\n",
    "\n",
    "This example provides an implementation of the discrete time quantum random walk algorithm (hereafter referred to as quantum walk) using the Amazon Braket SDK. \n",
    "One application of quantum walks is to search and find graph properties, including marked vertices, maximal cliques, just to name a few. These problems can be solved via quantum walks provided an oracle that knows the properties of the graph is given. \n",
    "We will not get into the details of solving such problems; instead, we will illustrate how to transverse a circular graph, a graph consists of a single cycle, with a quantum walker. \n",
    "\n",
    "\n",
    "The quantum walk is very similar to the classical walk, where a walker observes a random process, such as flipping a coin, followed by deciding its next step conditioned on the outcome of the random process. For quantum walker, it observes a quantum process instead, and the subsequent steps are superpositions of the possible steps that a classical walker would take. For example, in this notebook, we consider a quantum walker on a cycle with $N=2^n$ nodes, where each node has only two neighbors. As a result, a classical walker would have two choices at each step, either going to the left or right neighbor, whereas it would be a superposition of the two for the quantum walker. \n",
    "More concretely, we will use $|q, k\\rangle$ to denote the state of a quantum walker where $q=0,1$ is the state of the coin, and $k\\in\\left\\{0,..,N-1\\right\\}$ labels the node. \n",
    "Each step of the quantum walk is a product of the coin operator (C) and the shift operator (S). \n",
    "The coin operator could in principle be any unitary that mixes the coin states, but for simplicity, we use the Hadamard gate \n",
    "\\begin{align}\n",
    "C|q,k\\rangle = H\\otimes I|q,k\\rangle = \\frac{1}{\\sqrt{2}}(|0,k\\rangle + (-1)^q|1,k\\rangle).\n",
    "\\end{align}\n",
    "The shift operator moves the walker to the neighboring nodes conditioned on the outcomes of the toss coin followed by flipping the coin state\n",
    "\\begin{align}\n",
    "S|q,k\\rangle = |q\\oplus 1, k\\oplus(-1)^q\\rangle\n",
    "\\end{align}\n",
    "where $\\oplus$ denotes the modular addition: for the first register, the addition is modulo 2 since there are only 2 states for the coin, whereas the addition is modulo $N$ for the second register. \n",
    "The quantum walk then proceeds by applying these two operators in alternation, and a $p$ step quantum walk is just the operator $(SC)^p$. \n",
    "\n",
    "How do we realize a quantum circuit to perform conditional modular addition? In order to perform modular addition and subtraction, one could invoke the quantum Fourier transform (QFT) adder introduced in Ref. [2](https://arxiv.org/abs/1411.5949). Here we will modify the QFT adder for performing conditional modular addition for the quantum walker. \n",
    "\n",
    "\n",
    "\n",
    "# References\n",
    "\n",
    "[[1] Quantum Algorithm Implementations for Beginners](https://arxiv.org/abs/1804.03719)\n",
    "\n",
    "[[2] Quantum arithmetic with the Quantum Fourier Transform](https://arxiv.org/abs/1411.5949)\n",
    "\n",
    "We start by importing the necessary functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6e5ec8f7",
   "metadata": {},
   "outputs": [],
   "source": [
    "from collections import Counter\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from braket.devices import LocalSimulator\n",
    "from braket.tracking import Tracker\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "from braket.experimental.algorithms.quantum_walk import quantum_walk, run_quantum_walk\n",
    "\n",
    "tracker = Tracker().start()  # to keep track of Braket costs\""
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd689dd2",
   "metadata": {},
   "source": [
    "# Quantum walk on a graph with four nodes\n",
    "\n",
    "As an example, we consider the quantum walk on a graph with 4 nodes as shown in the figure below. Without loss of generality, we shall assume that the walker starts from the vertex labeled as 0. We will demonstrate, step by step, how the walker transverses around the graph and arrive at the opposite corner of the graph.\n",
    "\n",
    "For that, we define the quantum circuits for quantum walks with 1, 2, 3 and 4 steps."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f9d753bd",
   "metadata": {},
   "outputs": [],
   "source": [
    "quantum_walk_circuit_4_nodes_1_step = quantum_walk(4, 1)\n",
    "quantum_walk_circuit_4_nodes_2_steps = quantum_walk(4, 2)\n",
    "quantum_walk_circuit_4_nodes_3_steps = quantum_walk(4, 3)\n",
    "quantum_walk_circuit_4_nodes_4_steps = quantum_walk(4, 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b70b3e9",
   "metadata": {},
   "source": [
    "We can visualize the quantum circuit for one-step quantum walk as follows. The $n$-step quantum walk for the same graph is simply the $n$-time repetition of the same circuit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "66292969",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "T  : │  0  │       1       │           2           │       3        │        4        │       5        │  6  │       7        │  8  │\n",
      "      ┌───┐                                                                                             ┌───┐                        \n",
      "q0 : ─┤ H ├───────────────────────────────●────────────────●─────────────────●────────────────●─────────┤ X ├────────────────────────\n",
      "      └───┘                               │                │                 │                │         └───┘                        \n",
      "                            ┌───┐         │                │         ┌───────┴───────┐ ┌──────┴───────┐ ┌───┐                        \n",
      "q1 : ──────────────●────────┤ H ├─────────┼────────────────┼─────────┤ PHASE01(3.14) ├─┤ PHASE(-3.14) ├─┤ H ├────────●───────────────\n",
      "                   │        └───┘         │                │         └───────────────┘ └──────────────┘ └───┘        │               \n",
      "      ┌───┐ ┌──────┴──────┐       ┌───────┴───────┐ ┌──────┴───────┐                                          ┌──────┴───────┐ ┌───┐ \n",
      "q2 : ─┤ H ├─┤ PHASE(1.57) ├───────┤ PHASE01(1.57) ├─┤ PHASE(-1.57) ├──────────────────────────────────────────┤ PHASE(-1.57) ├─┤ H ├─\n",
      "      └───┘ └─────────────┘       └───────────────┘ └──────────────┘                                          └──────────────┘ └───┘ \n",
      "T  : │  0  │       1       │           2           │       3        │        4        │       5        │  6  │       7        │  8  │\n"
     ]
    }
   ],
   "source": [
    "print(quantum_walk_circuit_4_nodes_1_step)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d2ac982",
   "metadata": {},
   "source": [
    "Before proceeding, let us explicitly calculate the states after each step of the quantum walk. Recall that we start with the $0$-th vertex and hence the initial state reads\n",
    "\\begin{align}\n",
    "|\\psi_0\\rangle = |00\\rangle\n",
    "\\end{align}\n",
    "where the first register is the coin qubit. After the first step, we have\n",
    "\\begin{align}\n",
    "|\\psi_1\\rangle \\equiv (SC)|\\psi_0\\rangle = S\\frac{1}{\\sqrt{2}}\\left(|0\\rangle+|1\\rangle\\right)\\otimes|0\\rangle = \\frac{1}{\\sqrt{2}}\\left(|11\\rangle+|03\\rangle\\right)\n",
    "\\end{align}\n",
    "where we have used the fact $-1\\equiv3\\text{ mod }4$. After this step, the walker will be in the superposition of the first and third nodes. For the second step we have\n",
    "\\begin{align}\n",
    "|\\psi_2\\rangle \\equiv (SC)|\\psi_1\\rangle = \\frac{1}{2}\\left[|12\\rangle-|00\\rangle+|10\\rangle+|02\\rangle\\right]\n",
    "\\end{align}\n",
    "where the walker will be in the superposition of the zeroth and the second nodes. For the third step we have \\begin{align}\n",
    "|\\psi_3\\rangle \\equiv (SC)|\\psi_2\\rangle = \\frac{1}{\\sqrt{2}}(|1\\rangle-|0\\rangle)|3\\rangle\n",
    "\\end{align}\n",
    "where the walker will be in the third node deterministically after the third step. Finally, after the fourth step, we have\n",
    "\\begin{align}\n",
    "|\\psi_4\\rangle \\equiv (SC)|\\psi_3\\rangle = -|02\\rangle\n",
    "\\end{align}\n",
    "where the walker end up in the opposite corner, namely the second node, deterministically after the fourth step."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "79064b7a",
   "metadata": {},
   "source": [
    "# Run the quantum walk on a local simulator\n",
    "\n",
    "Let's now run the 4 quantum circuits on a local simulator, followed by measuring the qubits for the nodes. The measurement results are shown with bar plots. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "cf274bd6",
   "metadata": {},
   "outputs": [],
   "source": [
    "local_sim = LocalSimulator()\n",
    "counts_4_nodes_1_step_sim = run_quantum_walk(quantum_walk_circuit_4_nodes_1_step, local_sim)\n",
    "counts_4_nodes_2_steps_sim = run_quantum_walk(quantum_walk_circuit_4_nodes_2_steps, local_sim)\n",
    "counts_4_nodes_3_steps_sim = run_quantum_walk(quantum_walk_circuit_4_nodes_3_steps, local_sim)\n",
    "counts_4_nodes_4_steps_sim = run_quantum_walk(quantum_walk_circuit_4_nodes_4_steps, local_sim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c0389139",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_bitstrings(counts: Counter, title: str = None):\n",
    "    plt.bar(counts.keys(), counts.values())\n",
    "    plt.xticks(list(counts.keys()))\n",
    "    plt.xlabel(\"index of the nodes\")\n",
    "    plt.ylabel(\"probability\")\n",
    "    plt.title(title)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "ea6b1c25",
   "metadata": {},
   "outputs": [
    {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_bitstrings(counts_4_nodes_1_step_sim[\"quantum_walk_measurement_counts\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "1c3fad2e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_bitstrings(counts_4_nodes_2_steps_sim[\"quantum_walk_measurement_counts\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "ce8e39d2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_bitstrings(counts_4_nodes_3_steps_sim[\"quantum_walk_measurement_counts\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "3d5a24f8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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uF19++aU6dOige+65h8/BAQAAIeN33OTl5bXiGAAAAMHhd9xMnz69NecAAAAIihZ9iJ8kffHFF6qvr/dZ5nA4WrpbAACAgAR0Q3Ftba1mzZqlmJgYdenSRT169PB5AAAAhEpAcfPII4/oT3/6k1asWCG73a5nn31WCxcuVFxcnF588cVgzwgAAOC3gC5Lvf7663rxxRd14403Kj09Xddff7369++vK664Qr/97W915513BntOAAAAvwR05qa6ulpXXnmlpHP311RXV0uSvvvd72rHjh3Bmw4AAKCZAoqbK6+8UocPH5YkDRw4UC+99JKkc2d0vvoiTQAAgFAIKG7S09P13nvvSZIyMzOVn5+vyMhIzZ07Vw8//HBQBwQAAGiOgO65mTt3rvffKSkpKisrU0lJifr376+hQ4cGbTgAAIDmavHn3EiS0+mU0+kMxq4AAABaJKDLUpJUWFio2267TVdddZWuuuoq3XbbbXrzzTeDORsAAECzBRQ3zzzzjG6++WZ169ZNGRkZysjIkMPh0K233qr8/PxgzwgAAOC3gC5LLVmyRE899ZRmzZrlXTZnzhyNHTtWS5Ys0cyZM4M2IAAAQHMEdObm1KlTuvnmmxstHz9+vGpqalo8FAAAQKACiptJkybplVdeabT81Vdf1W233dbioQAAAALl92WpX/7yl95/Dxo0SI8//ri2b9+u5ORkSdI777yjXbt26aGHHgr+lAAAAH6yWZZl+bNiv379/NuhzaaPP/64RUO1JrfbraioKNXU1MjhcAR9/87MTUHfJwAA7cmRpROCvs/m/P32+8zNV1+3AAAA0JYF/Dk3X7EsS36e/AEAAGh1AcfNiy++qCFDhqhTp07q1KmThg4dql//+tfBnA0AAKDZAvqcm9zcXD366KOaNWuWxo4dK0nauXOn7r//flVVVfl89xQAAMDFFFDcPP3001qxYoXS0tK8yyZNmqRrr71Wjz32GHEDAABCJqDLUsePH9eYMWMaLR8zZoyOHz/e4qEAAAACFVDc9O/fXy+99FKj5Rs2bNCAAQNaPBQAAECgArostXDhQk2ZMkU7duzw3nOza9cuFRYWNhk9AAAAF0tAZ25uv/127dmzR9HR0SooKFBBQYGio6O1Z88e/eAHPwj2jAAAAH5r9pmbs2fP6j/+4z/06KOP6je/+U1rzAQAABCwZp+56dixo15++eXWmAUAAKDFArosNXnyZBUUFAR5FAAAgJYL6IbiAQMGaNGiRdq1a5cSExPVpUsXn9/PmTMnKMMBAAA0V0Bx89xzz6l79+4qLi5WcXGxz+9sNhtxAwAAQiaguPn6N4R/9aWZNpstOBMBAAC0QMBfnPncc89p8ODBioyMVGRkpAYPHqxnn302mLMBAAA0W0BnbrKyspSbm6vZs2crOTlZklRUVKS5c+eqvLxcixYtCuqQAAAA/gooblasWKHVq1dr6tSp3mWTJk3S0KFDNXv2bOIGAACETECXpc6ePatRo0Y1Wp6YmKgvv/yyxUMBAAAEKqC4ufvuu7VixYpGy1etWqU777yzxUMBAAAEKqDLUtK5G4rfeOMNfec735EkvfvuuyovL1daWppcLpd3vdzc3JZPCQAA4KeA4mb//v0aOXKkJOnQoUOSpOjoaEVHR2v//v3e9Xh7OAAAuNgCipu33nor2HMAAAAERcCfcxNM+fn5cjqdioyMVFJSkvbs2ePXduvXr5fNZtPkyZNbd0AAANBuhDxuNmzYIJfLpezsbJWUlGjYsGFKTU3ViRMnLrjdkSNH9NOf/lTXX3/9RZoUAAC0ByGPm9zcXM2YMUPp6ekaNGiQVq5cqc6dO+v5558/7zYNDQ268847tXDhQl155ZUXcVoAANDWhTRu6uvrVVxcrJSUFO+ysLAwpaSkqKio6LzbLVq0SDExMbr33nu/9Tnq6urkdrt9HgAAwFwhjZuqqio1NDQoNjbWZ3lsbKwqKiqa3Gbnzp167rnntHr1ar+eIycnR1FRUd5HfHx8i+cGAABtV8gvSzXH6dOndffdd2v16tWKjo72a5t58+appqbG+zh69GgrTwkAAEIp4A/xC4bo6GiFh4ersrLSZ3llZaV69erVaP1Dhw7pyJEjmjhxoneZx+ORJHXo0EEHDx7UVVdd5bON3W6X3W5vhekBAEBbFNIzNxEREUpMTFRhYaF3mcfjUWFhoffbxr9u4MCB+utf/6rS0lLvY9KkSRo3bpxKS0u55AQAAEJ75kaSXC6Xpk+frlGjRmn06NHKy8tTbW2t0tPTJUlpaWnq06ePcnJyFBkZqcGDB/ts3717d0lqtBwAAFyaQh43U6ZM0cmTJ5WVlaWKigoNHz5cW7Zs8d5kXF5errCwdnVrEAAACCGbZVlWqIe4mNxut6KiolRTUyOHwxH0/TszNwV9nwAAtCdHlk4I+j6b8/ebUyIAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3AADAKMQNAAAwCnEDAACMQtwAAACjEDcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3AADAKMQNAAAwCnEDAACMQtwAAACjEDcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3AADAKMQNAAAwCnEDAACMQtwAAACjEDcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3AADAKMQNAAAwCnEDAACMQtwAAACjEDcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3AADAKMQNAAAwCnEDAACMQtwAAACjtIm4yc/Pl9PpVGRkpJKSkrRnz57zrrt69Wpdf/316tGjh3r06KGUlJQLrg8AAC4tIY+bDRs2yOVyKTs7WyUlJRo2bJhSU1N14sSJJtffvn27pk6dqrfeektFRUWKj4/X+PHj9emnn17kyQEAQFtksyzLCuUASUlJuu6667R8+XJJksfjUXx8vGbPnq3MzMxv3b6hoUE9evTQ8uXLlZaW1uj3dXV1qqur8/7sdrsVHx+vmpoaORyO4B3I/3Nmbgr6PgEAaE+OLJ0Q9H263W5FRUX59fc7pGdu6uvrVVxcrJSUFO+ysLAwpaSkqKioyK99nDlzRmfPntVll13W5O9zcnIUFRXlfcTHxwdldgAA0DaFNG6qqqrU0NCg2NhYn+WxsbGqqKjwax8/+9nPFBcX5xNIXzdv3jzV1NR4H0ePHm3x3AAAoO3qEOoBWmLp0qVav369tm/frsjIyCbXsdvtstvtF3kyAAAQKiGNm+joaIWHh6uystJneWVlpXr16nXBbZ988kktXbpUb775poYOHdqaYwIAgHYkpJelIiIilJiYqMLCQu8yj8ejwsJCJScnn3e7X/ziF1q8eLG2bNmiUaNGXYxRAQBAOxHyy1Iul0vTp0/XqFGjNHr0aOXl5am2tlbp6emSpLS0NPXp00c5OTmSpCeeeEJZWVlat26dnE6n996crl27qmvXriE7DgAA0DaEPG6mTJmikydPKisrSxUVFRo+fLi2bNnivcm4vLxcYWH/PMG0YsUK1dfX64477vDZT3Z2th577LGLOToAAGiDQv45Nxdbc94nHwg+5wYAcKm7pD/nBgAAINiIGwAAYBTiBgAAGIW4AQAARiFuAACAUYgbAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGIW4AQAARiFuAACAUYgbAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGIW4AQAARiFuAACAUYgbAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGIW4AQAARiFuAACAUYgbAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGIW4AQAARiFuAACAUYgbAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABilTcRNfn6+nE6nIiMjlZSUpD179lxw/d///vcaOHCgIiMjNWTIEG3evPkiTQoAANq6kMfNhg0b5HK5lJ2drZKSEg0bNkypqak6ceJEk+vv3r1bU6dO1b333qt9+/Zp8uTJmjx5svbv33+RJwcAAG2RzbIsK5QDJCUl6brrrtPy5cslSR6PR/Hx8Zo9e7YyMzMbrT9lyhTV1tbqj3/8o3fZd77zHQ0fPlwrV6781udzu92KiopSTU2NHA5H8A7k/zkzNwV9nwAAtCdHlk4I+j6b8/e7Q9CfvRnq6+tVXFysefPmeZeFhYUpJSVFRUVFTW5TVFQkl8vlsyw1NVUFBQVNrl9XV6e6ujrvzzU1NZLOvUitwVN3plX2CwBAe9Eaf2O/2qc/52RCGjdVVVVqaGhQbGysz/LY2FgdOHCgyW0qKiqaXL+ioqLJ9XNycrRw4cJGy+Pj4wOcGgAAXEhUXuvt+/Tp04qKirrgOiGNm4th3rx5Pmd6PB6Pqqurdfnll8tms4VwMgDB5na7FR8fr6NHj7bKZWcAoWNZlk6fPq24uLhvXTekcRMdHa3w8HBVVlb6LK+srFSvXr2a3KZXr17NWt9ut8tut/ss6969e+BDA2jzHA4HcQMY6NvO2HwlpO+WioiIUGJiogoLC73LPB6PCgsLlZyc3OQ2ycnJPutL0rZt2867PgAAuLSE/LKUy+XS9OnTNWrUKI0ePVp5eXmqra1Venq6JCktLU19+vRRTk6OJCkjI0M33HCDli1bpgkTJmj9+vXau3evVq1aFcrDAAAAbUTI42bKlCk6efKksrKyVFFRoeHDh2vLli3em4bLy8sVFvbPE0xjxozRunXrtGDBAv385z/XgAEDVFBQoMGDB4fqEAC0EXa7XdnZ2Y0uRQO4tIT8c24AAACCKeSfUAwAABBMxA0AADAKcQMAAIxC3AAAAKMQNwDavZycHF133XXq1q2bYmJiNHnyZB08eDDUYwEIEeIGQLv39ttva+bMmXrnnXe0bds2nT17VuPHj1dtbW2oRwMQArwVHIBxTp48qZiYGL399tv613/911CPA+Ai48wNAOPU1NRIki677LIQTwIgFDhzA8AoHo9HkyZN0qlTp7Rz585QjwMgBEL+9QsAEEwzZ87U/v37CRvgEkbcADDGrFmz9Mc//lE7duxQ3759Qz0OgBAhbgC0e5Zlafbs2XrllVe0fft29evXL9QjAQgh4gZAuzdz5kytW7dOr776qrp166aKigpJUlRUlDp16hTi6QBcbNxQDKDds9lsTS5fs2aNfvKTn1zcYQCEHGduALR7/B8NwNfxOTcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3gIFuvPFGPfjggy3ax5EjR2Sz2VRaWhqUmQJ15swZ3X777XI4HLLZbDp16pTf29psNhUUFLTabMHUVl5vwAR8QjFgoI0bN6pjx46hHiMoXnjhBf35z3/W7t27FR0draioqEbrPPbYYyooKCAMAEgibgAjXXbZZaEeIWgOHTqkhIQEDR48ONSjAGgnuCwFGOibl6WcTqeWLFmie+65R926ddO//Mu/aNWqVT7b7NmzRyNGjFBkZKRGjRqlffv2Ndrv/v37dcstt6hr166KjY3V3XffraqqKknS9u3bFRERoT//+c/e9X/xi18oJiZGlZWV55315Zdf1rXXXiu73S6n06lly5b5HMeyZcu0Y8cO2Ww23XjjjY22X7t2rRYuXKj33ntPNptNNptNa9eu9f6+qqpKP/jBD9S5c2cNGDBAr732mt/H1JS1a9eqe/fu2rp1qxISEtS1a1fdfPPNOn78uHcdj8ejRYsWqW/fvrLb7Ro+fLi2bNnis5+Wvt6S9Ic//EFDhgxRp06ddPnllyslJUW1tbXnnR24ZFgAjHPDDTdYGRkZ3p+vuOIK67LLLrPy8/OtDz/80MrJybHCwsKsAwcOWJZlWadPn7Z69uxpTZs2zdq/f7/1+uuvW1deeaUlydq3b59lWZb197//3erZs6c1b948q6yszCopKbFuuukma9y4cd7nefjhh60rrrjCOnXqlFVSUmJFRERYr7766nnn3Lt3rxUWFmYtWrTIOnjwoLVmzRqrU6dO1po1ayzLsqzPPvvMmjFjhpWcnGwdP37c+uyzzxrt48yZM9ZDDz1kXXvttdbx48et48ePW2fOnLEsy7IkWX379rXWrVtnffjhh9acOXOsrl27evfjzzF905o1a6yOHTtaKSkp1l/+8heruLjYSkhIsKZNm+ZdJzc313I4HNbvfvc768CBA9YjjzxidezY0frggw+C9nofO3bM6tChg5Wbm2sdPnzYev/99638/Hzr9OnT550duFQQN4CBmoqbu+66y/uzx+OxYmJirBUrVliWZVm/+tWvrMsvv9z6xz/+4V1nxYoVPn9sFy9ebI0fP97neY4ePWpJsg4ePGhZlmXV1dVZw4cPt3784x9bgwYNsmbMmHHBOadNm2bddNNNPssefvhha9CgQd6fMzIyrBtuuOGC+8nOzraGDRvWaLkka8GCBd6fP//8c0uS9d///d9+H9M3rVmzxpJkffTRR95l+fn5VmxsrPfnuLg46/HHH/fZ7rrrrrMeeOABy7KC83oXFxdbkqwjR46c72UBLllclgIuEUOHDvX+22azqVevXjpx4oQkqaysTEOHDlVkZKR3neTkZJ/t33vvPb311lvq2rWr9zFw4EBJ5+6LkaSIiAj99re/1csvv6wvvvhCTz311AVnKisr09ixY32WjR07Vh9++KEaGhoCP9iv+fpxd+nSRQ6Hw3vc/hxTUzp37qyrrrrK+3Pv3r29+3S73Tp27FiTx1VWViYpOK/3sGHD9G//9m8aMmSIfvSjH2n16tX6+9//3uzXBzARNxQDl4hvvnvKZrPJ4/H4vf3nn3+uiRMn6oknnmj0u969e3v/vXv3bklSdXW1qqur1aVLlwAnDo4LHbe/x+TPPi3LCsK0//Rts4WHh2vbtm3avXu33njjDT399NOaP3++3n33XfXr1y+oswDtDWduACghIUHvv/++vvjiC++yd955x2edkSNH6n/+53/kdDrVv39/n8dXAXPo0CHNnTtXq1evVlJSkqZPn37BgEpISNCuXbt8lu3atUtXX321wsPD/Z4/IiIioDM9/hxTczkcDsXFxTV5XIMGDZIUvNfbZrNp7NixWrhwofbt26eIiAi98sorAc0NmIS4AaBp06bJZrNpxowZ+t///V9t3rxZTz75pM86M2fOVHV1taZOnaq//OUvOnTokLZu3ar09HQ1NDSooaFBd911l1JTU5Wenq41a9bo/fff93n30zc99NBDKiws1OLFi/XBBx/ohRde0PLly/XTn/60WfM7nU4dPnxYpaWlqqqqUl1dnV/bfdsxBerhhx/WE088oQ0bNujgwYPKzMxUaWmpMjIyJAXn9X733Xe1ZMkS7d27V+Xl5dq4caNOnjyphISEgOcGTEHcAFDXrl31+uuv669//atGjBih+fPnN7oc8tXZiIaGBo0fP15DhgzRgw8+qO7duyssLEyPP/64PvnkE/3qV7+SdO7SyapVq7RgwQK99957TT7vyJEj9dJLL2n9+vUaPHiwsrKytGjRIv3kJz9p1vy33367br75Zo0bN049e/bU7373O7+2+7ZjCtScOXPkcrn00EMPaciQIdqyZYtee+01DRgwQFJwXm+Hw6EdO3bo1ltv1dVXX60FCxZo2bJluuWWWwKeGzCFzQr2hWIAAIAQ4swNAAAwCnEDAACMQtwAAACjEDcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAo/wf03hJ+wJ54rIAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_bitstrings(counts_4_nodes_4_steps_sim[\"quantum_walk_measurement_counts\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2774f306",
   "metadata": {},
   "source": [
    "We see that the bar plots are consistent with our calculations for each step. One may be puzzled after comparing the bar plots for the third and the fourth steps, why the quantum walker can be shifted from the third node to its neighbor deterministically with a single step. \n",
    "This seems to be in contrast to the very first step where the amplitude is evenly distributed to the neighbors of the zeroth node. \n",
    "The reason is that after the third step, despite that the walker is deterministically found to be on the third node, the coin qubit is in a superposition state, in contrast to the very first step, where the coin is in the state of $|0\\rangle$. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68b00402",
   "metadata": {},
   "source": [
    "# Run the quantum walk on a QPU\n",
    "\n",
    "Let's now run the same quantum circuits on the Rigetti device, followed by inspecting the measurement results with bar plots.\n",
    "\n",
    "Include estimated price for running in USD.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "d8d7c3f3",
   "metadata": {},
   "outputs": [],
   "source": [
    "from braket.aws import AwsDevice\n",
    "\n",
    "qpu = AwsDevice(\"arn:aws:braket:us-west-1::device/qpu/rigetti/Ankaa-2\")\n",
    "counts_4_nodes_1_step_qpu = run_quantum_walk(quantum_walk_circuit_4_nodes_1_step, qpu)\n",
    "counts_4_nodes_2_steps_qpu = run_quantum_walk(quantum_walk_circuit_4_nodes_2_steps, qpu)\n",
    "counts_4_nodes_3_steps_qpu = run_quantum_walk(quantum_walk_circuit_4_nodes_3_steps, qpu)\n",
    "counts_4_nodes_4_steps_qpu = run_quantum_walk(quantum_walk_circuit_4_nodes_4_steps, qpu)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "3f6fea66",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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nU8XFxY0a49ixYzpx4oS6d+/u115UVKQePXroiiuu0KRJk3Tw4MEzjlFTU6Oqqiq/AwAAtF+mBqCKigrV1dUpKirKrz0qKkoej6dRY/ziF79QbGysX4hKT0/X8uXL5Xa7NXfuXH388ccaPny46urqGhwjJydHERERviMuLi7wSQEAgFavg9kFNMWcOXO0YsUKFRUVKTQ01Nc+ZswY38+DBg1SQkKC+vTpo6KiIt1yyy2njZOZmSmXy+V7XFVVRQgCAKAdM3UFKDIyUsHBwfJ6vX7tXq9X0dHRZz13/vz5mjNnjj744AMlJCSctW/v3r0VGRmpb7/9tsHn7Xa7wsPD/Q4AANB+BRSAqqurm+XFQ0JClJSU5LeB+dSG5tTU1DOeN2/ePM2ePVuFhYVKTk4+5+vs2bNHBw8eVExMTLPUDQAA2raAAlBUVJTuu+8+rV27tskFuFwuLVq0SMuWLVN5ebkmTZqk6upqZWRkSJLGjRunzMxMX/+5c+dq5syZWrx4sRwOhzwejzwej44ePSpJOnr0qKZNm6b169dr586dcrvdGjFihPr27au0tLQm1wsAANq+gALQG2+8oUOHDumHP/yhLr/8cs2ZM0d79+4NqIDRo0dr/vz5ysrKUmJiojZt2qTCwkLfxuhdu3Zp3759vv4LFy5UbW2t7r77bsXExPiO+fPnS5KCg4O1efNm3Xnnnbr88ss1YcIEJSUl6ZNPPpHdbg+oRgAA0L7YDMMwAj35wIEDev3117V06VKVl5crLS1N9913n+6880516NB291dXVVUpIiJClZWV7AdCozmm/9nsEixr55zbzC4BbRT/b83RUv9nz+fvd5M2QV988cVyuVzavHmzcnNz9de//lV33323YmNjlZWVpWPHjjVleAAAgBbRpGUar9erZcuWaenSpfrHP/6hu+++WxMmTNCePXs0d+5crV+/Xh988EFz1QoAANAsAgpAq1at0pIlS/T+++8rPj5eDzzwgH7605+qa9euvj7XXHONBgwY0Fx1AgAANJuAAlBGRobGjBmjdevWaciQIQ32iY2N1RNPPNGk4gAAAFpCQAFo37596tix41n7hIWFKTs7O6CiAAAAWlJAm6C7dOmi/fv3n9Z+8OBBBQcHN7koAACAlhRQADrTJ+dramoUEhLSpIIAAABa2nldAluwYIEkyWaz6dVXX1Xnzp19z9XV1WnNmjXq379/81YIAADQzM4rAL3wwguSTq4AFRQU+F3uCgkJkcPhUEFBQfNWCAAA0MzOKwDt2LFDknTzzTdr1apV6tatW4sUBQAA0JIC+hTYRx991Nx1AAAAXDCNDkAul0uzZ89Wp06d5HK5zto3Nze3yYUBAAC0lEYHoM8//1wnTpzw/XwmNput6VUBAAC0oEYHoH+/7MUlMAAA0JY16W7wAAAAbVGjV4B+/OMfN3rQVatWBVQMAADAhdDoABQREdGSdQAAAFwwjQ5AS5Ysack6AAAALhj2AAEAAMtp9ArQ1VdfLbfbrW7duumqq64668fdy8rKmqU4AACAltDoADRixAjZ7XZJ0siRI1uqHgAAgBbX6ACUnZ3d4M8AAABtTUD3Ajtl48aNKi8vlyTFx8crKSmpWYoCAABoSQEFoD179mjs2LFat26dunbtKkk6fPiwrrnmGq1YsUKXXHJJc9YIAADQrAL6FNjPf/5znThxQuXl5Tp06JAOHTqk8vJy1dfX6+c//3lz1wgAANCsAloB+vjjj/Xpp5/qiiuu8LVdccUVeumll3T99dc3W3EAAAAtIaAVoLi4ON+d4f9dXV2dYmNjm1wUAABASwooAD333HN66KGHtHHjRl/bxo0bNWXKFM2fP7/ZigMAAGgJjb4E1q1bN78vP6yurlZKSoo6dDg5xL/+9S916NBB9913H98TBAAAWrVGB6C8vLwWLAMAAODCaXQAGj9+fEvWAQAAcME06YsQJen48eOqra31awsPD2/qsAAAAC0moE3Q1dXVmjx5snr06KFOnTqpW7dufgcAAEBrFlAAevzxx/W3v/1NCxculN1u16uvvqpZs2YpNjZWy5cvP+/x8vPz5XA4FBoaqpSUFJWUlJyx76JFi3T99df7wpbT6Tytv2EYysrKUkxMjMLCwuR0OrVt27bzrgsAALRPAQWgP/7xj/rVr36lUaNGqUOHDrr++us1Y8YMPfvss3rzzTfPa6yVK1fK5XIpOztbZWVlGjx4sNLS0rR///4G+xcVFWns2LH66KOPVFxcrLi4OA0bNkzfffedr8+8efO0YMECFRQUaMOGDerUqZPS0tJ0/PjxQKYLAADamYAC0KFDh9S7d29JJ/f7HDp0SJJ03XXXac2aNec1Vm5uriZOnKiMjAzFx8eroKBAHTt21OLFixvs/+abb+qBBx5QYmKi+vfvr1dffVX19fVyu92STq7+5OXlacaMGRoxYoQSEhK0fPly7d27V6tXrw5kugAAoJ0JKAD17t1bO3bskCT1799f77zzjqSTK0Onbo7aGLW1tSotLZXT6fy+oKAgOZ1OFRcXN2qMY8eO6cSJE+revbskaceOHfJ4PH5jRkREKCUl5Yxj1tTUqKqqyu8AAADtV0ABKCMjQ1988YUkafr06crPz1doaKgeeeQRTZs2rdHjVFRUqK6uTlFRUX7tUVFR8ng8jRrjF7/4hWJjY32B59R55zNmTk6OIiIifEdcXFyj5wAAANqegD4G/8gjj/h+djqdKi8vV1lZmfr27auEhIRmK+5c5syZoxUrVqioqEihoaEBj5OZmSmXy+V7XFVVRQgCAKAda/L3AEmSw+GQw+E47/MiIyMVHBwsr9fr1+71ehUdHX3Wc+fPn685c+bor3/9q1/oOnWe1+tVTEyM35iJiYkNjmW322W328+7fgAA0DYFdAlMktxut26//Xb16dNHffr00e23366//vWv5zVGSEiIkpKSfBuYJfk2NKempp7xvHnz5mn27NkqLCxUcnKy33O9evVSdHS035hVVVXasGHDWccEAADWEVAA+tWvfqX09HR16dJFU6ZM0ZQpUxQeHq4f/ehHys/PP6+xXC6XFi1apGXLlqm8vFyTJk1SdXW1MjIyJEnjxo1TZmamr//cuXM1c+ZMLV68WA6HQx6PRx6PR0ePHpUk2Ww2TZ06VU8//bT+8Ic/6Msvv9S4ceMUGxvLTVoBAICkAC+BPfvss3rhhRc0efJkX9vDDz+sa6+9Vs8++6wefPDBRo81evRoHThwQFlZWfJ4PEpMTFRhYaFvE/OuXbsUFPR9Tlu4cKFqa2t19913+42TnZ2tJ598UtLJL2qsrq7W/fffr8OHD+u6665TYWFhk/YJAQCA9sNmGIZxvid17txZmzZtUt++ff3at23bpquuusq3GtNWVVVVKSIiQpWVldzXDI3mmP5ns0uwrJ1zbjO7BLRR/L81R0v9nz2fv98BXQK788479e67757W/vvf/1633357IEMCAABcMI2+BLZgwQLfz/Hx8XrmmWdUVFTk21i8fv16rVu3To8++mjzVwkAANCMGh2AXnjhBb/H3bp109dff62vv/7a19a1a1ctXrxYM2bMaL4KAQAAmlmjA9CpW18AAAC0dQF/D9AphmEogH3UAAAApgk4AC1fvlyDBg1SWFiYwsLClJCQoNdff705awMAAGgRAX0PUG5urmbOnKnJkyfr2muvlSStXbtW//d//6eKigq/e4UBAAC0NgEFoJdeekkLFy7UuHHjfG133nmnrrzySj355JMEIAAA0KoFdAls3759uuaaa05rv+aaa7Rv374mFwUAANCSAgpAffv21TvvvHNa+8qVK9WvX78mFwUAANCSAroENmvWLI0ePVpr1qzx7QFat26d3G53g8EIAACgNQloBWjUqFEqKSlRZGSkVq9erdWrVysyMlIlJSW66667mrtGAACAZnXeK0AnTpzQ//7v/2rmzJl64403WqImAACAFnXeK0AXXXSRfve737VELQAAABdEQJfARo4cqdWrVzdzKQAAABdGQJug+/Xrp6eeekrr1q1TUlKSOnXq5Pf8ww8/3CzFAQAAtISAAtBrr72mrl27qrS0VKWlpX7P2Ww2AhAAAGjVAgpA/35n+FM3QrXZbM1TEQAAQAsL+Gaor732mgYOHKjQ0FCFhoZq4MCBevXVV5uzNgAAgBYR0ApQVlaWcnNz9dBDDyk1NVWSVFxcrEceeUS7du3SU0891axFAgAANKeAAtDChQu1aNEijR071td25513KiEhQQ899BABCAAAtGoBXQI7ceKEkpOTT2tPSkrSv/71ryYXBQAA0JICCkA/+9nPtHDhwtPaX3nlFf3P//xPk4sCAABoSQFdApNOboL+4IMP9F//9V+SpA0bNmjXrl0aN26cXC6Xr19ubm7TqwQAAGhGAQWgLVu26Oqrr5Ykbd++XZIUGRmpyMhIbdmyxdePj8YDAIDWKKAA9NFHHzV3HQAAABdMwN8DBAAA0FYRgAAAgOUQgAAAgOUQgAAAgOUQgAAAgOUQgAAAgOUQgAAAgOUQgAAAgOWYHoDy8/PlcDgUGhqqlJQUlZSUnLHvV199pVGjRsnhcMhmsykvL++0Pk8++aRsNpvf0b9//xacAQAAaGtMDUArV66Uy+VSdna2ysrKNHjwYKWlpWn//v0N9j927Jh69+6tOXPmKDo6+ozjXnnlldq3b5/vWLt2bUtNAQAAtEGmBqDc3FxNnDhRGRkZio+PV0FBgTp27KjFixc32H/IkCF67rnnNGbMGNnt9jOO26FDB0VHR/uOyMjIs9ZRU1OjqqoqvwMAALRfAd8Nvqlqa2tVWlqqzMxMX1tQUJCcTqeKi4ubNPa2bdsUGxur0NBQpaamKicnR5deeukZ++fk5GjWrFlNes3z4Zj+5wv2WvC3c85tZpcAAGgFTFsBqqioUF1dnaKiovzao6Ki5PF4Ah43JSVFS5cuVWFhoRYuXKgdO3bo+uuv15EjR854TmZmpiorK33H7t27A359AADQ+pm2AtRShg8f7vs5ISFBKSkpuuyyy/TOO+9owoQJDZ5jt9vPekkNAAC0L6atAEVGRio4OFher9ev3ev1nnWD8/nq2rWrLr/8cn377bfNNiYAAGjbTAtAISEhSkpKktvt9rXV19fL7XYrNTW12V7n6NGj2r59u2JiYpptTAAA0LaZegnM5XJp/PjxSk5O1tChQ5WXl6fq6mplZGRIksaNG6eePXsqJydH0smN019//bXv5++++06bNm1S586d1bdvX0nSY489pjvuuEOXXXaZ9u7dq+zsbAUHB2vs2LHmTBIAALQ6pgag0aNH68CBA8rKypLH41FiYqIKCwt9G6N37dqloKDvF6n27t2rq666yvd4/vz5mj9/vm688UYVFRVJkvbs2aOxY8fq4MGDuvjii3Xddddp/fr1uvjiiy/o3AAAQOtl+iboyZMna/LkyQ0+dyrUnOJwOGQYxlnHW7FiRXOVBgAA2inTb4UBAABwoRGAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5RCAAACA5XQwuwAAaO0c0/9sdgmWtXPObWaXgHaKFSAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5BCAAAGA5pgeg/Px8ORwOhYaGKiUlRSUlJWfs+9VXX2nUqFFyOByy2WzKy8tr8pgAAMB6TA1AK1eulMvlUnZ2tsrKyjR48GClpaVp//79DfY/duyYevfurTlz5ig6OrpZxgQAANZjagDKzc3VxIkTlZGRofj4eBUUFKhjx45avHhxg/2HDBmi5557TmPGjJHdbm+WMQEAgPWYFoBqa2tVWloqp9P5fTFBQXI6nSouLr6gY9bU1KiqqsrvAAAA7ZdpAaiiokJ1dXWKiorya4+KipLH47mgY+bk5CgiIsJ3xMXFBfT6AACgbTB9E3RrkJmZqcrKSt+xe/dus0sCAAAtqINZLxwZGang4GB5vV6/dq/Xe8YNzi01pt1uP+OeIgAA0P6YtgIUEhKipKQkud1uX1t9fb3cbrdSU1NbzZgAAKD9MW0FSJJcLpfGjx+v5ORkDR06VHl5eaqurlZGRoYkady4cerZs6dycnIkndzk/PXXX/t+/u6777Rp0yZ17txZffv2bdSYAAAApgag0aNH68CBA8rKypLH41FiYqIKCwt9m5h37dqloKDvF6n27t2rq666yvd4/vz5mj9/vm688UYVFRU1akwAAABTA5AkTZ48WZMnT27wuVOh5hSHwyHDMJo0JgAAAJ8CAwAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAlkMAAgAAltMqAlB+fr4cDodCQ0OVkpKikpKSs/b/zW9+o/79+ys0NFSDBg3Se++95/f8vffeK5vN5nekp6e35BQAAEAbYnoAWrlypVwul7Kzs1VWVqbBgwcrLS1N+/fvb7D/p59+qrFjx2rChAn6/PPPNXLkSI0cOVJbtmzx65eenq59+/b5jrfffvtCTAcAALQBpgeg3NxcTZw4URkZGYqPj1dBQYE6duyoxYsXN9j/xRdfVHp6uqZNm6YBAwZo9uzZuvrqq/Xyyy/79bPb7YqOjvYd3bp1O2MNNTU1qqqq8jsAAED7ZWoAqq2tVWlpqZxOp68tKChITqdTxcXFDZ5TXFzs11+S0tLSTutfVFSkHj166IorrtCkSZN08ODBM9aRk5OjiIgI3xEXF9eEWQEAgNbO1ABUUVGhuro6RUVF+bVHRUXJ4/E0eI7H4zln//T0dC1fvlxut1tz587Vxx9/rOHDh6uurq7BMTMzM1VZWek7du/e3cSZAQCA1qyD2QW0hDFjxvh+HjRokBISEtSnTx8VFRXplltuOa2/3W6X3W6/kCUCAAATmboCFBkZqeDgYHm9Xr92r9er6OjoBs+Jjo4+r/6S1Lt3b0VGRurbb79tetEAAKDNMzUAhYSEKCkpSW6329dWX18vt9ut1NTUBs9JTU316y9JH3744Rn7S9KePXt08OBBxcTENE/hAACgTTP9U2Aul0uLFi3SsmXLVF5erkmTJqm6uloZGRmSpHHjxikzM9PXf8qUKSosLNTzzz+vb775Rk8++aQ2btyoyZMnS5KOHj2qadOmaf369dq5c6fcbrdGjBihvn37Ki0tzZQ5AgCA1sX0PUCjR4/WgQMHlJWVJY/Ho8TERBUWFvo2Ou/atUtBQd/ntGuuuUZvvfWWZsyYoV/+8pfq16+fVq9erYEDB0qSgoODtXnzZi1btkyHDx9WbGyshg0bptmzZ7PPBwAASGoFAUiSJk+e7FvB+U9FRUWntf3kJz/RT37ykwb7h4WF6f3332/O8gAAQDtj+iUwAACAC40ABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALIcABAAALKdVBKD8/Hw5HA6FhoYqJSVFJSUlZ+3/m9/8Rv3791doaKgGDRqk9957z+95wzCUlZWlmJgYhYWFyel0atu2bS05BQAA0IaYHoBWrlwpl8ul7OxslZWVafDgwUpLS9P+/fsb7P/pp59q7NixmjBhgj7//HONHDlSI0eO1JYtW3x95s2bpwULFqigoEAbNmxQp06dlJaWpuPHj1+oaQEAgFbM9ACUm5uriRMnKiMjQ/Hx8SooKFDHjh21ePHiBvu/+OKLSk9P17Rp0zRgwADNnj1bV199tV5++WVJJ1d/8vLyNGPGDI0YMUIJCQlavny59u7dq9WrV1/AmQEAgNaqg5kvXltbq9LSUmVmZvragoKC5HQ6VVxc3OA5xcXFcrlcfm1paWm+cLNjxw55PB45nU7f8xEREUpJSVFxcbHGjBlz2pg1NTWqqanxPa6srJQkVVVVBTy3s6mvOdYi4+LcWuo9lXhfzdSS76vEe2sm3tv2qaXe11PjGoZxzr6mBqCKigrV1dUpKirKrz0qKkrffPNNg+d4PJ4G+3s8Ht/zp9rO1Oc/5eTkaNasWae1x8XFNW4iaDMi8syuAC2B97X94r1tn1r6fT1y5IgiIiLO2sfUANRaZGZm+q0q1dfX69ChQ/rBD34gm81mYmWtS1VVleLi4rR7926Fh4ebXQ6aEe9t+8T72n7x3jbMMAwdOXJEsbGx5+xragCKjIxUcHCwvF6vX7vX61V0dHSD50RHR5+1/6l/vV6vYmJi/PokJiY2OKbdbpfdbvdr69q16/lMxVLCw8P5D9dO8d62T7yv7Rfv7enOtfJziqmboENCQpSUlCS32+1rq6+vl9vtVmpqaoPnpKam+vWXpA8//NDXv1evXoqOjvbrU1VVpQ0bNpxxTAAAYC2mXwJzuVwaP368kpOTNXToUOXl5am6uloZGRmSpHHjxqlnz57KycmRJE2ZMkU33nijnn/+ed12221asWKFNm7cqFdeeUWSZLPZNHXqVD399NPq16+fevXqpZkzZyo2NlYjR440a5oAAKAVMT0AjR49WgcOHFBWVpY8Ho8SExNVWFjo28S8a9cuBQV9v1B1zTXX6K233tKMGTP0y1/+Uv369dPq1as1cOBAX5/HH39c1dXVuv/++3X48GFdd911KiwsVGho6AWfX3tit9uVnZ192uVCtH28t+0T72v7xXvbdDajMZ8VAwAAaEdM/yJEAACAC40ABAAALIcABAAALIcABAAALIcAhEbLz8+Xw+FQaGioUlJSVFJSYnZJaKI1a9bojjvuUGxsrGw2GzcMbidycnI0ZMgQdenSRT169NDIkSO1detWs8tCM1i4cKESEhJ8X4CYmpqqv/zlL2aX1SYRgNAoK1eulMvlUnZ2tsrKyjR48GClpaVp//79ZpeGJqiurtbgwYOVn59vdiloRh9//LEefPBBrV+/Xh9++KFOnDihYcOGqbq62uzS0ESXXHKJ5syZo9LSUm3cuFE//OEPNWLECH311Vdml9bm8DF4NEpKSoqGDBmil19+WdLJb+yOi4vTQw89pOnTp5tcHZqDzWbTu+++yxeGtkMHDhxQjx499PHHH+uGG24wuxw0s+7du+u5557ThAkTzC6lTWEFCOdUW1ur0tJSOZ1OX1tQUJCcTqeKi4tNrAxAY1RWVko6+YcS7UddXZ1WrFih6upqbvUUANO/CRqtX0VFherq6nzfzn1KVFSUvvnmG5OqAtAY9fX1mjp1qq699lq/b8xH2/Xll18qNTVVx48fV+fOnfXuu+8qPj7e7LLaHAIQALRjDz74oLZs2aK1a9eaXQqayRVXXKFNmzapsrJSv/3tbzV+/Hh9/PHHhKDzRADCOUVGRio4OFher9ev3ev1Kjo62qSqAJzL5MmT9ac//Ulr1qzRJZdcYnY5aCYhISHq27evJCkpKUmfffaZXnzxRf361782ubK2hT1AOKeQkBAlJSXJ7Xb72urr6+V2u7nuDLRChmFo8uTJevfdd/W3v/1NvXr1MrsktKD6+nrV1NSYXUabwwoQGsXlcmn8+PFKTk7W0KFDlZeXp+rqamVkZJhdGprg6NGj+vbbb32Pd+zYoU2bNql79+669NJLTawMTfHggw/qrbfe0u9//3t16dJFHo9HkhQREaGwsDCTq0NTZGZmavjw4br00kt15MgRvfXWWyoqKtL7779vdmltDh+DR6O9/PLLeu655+TxeJSYmKgFCxYoJSXF7LLQBEVFRbr55ptPax8/fryWLl164QtCs7DZbA22L1myRPfee++FLQbNasKECXK73dq3b58iIiKUkJCgX/ziF7r11lvNLq3NIQABAADLYQ8QAACwHAIQAACwHAIQAACwHAIQAACwHAIQAACwHAIQAACwHAIQAACwHAIQAACwHAIQYEE33XSTpk6d2qQxdu7cKZvNpk2bNjVLTYE6duyYRo0apfDwcNlsNh0+fLjR59psNq1evbrFamtOreX3DbQX3AsMsKBVq1bpoosuMruMZrFs2TJ98skn+vTTTxUZGamIiIjT+jz55JNavXo14QGADwEIsKDu3bubXUKz2b59uwYMGKCBAweaXQqANoRLYIAF/eclMIfDoWeffVb33XefunTpoksvvVSvvPKK3zklJSW66qqrFBoaquTkZH3++eenjbtlyxYNHz5cnTt3VlRUlH72s5+poqJC0skbr4aEhOiTTz7x9Z83b5569Oghr9d7xlp/97vf6corr5TdbpfD4dDzzz/vN4/nn39ea9askc1m00033XTa+UuXLtWsWbP0xRdfyGazyWaz+d3otaKiQnfddZc6duyofv366Q9/+EOj59SQpUuXqmvXrnr//fc1YMAAde7cWenp6dq3b5+vT319vZ566ildcsklstvtSkxMVGFhod84Tf19S9Jvf/tbDRo0SGFhYfrBD34gp9Op6urqM9YOWIoBwHJuvPFGY8qUKb7Hl112mdG9e3cjPz/f2LZtm5GTk2MEBQUZ33zzjWEYhnHkyBHj4osvNu655x5jy5Ytxh//+Eejd+/ehiTj888/NwzDMP75z38aF198sZGZmWmUl5cbZWVlxq233mrcfPPNvteZNm2acdlllxmHDx82ysrKjJCQEOP3v//9GevcuHGjERQUZDz11FPG1q1bjSVLlhhhYWHGkiVLDMMwjIMHDxoTJ040UlNTjX379hkHDx48bYxjx44Zjz76qHHllVca+/btM/bt22ccO3bMMAzDkGRccsklxltvvWVs27bNePjhh43OnTv7xmnMnP7TkiVLjIsuushwOp3GZ599ZpSWlhoDBgww7rnnHl+f3NxcIzw83Hj77beNb775xnj88ceNiy66yPj73//ebL/vvXv3Gh06dDByc3ONHTt2GJs3bzby8/ONI0eOnLF2wEoIQIAFNRSAfvrTn/oe19fXGz169DAWLlxoGIZh/PrXvzZ+8IMfGP/v//0/X5+FCxf6/UGePXu2MWzYML/X2b17tyHJ2Lp1q2EYhlFTU2MkJiYa//3f/23Ex8cbEydOPGud99xzj3Hrrbf6tU2bNs2Ij4/3PZ4yZYpx4403nnWc7OxsY/Dgwae1SzJmzJjhe3z06FFDkvGXv/yl0XP6T0uWLDEkGd9++62vLT8/34iKivI9jo2NNZ555hm/84YMGWI88MADhmE0z++7tLTUkGTs3LnzTL8WwNK4BAZAkpSQkOD72WazKTo6Wvv375cklZeXKyEhQaGhob4+qampfud/8cUX+uijj9S5c2ff0b9/f0kn9+lIUkhIiN5880397ne/0/Hjx/XCCy+ctaby8nJde+21fm3XXnuttm3bprq6usAn+2/+fd6dOnVSeHi4b96NmVNDOnbsqD59+vgex8TE+MasqqrS3r17G5xXeXm5pOb5fQ8ePFi33HKLBg0apJ/85CdatGiR/vnPf5737wdor9gEDUCSTvtUmM1mU319faPPP3r0qO644w7NnTv3tOdiYmJ8P3/66aeSpEOHDunQoUPq1KlTgBU3j7PNu7FzasyYhmE0Q7XfO1dtwcHB+vDDD/Xpp5/qgw8+0EsvvaQnnnhCGzZsUK9evZq1FqAtYgUIwDkNGDBAmzdv1vHjx31t69ev9+tz9dVX66uvvpLD4VDfvn39jlMhZ/v27XrkkUe0aNEipaSkaPz48WcNWQMGDNC6dev82tatW6fLL79cwcHBja4/JCQkoBWjxszpfIWHhys2NrbBecXHx0tqvt+3zWbTtddeq1mzZunzzz9XSEiI3n333YDqBtobAhCAc7rnnntks9k0ceJEff3113rvvfc0f/58vz4PPvigDh06pLFjx+qzzz7T9u3b9f777ysjI0N1dXWqq6vTT3/6U6WlpSkjI0NLlizR5s2b/T7V9Z8effRRud1uzZ49W3//+9+1bNkyvfzyy3rsscfOq36Hw6EdO3Zo06ZNqqioUE1NTaPOO9ecAjVt2jTNnTtXK1eu1NatWzV9+nRt2rRJU6ZMkdQ8v+8NGzbo2Wef1caNG7Vr1y6tWrVKBw4c0IABAwKuG2hPCEAAzqlz58764x//qC+//FJXXXWVnnjiidMuvZxa1airq9OwYcM0aNAgTZ06VV27dlVQUJCeeeYZ/eMf/9Cvf/1rSScv07zyyiuaMWOGvvjiiwZf9+qrr9Y777yjFStWaODAgcrKytJTTz2le++997zqHzVqlNLT03XzzTfr4osv1ttvv92o8841p0A9/PDDcrlcevTRRzVo0CAVFhbqD3/4g/r16yepeX7f4eHhWrNmjX70ox/p8ssv14wZM/T8889r+PDhAdcNtCc2o7kvTAMAALRyrAABAADLIQABAADLIQABAADLIQABAADLIQABAADLIQABAADLIQABAADLIQABAADLIQABAADLIQABAADLIQABAADL+f//DxhW1IMiPAAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_bitstrings(counts_4_nodes_1_step_qpu[\"quantum_walk_measurement_counts\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "9ca18304",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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u3aoZM2ZoyZIlARsOAACgrXw+LdW7d+8Wb9zX2Nio9PR0deny7V1888036tKli+68807eBwcAANjG57gpLi4+h2MAAAAEhs9xk52dfS7nAAAACIh2vYmfJH399dc6efJki7Vz8YGUAAAAvvDrguLGxkbl5uaqb9++6tGjh3r37t3iBgAAYBe/4uaXv/yl3nzzTZWWlio0NFRPPfWU5s6dq9jYWD377LOBnhEAAMBnfp2Wevnll/Xss89q3LhxcjqduvLKKzV48GANGDBAzz//vG677bZAzwkAAOATv47cHDlyRAMHDpT07fU1R44ckST927/9mzZu3Bi46QAAANrIr7gZOHCg9u7dK0kaOnSoXnjhBUnfHtH57oM0AQAA7OBX3DidTr3//vuSpNmzZ6ukpERhYWG6//77NWvWrIAOCAAA0BZ+XXNz//33e7/OzMzUzp07VVVVpcGDByspKSlgwwEAALRVu9/nRpLi4+MVHx8fiLsCAABoF79OS0mS2+3Wddddp0GDBmnQoEG67rrr9MYbbwRyNgAAgDbzK25+97vfaeLEierZs6dmzJihGTNmKCIiQj/5yU9UUlIS6BkBAAB85tdpqQULFuiRRx5Rbm6ud+3ee+/V2LFjtWDBAk2fPj1gAwIAALSFX0dujh49qokTJ562PmHCBNXX17d7KAAAAH/5FTc33HCD1qxZc9r6Sy+9pOuuu67dQwEAAPjL59NSjz76qPfrxMREPfTQQ9qwYYMyMjIkSe+88442bdqkmTNnBn5KAAAAH/kcN4888kiL73v37q2PP/5YH3/8sXetV69eWrZsmebMmRO4CQEAANrA57j57uMWAAAAOjK/3+fmO5ZlybKsQMwCAADQbn7HzbPPPqvhw4erW7du6tatm5KSkvT73/8+kLMBAAC0mV/vc1NUVKRf//rXys3N1dixYyVJb7/9tn7+85+rrq6uxWdPAQAAnE9+xc1jjz2m0tJSTZ061bt2ww036IorrtCDDz5I3AAAANv4dVqqurpaY8aMOW19zJgxqq6ubvdQAAAA/vIrbgYPHqwXXnjhtPXy8nINGTKk3UMBAAD4y6/TUnPnztXkyZO1ceNG7zU3mzZtktvtbjV6AAAAzhe/jtzcfPPN2rJli6KiorR27VqtXbtWUVFR2rJli2666aZAzwgAAOCzNh+5OXXqlP77v/9bv/71r/Xcc8+di5kAAAD81uYjN127dtWLL754LmYBAABoN79OS02aNElr164N8CgAAADt59cFxUOGDNG8efO0adMmpaamqkePHi1+fu+99wZkOAAAgLbyK26efvpp9erVS5WVlaqsrGzxM4fDQdwAAADb+BU3//wJ4d99aKbD4QjMRAAAAO3g9wdnPv300xo2bJjCwsIUFhamYcOG6amnngrkbAAAAG3m15Gb/Px8FRUV6Z577lFGRoYkqaKiQvfff7/279+vefPmBXRIAAAAX/kVN6WlpVq6dKmmTJniXbvhhhuUlJSke+65h7gBAAC28eu01KlTp5SWlnbaempqqr755pt2DwUAAOAvv+LmjjvuUGlp6WnrTz75pG677bZ2DwUAAOAvv05LSd9eUPzaa6/phz/8oSTp3Xff1f79+zV16lS5XC7vdkVFRe2fEgAAwEd+xc2OHTs0cuRISdKePXskSVFRUYqKitKOHTu82/HycAAAcL75FTfr168P9BwAAAAB4ff73AAAAHRExA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMEqHiJuSkhLFx8crLCxM6enp2rJlyxm3Xbp0qa688kr17t1bvXv3VmZm5lm3BwAAFxbb46a8vFwul0sFBQWqqqpScnKysrKydOjQoVa337Bhg6ZMmaL169eroqJCcXFxmjBhgr744ovzPDkAAOiIbI+boqIi5eTkyOl0KjExUWVlZerevbuWLVvW6vbPP/+87r77bqWkpGjo0KF66qmn5PF45Ha7W92+qalJDQ0NLW4AAMBctsbNyZMnVVlZqczMTO9aUFCQMjMzVVFR4dN9nDhxQqdOnVKfPn1a/XlhYaEiIyO9t7i4uIDMDgAAOiZb46aurk7Nzc2Kjo5usR4dHa2amhqf7uNXv/qVYmNjWwTSP8vLy1N9fb33duDAgXbPDQAAOi6/PxW8I1i4cKFWrVqlDRs2KCwsrNVtQkNDFRoaep4nAwAAdrE1bqKiohQcHKza2toW67W1tYqJiTnrvkuWLNHChQv1xhtvKCkp6VyOCQAAOhFbT0uFhIQoNTW1xcXA310cnJGRccb9fvOb32j+/Plat26d0tLSzseoAACgk7D9tJTL5VJ2drbS0tI0evRoFRcXq7GxUU6nU5I0depU9evXT4WFhZKkRYsWKT8/XytXrlR8fLz32pzw8HCFh4fb9jgAAEDHYHvcTJ48WYcPH1Z+fr5qamqUkpKidevWeS8y3r9/v4KC/nGAqbS0VCdPntQtt9zS4n4KCgr04IMPns/RAQBAB2R73EhSbm6ucnNzW/3Zhg0bWny/b9++cz8QAADotGx/Ez8AAIBAIm4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGIW4AQAARiFuAACAUYgbAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGIW4AQAARiFuAACAUYgbAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGIW4AQAARiFuAACAUYgbAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGIW4AQAARiFuAACAUYgbAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGMX2uCkpKVF8fLzCwsKUnp6uLVu2nHHbjz76SDfffLPi4+PlcDhUXFx8/gYFAACdgq1xU15eLpfLpYKCAlVVVSk5OVlZWVk6dOhQq9ufOHFCAwcO1MKFCxUTE3OepwUAAJ2BrXFTVFSknJwcOZ1OJSYmqqysTN27d9eyZcta3X7UqFFavHixfvrTnyo0NPQ8TwsAADoD2+Lm5MmTqqysVGZm5j+GCQpSZmamKioqAvZ7mpqa1NDQ0OIGAADMZVvc1NXVqbm5WdHR0S3Wo6OjVVNTE7DfU1hYqMjISO8tLi4uYPcNAAA6HtsvKD7X8vLyVF9f770dOHDA7pEAAMA51MWuXxwVFaXg4GDV1ta2WK+trQ3oxcKhoaFcnwMAwAXEtiM3ISEhSk1Nldvt9q55PB653W5lZGTYNRYAAOjkbDtyI0kul0vZ2dlKS0vT6NGjVVxcrMbGRjmdTknS1KlT1a9fPxUWFkr69iLkjz/+2Pv1F198oe3btys8PFyDBw+27XEAAICOw9a4mTx5sg4fPqz8/HzV1NQoJSVF69at815kvH//fgUF/ePg0sGDBzVixAjv90uWLNGSJUt01VVXacOGDed7fAAA0AHZGjeSlJubq9zc3FZ/9q/BEh8fL8uyzsNUAACgszL+1VIAAODCQtwAAACjEDcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3AADAKMQNAAAwCnEDAACMQtwAAACjEDcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3AADAKMQNAAAwCnEDAACMQtwAAACjEDcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3AADAKMQNAAAwCnEDAACMQtwAAACjEDcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCjEDQAAMApxAwAAjELcAAAAoxA3AADAKMQNAAAwShe7BwA6g/jZf7F7hAvWvoXX2j0CgE6GIzcAAMAoxA0AADAKcQMAAIxC3AAAAKMQNwAAwCi8WgoAYCRe5Wgfu1/lyJEbAABgFOIGAAAYpUPETUlJieLj4xUWFqb09HRt2bLlrNv/7//+r4YOHaqwsDANHz5cr7zyynmaFAAAdHS2x015eblcLpcKCgpUVVWl5ORkZWVl6dChQ61uv3nzZk2ZMkXTpk3Ttm3bNGnSJE2aNEk7duw4z5MDAICOyPa4KSoqUk5OjpxOpxITE1VWVqbu3btr2bJlrW7/29/+VhMnTtSsWbOUkJCg+fPna+TIkXr88cfP8+QAAKAjsvXVUidPnlRlZaXy8vK8a0FBQcrMzFRFRUWr+1RUVMjlcrVYy8rK0tq1a1vdvqmpSU1NTd7v6+vrJUkNDQ3tnL51nqYT5+R+8f3O1XMq8bza6Vw+rxLPrZ14bs11Lp7b7+7Tsqzv3dbWuKmrq1Nzc7Oio6NbrEdHR2vXrl2t7lNTU9Pq9jU1Na1uX1hYqLlz5562HhcX5+fU6Kgii+2eAOcCz6u5eG7NdS6f22PHjikyMvKs2xj/Pjd5eXktjvR4PB4dOXJEF110kRwOh42TdSwNDQ2Ki4vTgQMHFBERYfc4CCCeW3Px3JqJ57V1lmXp2LFjio2N/d5tbY2bqKgoBQcHq7a2tsV6bW2tYmJiWt0nJiamTduHhoYqNDS0xVqvXr38H9pwERER/GMyFM+tuXhuzcTzerrvO2LzHVsvKA4JCVFqaqrcbrd3zePxyO12KyMjo9V9MjIyWmwvSa+//voZtwcAABcW209LuVwuZWdnKy0tTaNHj1ZxcbEaGxvldDolSVOnTlW/fv1UWFgoSZoxY4auuuoqPfzww7r22mu1atUqbd26VU8++aSdDwMAAHQQtsfN5MmTdfjwYeXn56umpkYpKSlat26d96Lh/fv3KyjoHweYxowZo5UrV2rOnDn6n//5Hw0ZMkRr167VsGHD7HoIRggNDVVBQcFpp/DQ+fHcmovn1kw8r+3nsHx5TRUAAEAnYfub+AEAAAQScQMAAIxC3AAAAKMQNwAAwCjEDSRJJSUlio+PV1hYmNLT07Vlyxa7R0I7bdy4Uddff71iY2PlcDjO+Plr6FwKCws1atQo9ezZU3379tWkSZO0e/duu8dCAJSWliopKcn75n0ZGRn661//avdYnRJxA5WXl8vlcqmgoEBVVVVKTk5WVlaWDh06ZPdoaIfGxkYlJyerpKTE7lEQQG+99ZamT5+ud955R6+//rpOnTqlCRMmqLGx0e7R0E79+/fXwoULVVlZqa1bt+pHP/qRbrzxRn300Ud2j9bp8FJwKD09XaNGjdLjjz8u6dt3iY6Li9M999yj2bNn2zwdAsHhcGjNmjWaNGmS3aMgwA4fPqy+ffvqrbfe0r//+7/bPQ4CrE+fPlq8eLGmTZtm9yidCkduLnAnT55UZWWlMjMzvWtBQUHKzMxURUWFjZMB8EV9fb2kb/8ThDmam5u1atUqNTY28vFCfrD9HYphr7q6OjU3N3vfEfo70dHR2rVrl01TAfCFx+PRfffdp7Fjx/Iu7Yb48MMPlZGRoa+//lrh4eFas2aNEhMT7R6r0yFuAKCTmj59unbs2KG3337b7lEQIJdffrm2b9+u+vp6/fGPf1R2drbeeustAqeNiJsLXFRUlIKDg1VbW9tivba2VjExMTZNBeD75Obm6s9//rM2btyo/v372z0OAiQkJESDBw+WJKWmpuq9997Tb3/7Wz3xxBM2T9a5cM3NBS4kJESpqalyu93eNY/HI7fbzXleoAOyLEu5ublas2aN3nzzTf3gBz+weyScQx6PR01NTXaP0elw5AZyuVzKzs5WWlqaRo8ereLiYjU2NsrpdNo9Gtrh+PHj+vTTT73f7927V9u3b1efPn106aWX2jgZ2mP69OlauXKlXnrpJfXs2VM1NTWSpMjISHXr1s3m6dAeeXl5uuaaa3TppZfq2LFjWrlypTZs2KBXX33V7tE6HV4KDknS448/rsWLF6umpkYpKSl69NFHlZ6ebvdYaIcNGzZo/Pjxp61nZ2drxYoV538gBITD4Wh1ffny5frZz352fodBQE2bNk1ut1vV1dWKjIxUUlKSfvWrX+nqq6+2e7ROh7gBAABG4ZobAABgFOIGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIG8BA48aN03333deu+9i3b58cDoe2b98ekJn8deLECd18882KiIiQw+HQ0aNHfd7X4XBo7dq152y2QOoof2/ABHy2FGCg1atXq2vXrnaPERDPPPOM/va3v2nz5s2KiopSZGTkads8+OCDWrt2LWEAQBJxAxipT58+do8QMHv27FFCQoKGDRtm9ygAOglOSwEG+tfTUvHx8VqwYIHuvPNO9ezZU5deeqmefPLJFvts2bJFI0aMUFhYmNLS0rRt27bT7nfHjh265pprFB4erujoaN1xxx2qq6uT9O0HdYaEhOhvf/ubd/vf/OY36tu3r2pra88464svvqgrrrhCoaGhio+P18MPP9zicTz88MPauHGjHA6Hxo0bd9r+K1as0Ny5c/X+++/L4XDI4XC0+GDQuro63XTTTerevbuGDBmiP/3pTz4/ptasWLFCvXr10quvvqqEhASFh4dr4sSJqq6u9m7j8Xg0b9489e/fX6GhoUpJSdG6deta3E97/96S9Mc//lHDhw9Xt27ddNFFFykzM1ONjY1nnB24YFgAjHPVVVdZM2bM8H4/YMAAq0+fPlZJSYn1ySefWIWFhVZQUJC1a9cuy7Is69ixY9bFF19s3XrrrdaOHTusl19+2Ro4cKAlydq2bZtlWZb11VdfWRdffLGVl5dn7dy506qqqrKuvvpqa/z48d7fM2vWLGvAgAHW0aNHraqqKiskJMR66aWXzjjn1q1braCgIGvevHnW7t27reXLl1vdunWzli9fblmWZX355ZdWTk6OlZGRYVVXV1tffvnlafdx4sQJa+bMmdYVV1xhVVdXW9XV1daJEycsy7IsSVb//v2tlStXWp988ol17733WuHh4d778eUx/avly5dbXbt2tTIzM6333nvPqqystBISEqxbb73Vu01RUZEVERFh/eEPf7B27dpl/fKXv7S6du1q/f3vfw/Y3/vgwYNWly5drKKiImvv3r3WBx98YJWUlFjHjh074+zAhYK4AQzUWtzcfvvt3u89Ho/Vt29fq7S01LIsy3riiSesiy66yPq///s/7zalpaUt/rOdP3++NWHChBa/58CBA5Yka/fu3ZZlWVZTU5OVkpJi/dd//ZeVmJho5eTknHXOW2+91br66qtbrM2aNctKTEz0fj9jxgzrqquuOuv9FBQUWMnJyaetS7LmzJnj/f748eOWJOuvf/2rz4/pXy1fvtySZH366afetZKSEis6Otr7fWxsrPXQQw+12G/UqFHW3XffbVlWYP7elZWVliRr3759Z/qzABcsTksBF4ikpCTv1w6HQzExMTp06JAkaefOnUpKSlJYWJh3m4yMjBb7v//++1q/fr3Cw8O9t6FDh0r69roYSQoJCdHzzz+vF198UV9//bUeeeSRs860c+dOjR07tsXa2LFj9cknn6i5udn/B/tP/vlx9+jRQxEREd7H7ctjak337t01aNAg7/eXXHKJ9z4bGhp08ODBVh/Xzp07JQXm752cnKwf//jHGj58uP7zP/9TS5cu1VdffdXmvw9gIi4oBi4Q//rqKYfDIY/H4/P+x48f1/XXX69Fixad9rNLLrnE+/XmzZslSUeOHNGRI0fUo0cPPycOjLM9bl8fky/3aVlWAKb9h++bLTg4WK+//ro2b96s1157TY899pgeeOABvfvuu/rBD34Q0FmAzoYjNwCUkJCgDz74QF9//bV37Z133mmxzciRI/XRRx8pPj5egwcPbnH7LmD27Nmj+++/X0uXLlV6erqys7PPGlAJCQnatGlTi7VNmzbpsssuU3BwsM/zh4SE+HWkx5fH1FYRERGKjY1t9XElJiZKCtzf2+FwaOzYsZo7d662bdumkJAQrVmzxq+5AZMQNwB06623yuFwKCcnRx9//LFeeeUVLVmypMU206dP15EjRzRlyhS999572rNnj1599VU5nU41NzerublZt99+u7KysuR0OrV8+XJ98MEHLV799K9mzpwpt9ut+fPn6+9//7ueeeYZPf744/rFL37Rpvnj4+O1d+9ebd++XXV1dWpqavJpv+97TP6aNWuWFi1apPLycu3evVuzZ8/W9u3bNWPGDEmB+Xu/++67WrBggbZu3ar9+/dr9erVOnz4sBISEvyeGzAFcQNA4eHhevnll/Xhhx9qxIgReuCBB047HfLd0Yjm5mZNmDBBw4cP13333adevXopKChIDz30kD777DM98cQTkr49dfLkk09qzpw5ev/991v9vSNHjtQLL7ygVatWadiwYcrPz9e8efP0s5/9rE3z33zzzZo4caLGjx+viy++WH/4wx982u/7HpO/7r33XrlcLs2cOVPDhw/XunXr9Kc//UlDhgyRFJi/d0REhDZu3Kif/OQnuuyyyzRnzhw9/PDDuuaaa/yeGzCFwwr0iWIAAAAbceQGAAAYhbgBAABGIW4AAIBRiBsAAGAU4gYAABiFuAEAAEYhbgAAgFGIGwAAYBTiBgAAGIW4AQAARiFuAACAUf4fBiXKc4LRoEkAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_bitstrings(counts_4_nodes_2_steps_qpu[\"quantum_walk_measurement_counts\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "aea10773",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_bitstrings(counts_4_nodes_3_steps_qpu[\"quantum_walk_measurement_counts\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "54d42e7d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_bitstrings(counts_4_nodes_4_steps_qpu[\"quantum_walk_measurement_counts\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84d345d5",
   "metadata": {},
   "source": [
    "Some comments of the noise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "a19a546e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Task Summary\n",
      "{'arn:aws:braket:us-west-1::device/qpu/rigetti/Ankaa-2': {'shots': 4000, 'tasks': {'COMPLETED': 4}}} \n",
      "\n",
      "Estimated cost to run this example: 4.80 USD\n"
     ]
    }
   ],
   "source": [
    "print(\"Task Summary\")\n",
    "print(f\"{tracker.quantum_tasks_statistics()} \\n\")\n",
    "print(\n",
    "    f\"Estimated cost to run this example: {tracker.qpu_tasks_cost() + tracker.simulator_tasks_cost():.2f} USD\"\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a7484c3",
   "metadata": {},
   "source": [
    "Note: Charges shown are estimates based on your Amazon Braket simulator and quantum processing unit (QPU) task usage. Estimated charges shown may differ from your actual charges. Estimated charges do not factor in any discounts or credits, and you may experience additional charges based on your use of other services such as Amazon Elastic Compute Cloud (Amazon EC2)."
   ]
  }
 ],
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    "name": "ipython",
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