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---------------C-------------C------------C-------------C------------X----------------\n",
      "                      |             |            |             |                             \n",
      "q1 : ---C-----------H-|-------------|------------PHASE01(3.14)-PHASE(-3.14)-H-C--------------\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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",
      "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": 16,
   "id": "d8d7c3f3",
   "metadata": {},
   "outputs": [],
   "source": [
    "from braket.aws import AwsDevice\n",
    "\n",
    "qpu = AwsDevice(\"arn:aws:braket:us-west-1::device/qpu/rigetti/Aspen-M-3\")\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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",
      "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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",
      "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": 17,
   "id": "a19a546e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Task Summary\n",
      "{'arn:aws:braket:us-west-1::device/qpu/rigetti/Aspen-M-3': {'shots': 4000, 'tasks': {'COMPLETED': 4}}} \n",
      "\n",
      "Estimated cost to run this example: 2.60 USD\n"
     ]
    }
   ],
   "source": [
    "print(\"Task Summary\")\n",
    "print(f\"{tracker.quantum_tasks_statistics()} \\n\")\n",
    "print(f\"Estimated cost to run this example: {tracker.qpu_tasks_cost() + tracker.simulator_tasks_cost():.2f} USD\")"
   ]
  },
  {
   "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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