2_quantum_fourier_transform.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "f6146cef-41da-4219-be7b-f44ed09c9457",
   "metadata": {},
   "source": [
    "# Quantum Fourier Transform\n",
    "\n",
    "The Quantum Fourier Transform (QFT) is an important subroutine to many quantum algorithms, most famously Shor's algorithm for factoring and the quantum phase estimation (QPE) algorithm for estimating the eigenvalues of a unitary operator [1, 2]. The QFT can be performed efficiently on a quantum computer, using Hadamard gates, controlled phase shift gates and swap gates.\n",
    "\n",
    "## Reference\n",
    "\n",
    "[1] Wikipedia: https://en.wikipedia.org/wiki/Quantum_Fourier_transform\n",
    "\n",
    "[2] More Details: https://young.physics.ucsc.edu/150/QFT-FFT.pdf\n",
    "\n",
    "[3] Nielsen, Michael A., Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (2nd ed.). Cambridge: Cambridge University Press."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "47e6bff8-4e44-4f76-a3d6-f29c34bfa4ed",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "from utils.notebook_plotting import plot_bitstrings_formatted\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "import math\n",
    "import numpy as np\n",
    "from braket.circuits import Circuit\n",
    "from braket.devices import LocalSimulator\n",
    "\n",
    "from braket.experimental.algorithms.quantum_fourier_transform import (\n",
    "    quantum_fourier_transform as qft_module\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d7d0fbfe-8c10-479a-89b3-dfe73ba960c9",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Circuit"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ee77c35-45c9-4073-93be-c2b9bd5b37a7",
   "metadata": {},
   "source": [
    "![image info](./images/qft.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50b0f18f-38fc-4570-bb42-36848d571754",
   "metadata": {
    "tags": []
   },
   "source": [
    "### Hints:\n",
    "\n",
    "<details>\n",
    "<summary>$\\textbf{Hint 1}$</summary>\n",
    "$H$ is the Hadamard gate, implemented using the function $\\textbf{Circuit.h(qubit)}$. Circuit.h(0) applies Hadamard gate to the first input qubit ($x_1$ in the above sketch)\n",
    "</details>\n",
    "\n",
    "<details>\n",
    "<summary>$\\textbf{Hint 2}$</summary>\n",
    "$R_{k}$ is the controlled phase gate with $R_{k}$ = \n",
    "$\\begin{pmatrix}\n",
    "    1 & 0 \\\\\n",
    "    0 & e^{i2\\pi / 2^k}\n",
    "\\end{pmatrix}$\n",
    "where the rotation angle is defined by $2\\pi / 2^k$. In the above sketch, circled $R_{k}$ is the operation acting on the target qubit, where the channel denoted with $\\bullet$ is the control qubit.\n",
    "</details>\n",
    "\n",
    "<details>\n",
    "<summary>$\\textbf{Hint 3}$</summary>\n",
    "$R_{k}$ is implemented using the function $\\textbf{Circuit.cphaseshift(control_qubit, target_qubit, angle)}$. Circuit.cphaseshift(0, 2, 2$\\pi$) applies 2$\\pi$ rotation to third input qubit (target=2) using first input qubit (control=0) as control qubit.\n",
    "</details>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "49957f8c-4ac4-44cc-a092-62bbf07ad351",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def quantum_fourier_transform_circuit(num_qubits: int) -> Circuit:\n",
    "    \"\"\"Construct a circuit object corresponding to the Quantum Fourier Transform (QFT)\n",
    "    algorithm, applied to the argument qubits.  Does not use recursion to generate the QFT.\n",
    "\n",
    "    Args:\n",
    "        num_qubits (int): number of qubits on which to apply the QFT\n",
    "\n",
    "    Returns:\n",
    "        Circuit: qft circuit\n",
    "    \"\"\"\n",
    "\n",
    "    qft_circ = Circuit()\n",
    "    qubits = list(range(num_qubits))\n",
    "    \n",
    "    # Loop over the qubits\n",
    "    for k in range(num_qubits):\n",
    "        # First add a Hadamard gate\n",
    "        \"\"\"\n",
    "        write code here\n",
    "        \"\"\"\n",
    "\n",
    "        # Then apply the controlled rotations, with weights (angles) defined by the distance\n",
    "        # to the control qubit. Start on the qubit after qubit k, and iterate until the end.\n",
    "        for j in range(1, num_qubits - k):\n",
    "            # define angle\n",
    "            \"\"\"\n",
    "            write code here\n",
    "            \"\"\"\n",
    "            # add controlled phase gate\n",
    "            \"\"\"\n",
    "            write code here\n",
    "            \"\"\"\n",
    "\n",
    "    # Then add SWAP gates to reverse the order of the qubits:\n",
    "    # IMPORTANT: Please ignore the swap operations, they are extra steps\n",
    "    #            to our circuit implementation, in order to make it comparable \n",
    "    #            to the QFT algorithm within braket.\n",
    "    for i in range(math.floor(num_qubits / 2)):\n",
    "        qft_circ.swap(qubits[i], qubits[-i - 1])\n",
    "\n",
    "    return qft_circ"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98d286ff-851b-4866-9e4a-b94a911ee50c",
   "metadata": {},
   "source": [
    "# Evaluation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84376660-abbc-4793-8554-3b20333b6583",
   "metadata": {},
   "source": [
    "Evaluate your results:\n",
    "- We implemented two test cases to implement your qft algorithm on. For the first one, we simple use $|0\\rangle^{\\otimes n}$ (all input channels are $|0\\rangle$ state). For the second test case, we apply a state preparation circuit prior to qft. We therefore apply a hadamard gate to one of the input qubits (denoted by \"h_tgt\"). Try changing the \"h_tgt\" parameter and observe how the qft of the input state changes.\n",
    "- Test your implementation by comparing the probability plots of your qft implementation and the groudtruth by braket. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2b8a4473-2de5-412e-910c-b4f83e011f7d",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "n_qubits = 4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "568de14e-ea2b-4b7d-869d-9153f417c338",
   "metadata": {},
   "source": [
    "## 1) Input without State Preparation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "2bef3ba2-05d4-4f0c-b108-87e642b7db31",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.0664 0.0628 0.0597 0.0616 0.0668 0.0613 0.0591 0.0604 0.0651 0.0625\n",
      " 0.0657 0.0636 0.0608 0.0622 0.0599 0.0621]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0.0, 1.0)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Groundtruth\n",
    "task = qft_module.run_quantum_fourier_transform(\n",
    "    qubits=range(n_qubits), \n",
    "    n_shots=10000, \n",
    "    device=LocalSimulator()\n",
    ")\n",
    "probabilities = task.result().values[0]\n",
    "print(probabilities)\n",
    "plot_bitstrings_formatted(probabilities)\n",
    "plt.ylim([0, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b8f1e53e-bd02-45ca-b0f7-43abb79f262c",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Your implementation\n",
    "circuit = quantum_fourier_transform_circuit(n_qubits)\n",
    "device = LocalSimulator()\n",
    "\n",
    "circuit.probability()\n",
    "task = device.run(circuit, shots=10000)\n",
    "\n",
    "probabilities = task.result().values[0]\n",
    "print(probabilities)\n",
    "plot_bitstrings_formatted(probabilities)\n",
    "plt.ylim([0, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "88119887-60b0-4758-9d1c-97954770ee9f",
   "metadata": {
    "tags": []
   },
   "source": [
    "## 1) Input with State Preparation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f1727a34-2662-44df-994c-fbe6b548dbc5",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "h_tgt = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b7514905-7c7e-4859-8c65-326397403c84",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.1244 0.1054 0.063  0.021  0.     0.0172 0.0614 0.1087 0.1219 0.1031\n",
      " 0.0624 0.021  0.     0.019  0.0648 0.1067]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0.0, 1.0)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Groundtruth\n",
    "state_prep_circ = Circuit().h(h_tgt) # Applying Hadamard to the third input qubit\n",
    "\n",
    "task = qft_module.run_quantum_fourier_transform(\n",
    "    qubits=range(n_qubits), \n",
    "    n_shots=10000, \n",
    "    state_prep_circ=state_prep_circ, \n",
    "    device=LocalSimulator()\n",
    ")\n",
    "\n",
    "probabilities = task.result().values[0]\n",
    "print(probabilities)\n",
    "plot_bitstrings_formatted(probabilities)\n",
    "plt.ylim([0, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cdcb1a91-2cc4-46f9-86fb-7ee8f6cf541b",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Your implementation\n",
    "circuit = state_prep_circ + quantum_fourier_transform_circuit(n_qubits)\n",
    "device = LocalSimulator()\n",
    "\n",
    "circuit.probability()\n",
    "task = device.run(circuit, shots=10000)\n",
    "\n",
    "probabilities = task.result().values[0]\n",
    "print(probabilities)\n",
    "plot_bitstrings_formatted(probabilities)\n",
    "plt.ylim([0, 1])"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "conda_braket",
   "language": "python",
   "name": "conda_braket"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}