01.CM_trotterization.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "7152f2bf-5b38-470a-a523-6398a51c2c16",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "<frozen importlib._bootstrap>:219: RuntimeWarning: scipy._lib.messagestream.MessageStream size changed, may indicate binary incompatibility. Expected 56 from C header, got 64 from PyObject\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Importing standard Qiskit libraries\n",
    "from qiskit import QuantumCircuit, QuantumRegister, transpile, Aer, IBMQ, execute\n",
    "from qiskit.tools.jupyter import *\n",
    "from qiskit.visualization import *\n",
    "#from ibm_quantum_widgets import *\n",
    "from qiskit.providers.aer import QasmSimulator\n",
    "from tqdm.notebook import tqdm\n",
    "\n",
    "from qiskit.providers.aer import QasmSimulator\n",
    "from qiskit.tools.monitor import job_monitor\n",
    "from qiskit.circuit import Parameter\n",
    "import qiskit.quantum_info as qi\n",
    "\n",
    "# Suppress warnings\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "attachments": {
    "tight-binding.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "3e8021b4",
   "metadata": {},
   "source": [
    "# Tight-binding Model\n",
    "\n",
    "<div>\n",
    "<img src=\"attachment:tight-binding.png\" width=\"400\"/>\n",
    "</div>\n",
    "\n",
    "The tight-binding model is a quantum mechanical picture used to describe the conductance of electrons in solid-state materials. In this model, each atom is treated as a lattice site and the particle needs energy $\\epsilon_i$ to occupy that site. The electron can tunnel from a site to its neighbors with an energy cost of $J$. The following Hamiltonian describes this model:\n",
    "\n",
    "$$H_{\\rm tb}/\\hbar = \\sum_i \\epsilon_i Z_i + J \\sum_{\\langle i,j \\rangle} (X_i X_j + Y_i Y_j)$$\n",
    "\n",
    "The first term in the Hamiltonian describes the on-site energy of each site, and the second term describes the interaction energy between neighboring lattice sites. In the presence of a periodic lattice potential, the wavefunction of a given quantum particle overlaps neighboring lattice sites, leading to extended Bloch wavefunctions. In a uniform lattice, where all the site energies are equal ($\\epsilon_i=0$), the electron propagation is linear in time and is described by a continuous-time quantum random walk. This is in contrast to classical diffusive transport, where the propagation is quadratically slower in time.\n",
    "\n",
    "The system Hamiltonian allows us to determine how the quantum state evolves with time. This time evolution is governed by the Schrodinger equation:\n",
    "\n",
    "$$ i \\hbar \\frac{\\partial}{\\partial t}|\\psi(t)\\rangle = H |\\psi(t)\\rangle $$\n",
    "\n",
    "In case of a time-independent Hamiltonian (a Hamiltonian that doesn't change with time), the solution to the Schrodinger equation will take the form below:\n",
    "\n",
    "$$ |\\psi(t)\\rangle = e^{-i H t / \\hbar} |\\psi(0)\\rangle $$\n",
    "\n",
    "In this exercise we study the time dynamics under the tight-binding Hamiltonian, and construct a Trotterized circuit for its time evolution."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "172dc11b",
   "metadata": {},
   "source": [
    "## 1. Time evolution under the tight-binding Hamiltonian\n",
    "\n",
    "First, let's consider time evolution under the tight-binding Hamiltonian with 3 sites. Here we'll set $J=1$, and $\\epsilon_i=0$ for a uniform lattice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c8034d3c-e2c4-4dda-aa78-e44b34ad494c",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import Pauli operators (I, X, Y, Z)\n",
    "from qiskit.opflow import I, X, Y, Z\n",
    "\n",
    "J = 1\n",
    "\n",
    "# tight-binding Hamiltonian\n",
    "def H_tb():\n",
    "    # Interactions (I is the identity matrix; X and Y are Pauli matricies; ^ is a tensor product)\n",
    "    XXs = (I^X^X) + (X^X^I)\n",
    "    YYs = (I^Y^Y) + (Y^Y^I)\n",
    "    \n",
    "    # Sum interactions\n",
    "    H = J*(XXs + YYs)\n",
    "    \n",
    "    # Return Hamiltonian\n",
    "    return H"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "36f59aec-f85d-4898-a69d-231f7b53e2a1",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Unitary evolution under the tight-binding Hamiltonian\n",
    "def U_tb(t):\n",
    "    H = H_tb()\n",
    "    return (t * H).exp_i()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b52f1e66",
   "metadata": {},
   "source": [
    "We initialize the state of our system to $|100\\rangle$, and track the probabilities of the states $|100\\rangle, |010\\rangle, |001\\rangle$. These values correspond to the probability of finding the particle on each different site of our lattice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "1a46069b-d9e6-4c73-bf3a-1fce5922b421",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Import qubit states Zero (|0>) and One (|1>)\n",
    "from qiskit.opflow import Zero, One\n",
    "\n",
    "# Define array of time points\n",
    "ts = np.linspace(0, 3, 100) # DO NOT CHANGE THIS\n",
    "\n",
    "initial_state=One^Zero^Zero\n",
    "\n",
    "state_t=[U_tb(float(t)) @ initial_state for t in ts]\n",
    "\n",
    "p_100= [np.abs( (~(One^Zero^Zero) @ state).eval() )**2 for state in state_t]\n",
    "p_010= [np.abs( (~(Zero^One^Zero) @ state).eval() )**2 for state in state_t]\n",
    "p_001= [np.abs( (~(Zero^Zero^One) @ state).eval() )**2 for state in state_t]\n",
    "\n",
    "plt.figure(facecolor='white')\n",
    "plt.plot(ts, p_100, label=r'$p_{100}$')\n",
    "plt.plot(ts, p_010, label=r'$p_{010}$')\n",
    "plt.plot(ts, p_001, label=r'$p_{001}$')\n",
    "plt.xlabel(r'Time (1/J)')\n",
    "plt.ylabel(r'Population')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19c6ab38",
   "metadata": {},
   "source": [
    "## 2. Trotterization\n",
    "\n",
    "To execute the unitary time evolution on a [circuit model](https://qiskit.org/documentation/apidoc/circuit.html) quantum computer, we must decompose $U_{\\text{tb}}(t)$ into a product of single and two-qubit gates that are native to the quantum computer. One method to accomplish this is [Trotterization](https://en.wikipedia.org/wiki/Hamiltonian_simulation#Product_Formulas), also refered to as Trotter-Suzuki decomposition. \n",
    "\n",
    "Below, we show an example Trotterization as outlined in \\[1-2\\].\n",
    "\n",
    "Since the [Pauli operators do not commute](https://en.wikipedia.org/wiki/Pauli_matrices#Commutation_relations) with each other, the exponential $U_{\\text{tb}}(t)$ cannot be split into a product of simpler exponentials. However, we can approximate $U_{\\text{tb}}(t)$ as a product of simpler exponentials through Trotterization. Consider a subsystem of 2 spin-1/2 particles within the larger 3 spin system. The Hamiltonian on spins $i$ and $j$ ($i,j \\in \\{0,1,2\\}$) would be $H^{(i,j)}_{\\text{tb}} = \\sigma_x^{(i)}\\sigma_x^{(j)} + \\sigma_y^{(i)}\\sigma_y^{(j)} + \\sigma_z^{(i)}\\sigma_z^{(j)}$. Rewritting $U_{\\text{tb}}(t)$ in terms of the two possible subsystems within the total $N=3$ system you will simulate,\n",
    "\n",
    "$$\n",
    "U_{\\text{tb}}(t) = \\exp\\left[-i t \\left(H^{(0,1)}_{\\text{tb}} + H^{(1,2)}_{\\text{tb}} \\right)\\right].\n",
    "$$\n",
    "\n",
    "$H^{(0,1)}_{\\text{tb}}$ and $H^{(1,2)}_{\\text{tb}}$ do not commute, so $U_{\\text{tb}}(t) \\neq \\exp\\left(-i t H^{(0,1)}_{\\text{tb}}\\right) \\exp\\left(-i t H^{(1,2)}_{\\text{tb}} \\right)$. But, this product decomposition can be approximated with Trotterization which says $U_{\\text{tb}}(t)$ is approximately a short evolution of $H^{(0,1)}_{\\text{tb}}$ (time = $t/n$) and followed by a short evolution of $H^{(1,2)}_{\\text{tb}}$ (time = $t/n$) repeated $n$ times\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "U_{\\text{tb}}(t) &= \\exp\\left[-i t \\left(H^{(0,1)}_{\\text{tb}} + H^{(1,2)}_{\\text{tb}} \\right)\\right] \\\\\n",
    "U_{\\text{tb}}(t) &\\approx \\left[\\exp\\left(\\dfrac{-it}{n}H^{(0,1)}_{\\text{tb}}\\right) \\exp\\left(\\dfrac{-it}{n}H^{(1,2)}_{\\text{tb}} \\right)\\right]^n.\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "$n$ is the number of Trotter steps, and as $n$ increases, the approximation becomes more accurate. (Note that how the unitary operator is split up into subsystems for Trotterization is not necessarily unique.) The decomposition goes further. Within each 2 spin subsystems, the Pauli operator pairs ($\\sigma_x^{(i)}\\sigma_x^{(j)}$, $\\sigma_y^{(i)}\\sigma_y^{(j)}$, and $\\sigma_z^{(i)}\\sigma_z^{(j)}$) commute. This means we can decompose the exponential of a subsystem Hamiltonian ($H^{(i,j)}_{\\text{tb}}$) into a product of even simpler exponentials getting us closer to a gate implementation of $U_{\\text{tb}}(t)$\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "U_{\\text{tb}}(t) &\\approx \\left[\\exp\\left(\\dfrac{-it}{n}H^{(0,1)}_{\\text{tb}}\\right) \\exp\\left(\\dfrac{-it}{n}H^{(1,2)}_{\\text{tb}} \\right)\\right]^n \\\\\n",
    "U_{\\text{tb}}(t) &\\approx \\left[\\exp\\left(\\dfrac{-it}{n}\\left(\\sigma_x^{(0)}\\sigma_x^{(1)} + \\sigma_y^{(0)}\\sigma_y^{(1)} \\right)\\right) \\exp\\left(\\dfrac{-it}{n}\\left(\\sigma_x^{(1)}\\sigma_x^{(2)} + \\sigma_y^{(1)}\\sigma_y^{(2)} \\right)\\right)\\right]^{n} \\\\\n",
    "U_{\\text{tb}}(t) &\\approx \\left[\\exp\\left(\\dfrac{-it}{n}\\sigma_x^{(0)}\\sigma_x^{(1)}\\right) \\exp\\left(\\dfrac{-it}{n}\\sigma_y^{(0)}\\sigma_y^{(1)}\\right) \\exp\\left(\\dfrac{-it}{n}\\sigma_x^{(1)}\\sigma_x^{(2)}\\right) \\exp\\left(\\dfrac{-it}{n}\\sigma_y^{(1)}\\sigma_y^{(2)}\\right) \\right]^{n}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "For simplicity, and to use more common notation, let's rename the products with $XX(2t) = \\exp\\left(-it \\sigma_x\\sigma_x\\right)$, and $YY(2t) = \\exp\\left(-it \\sigma_y\\sigma_y\\right)$, and rewrite the Trotterized $U_{\\text{tb}}(t)$\n",
    "$$\n",
    "U_{\\text{tb}}(t) \\approx \\left[XX\\left(\\frac{2t}{n}\\right)^{(0,1)} YY\\left(\\frac{2t}{n}\\right)^{(0,1)}  XX\\left(\\frac{2t}{n}\\right)^{(1,2)} YY\\left(\\frac{2t}{n}\\right)^{(1,2)}\\right]^{n}\n",
    "$$\n",
    "And that's it! We have now approximately decomposed $U_{\\text{tb}}(t)$ into two-qubit gates $XX(t)$ and $YY(t)$. These gates are not native to superconducting qubits, but in Section 2, they will be further decomposed into native single and two-qubit gates with *further details found in the supplementary material.*\n",
    "\n",
    "\n",
    "\\[1\\] Y. Salathe, et al., *Digital Quantum Simulation of Spin Models with Circuit Quantum Electrodynamics*, [Phys. Rev. X **5**, 021027 (2015)](https://link.aps.org/doi/10.1103/PhysRevX.5.021027)\n",
    "\n",
    "\\[2\\] F. Tacchino, et al., *Quantum Computers as Universal Quantum Simulators: State-of-the-Art and Perspectives*, [Adv. Quantum Technol. *3* 3 (2020)](https://doi.org/10.1002/qute.201900052) \\[[free arxiv version](https://arxiv.org/abs/1907.03505)\\]\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e7c574e",
   "metadata": {},
   "source": [
    "### 2.1 Building the individual Pauli unitaries\n",
    "\n",
    "In this section we build ZZ(t), XX(t), and YY(t) operations using single- and two-qubit gates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "6d790e43-7e59-40b3-9a3a-806e7df15b70",
   "metadata": {},
   "outputs": [],
   "source": [
    "t = Parameter('t')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "0c98587c-7d60-4e9e-9ccb-470381997673",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 280.033x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Build a subcircuit for ZZ(t) from single- and two-qubit gates\n",
    "\n",
    "ZZ_qr = QuantumRegister(2)\n",
    "ZZ_qc = QuantumCircuit(ZZ_qr, name='ZZ')\n",
    "\n",
    "ZZ_qc.cnot(0,1)\n",
    "ZZ_qc.rz(2 * t, 1)\n",
    "ZZ_qc.cnot(0,1)\n",
    "\n",
    "# Convert custom quantum circuit into a gate\n",
    "ZZ = ZZ_qc.to_instruction()\n",
    "\n",
    "ZZ_qc.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11c706da",
   "metadata": {},
   "source": [
    "### 2.2 Clifford gates\n",
    "\n",
    "Clifford gates are quantum operators that map Pauli operators to other Pauli operators. The Hadamard gate ($H$) and the phase gate ($S$) are examples of single-qubit Clifford gates.\n",
    "\n",
    "$$H=\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & 1 \\\\1 & -1\\\\ \\end{pmatrix}$$\n",
    "\n",
    "$HZH^\\dagger=X$, and $HXH^\\dagger=Z$. Since the Hadamard gate is a Hermitian operator, $H=H^\\dagger$.\n",
    "\n",
    "$$S=\\begin{pmatrix} 1 & 0 \\\\0 & i\\\\ \\end{pmatrix}$$\n",
    "\n",
    "$SXS^\\dagger=Y$, and $SYS^\\dagger=-X$.\n",
    "\n",
    "Using the Clifford gates, we can transform the $e^{i ZZ t}$ to $e^{i XX t}$ and $e^{i YY t}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00ea5c84",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "<b>Challenge question 1a</b> \n",
    "\n",
    "Build a subcircuit for XX(t) from single- and two-qubit gates\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "c08bcedb-6a0b-4141-810b-e5e4ab75b6c4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 293.304x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "XX_qr = QuantumRegister(2)\n",
    "XX_qc = QuantumCircuit(XX_qr, name='XX')\n",
    "\n",
    "###EDIT CODE BELOW (add Clifford operator)\n",
    "\n",
    "###DO NOT EDIT BELOW\n",
    "XX_qc.h(0)\n",
    "XX_qc.h(1)\n",
    "XX_qc.append(ZZ, [0,1])\n",
    "XX_qc.h(1)\n",
    "XX_qc.h(0)\n",
    "###EDIT CODE BELOW (add Clifford operator)\n",
    "\n",
    "\n",
    "###DO NOT EDIT BELOW\n",
    "\n",
    "# Convert custom quantum circuit into a gate\n",
    "XX = XX_qc.to_instruction()\n",
    "\n",
    "XX_qc.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "915c22d7-89f3-4099-9acd-fe6f8721eb9f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Submitting your answer. Please wait...\n",
      "Congratulations 🎉! Your answer is correct and has been submitted.\n"
     ]
    }
   ],
   "source": [
    "## Grade and submit your solution\n",
    "from qc_grader.challenges.spring_2022 import grade_ex1a\n",
    "\n",
    "grade_ex1a(XX_qc)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23d9093d",
   "metadata": {
    "tags": []
   },
   "source": [
    "# <div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "<b>Challenge question 1b</b> \n",
    "\n",
    "Build a subcircuit for YY(t) from single- and two-qubit gates\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "cc04f46e-da32-423e-8f0a-58ac9d6a526e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 413.849x144.48 with 1 Axes>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "YY_qr = QuantumRegister(2)\n",
    "YY_qc = QuantumCircuit(YY_qr, name='YY')\n",
    "\n",
    "###EDIT CODE BELOW (add Clifford operator)\n",
    "\n",
    "\n",
    "###DO NOT EDIT BELOW\n",
    "YY_qc.s(0)\n",
    "YY_qc.s(1)\n",
    "YY_qc.h(0)\n",
    "YY_qc.h(1)\n",
    "YY_qc.append(ZZ, [0,1])\n",
    "YY_qc.h(0)\n",
    "YY_qc.h(1)\n",
    "YY_qc.sdg(0)\n",
    "YY_qc.sdg(1)\n",
    "###EDIT CODE BELOW (add Clifford operator)\n",
    "\n",
    "\n",
    "###DO NOT EDIT BELOW\n",
    "\n",
    "# Convert custom quantum circuit into a gate\n",
    "YY = YY_qc.to_instruction()\n",
    "\n",
    "YY_qc.draw()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "1862a53a-221a-4b41-8cc7-8ceb90e8ae3d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Submitting your answer. Please wait...\n",
      "Congratulations 🎉! Your answer is correct and has been submitted.\n"
     ]
    }
   ],
   "source": [
    "## Grade and submit your solution\n",
    "from qc_grader.challenges.spring_2022 import grade_ex1b\n",
    "\n",
    "grade_ex1b(YY_qc)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1b5382b9",
   "metadata": {},
   "source": [
    "### 2.3 Building the trotterizated circuit\n",
    "\n",
    "$X_iX_j$ and $Y_iY_j$ operators commute:\n",
    "\n",
    "$$[X_iX_j, Y_iY_j]= X_iX_j.Y_iY_j - Y_iY_j.X_iX_j = Z_iZ_j-(-Z_i)(-Z_j)=0$$.\n",
    "\n",
    "Therefore, we can decompose $e^{i t (X_iX_j + Y_iY_j)}$ as $e^{i t X_iX_j} e^{i t Y_iY_j}$. Furthermore, if $i\\neq j \\neq k \\neq l$ then $[X_iX_j,X_kX_l]=0$. Based on this, we can decompose each trotterized evolution time step to two blocks:\n",
    "\n",
    "$$U(\\Delta t) \\approx \\Big(\\prod_{i \\in \\rm{odd}} e^{-i \\Delta t X_iX_{i+1}} e^{-i \\Delta t Y_iY_{i+1}} \\Big)  \\Big(\\prod_{i \\in \\rm{even}} e^{-i \\Delta t X_iX_{i+1}} e^{-i \\Delta t Y_iY_{i+1}} \\Big)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "85d24eb2-2a44-45cb-b5ab-70c46890a3fd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 353.795x204.68 with 1 Axes>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num_qubits = 3\n",
    "\n",
    "Trot_qr = QuantumRegister(num_qubits)\n",
    "Trot_qc = QuantumCircuit(Trot_qr, name='Trot')\n",
    "\n",
    "for i in range(0, num_qubits - 1):\n",
    "    Trot_qc.append(YY, [Trot_qr[i], Trot_qr[i+1]])\n",
    "    Trot_qc.append(XX, [Trot_qr[i], Trot_qr[i+1]])\n",
    "\n",
    "# Convert custom quantum circuit into a gate\n",
    "Trot_gate = Trot_qc.to_instruction()\n",
    "\n",
    "Trot_qc.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "896a2f87",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "<b>Challenge question 1c</b> \n",
    "\n",
    "Create the trotterized circuit and output the unitary associated with the circuit\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "d3bd5759-6fa6-4305-8795-c869c95926cc",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "def U_trotterize(t_target, trotter_steps):\n",
    "    qr = QuantumRegister(3)\n",
    "    qc = QuantumCircuit(qr)\n",
    "\n",
    "    ###EDIT CODE BELOW (Create the trotterized circuit with various number of trotter steps)\n",
    "    for i in range(trotter_steps):\n",
    "        qc.append(Trot_gate, qr[:])\n",
    "    \n",
    "    ###DO NOT EDIT BELOW\n",
    "        \n",
    "    qc = qc.bind_parameters({t: t_target/trotter_steps})\n",
    "    \n",
    "    return qi.Operator(qc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "9e642289-f3e4-472a-9c55-9af5134ed027",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "0907a32c630642dda6cc68eabf0ad968",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/50 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t_target = 0.5\n",
    "U_target = U_tb(t_target)\n",
    "\n",
    "steps=np.arange(1,101,2)  ## DO NOT EDIT\n",
    "\n",
    "fidelities=[]\n",
    "for n in tqdm(steps):\n",
    "    U_trotter = U_trotterize(t_target, trotter_steps=n)\n",
    "    fidelity = qi.process_fidelity(U_trotter, target=U_target)\n",
    "    fidelities.append(fidelity)\n",
    "\n",
    "plt.figure(facecolor='white')\n",
    "plt.loglog(steps, 1-np.array(fidelities))\n",
    "plt.ylabel('Trotter error')\n",
    "plt.xlabel('Trotter steps')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "83b36595-aea8-4a97-8e3b-43b0454b0edf",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Submitting your answer. Please wait...\n",
      "Congratulations 🎉! Your answer is correct and has been submitted.\n"
     ]
    }
   ],
   "source": [
    "## Grade and submit your solution\n",
    "from qc_grader.challenges.spring_2022 import grade_ex1c\n",
    "\n",
    "grade_ex1c(fidelities)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "cd7acdf9-1840-494a-b911-9e03768599eb",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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