03_trotter_step.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "0c67a9f2",
   "metadata": {},
   "source": [
    "# 1 Introduction\n",
    "First, we have to activate the environment. The first run may take a while as external libraries will be downloaded and compiled."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "dd6cfdc9",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m  Activating\u001b[22m\u001b[39m project at `~/code/OpenScience21.jl`\n"
     ]
    }
   ],
   "source": [
    "Pkg.activate(\".\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f5d07d17",
   "metadata": {},
   "outputs": [],
   "source": [
    "using QuantumCircuits\n",
    "using QuantumCircuits.QML\n",
    "using QuantumCircuits.QML.Optimization\n",
    "using QuantumCircuits.QCircuits.Circuit\n",
    "using QuantumCircuits.QCircuits.ComplexGates\n",
    "using QuantumCircuits.Execute"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6dc9a6fa",
   "metadata": {},
   "source": [
    "# 2 ZZ, XX, and YY gates\n",
    "Now we have to create ZZ, XX, and YY gates for use in simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0897adb0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "XX (generic function with 2 methods)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This code com from using OpenScience21.Simulation.Gates module.\n",
    "function ZZ(qc, q0, q1, t, usePulse=false)\n",
    "    if usePulse\n",
    "        qc.h(q1)\n",
    "        qc.rzx(q0, q1, t)\n",
    "        qc.x(q0)\n",
    "        qc.rzx(q0, q1, -t)\n",
    "        qc.x(q0)\n",
    "        qc.h(q1)\n",
    "    else\n",
    "        qc.cx(q0, q1)\n",
    "        qc.rz(q1, 2*t)\n",
    "        qc.cx(q0, q1)\n",
    "    end\n",
    "end\n",
    "\n",
    "function YY(qc, q0, q1, t, usePulse=false)\n",
    "    if usePulse\n",
    "        qc.sdg([q0, q1])\n",
    "        qc.h(q0)\n",
    "        qc.rzx(q0, q1, t)\n",
    "        qc.x(q0)\n",
    "        qc.rzx(q0, q1, -t)\n",
    "        qc.x(q0)\n",
    "        qc.h(q0)\n",
    "        qc.s([q0, q1])\n",
    "    else\n",
    "        qc.rx([q0, q1], π/2)\n",
    "        qc.cx(q0, q1)\n",
    "        qc.rz(q1, 2*t)\n",
    "        qc.cx(q0, q1)\n",
    "        qc.rx([q0, q1], -π/2)\n",
    "    end\n",
    "end\n",
    "\n",
    "function XX(qc, q0, q1, t, usePulse=false)\n",
    "    if usePulse\n",
    "        qc.h(q0)\n",
    "        qc.rzx(q0, q1, t)\n",
    "        qc.x(q0)\n",
    "        qc.rzx(q0, q1, -t)\n",
    "        qc.x(q0)\n",
    "        qc.h(q0)\n",
    "    else\n",
    "        qc.ry([q0, q1], π/2)\n",
    "        qc.cx(q0, q1)\n",
    "        qc.rz(q1, 2*t)\n",
    "        qc.cx(q0, q1)\n",
    "        qc.ry([q0, q1], -π/2)\n",
    "    end\n",
    "end\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "397bcb2b",
   "metadata": {},
   "source": [
    "# 3 U4 - Cartan's KAK decomposition\n",
    "Any 2 qubits unitary can be decomposed used 3 cx gate (see the paper _\"Minimal Universal Two-qubit Quantum Circuits\"_ https://arxiv.org/abs/quant-ph/0308033). In QuantumCircuits library we have defined the U4 gate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c89dc693",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "q0_0: ┤ U3(2.2832,4.3098,1.6032) ├┤ X ├┤ Rz(3.6536) ├──■────────────────┤ X ├»\n",
       "      ├─────────────────────────┬┘└─┬─┘├────────────┤┌─┴─┐┌────────────┐└─┬─┘»\n",
       "q0_1: ┤ U3(4.263,1.3415,2.3539) ├───■──┤ Ry(5.2857) ├┤ X ├┤ Ry(4.1467) ├──■──»\n",
       "      └─────────────────────────┘      └────────────┘└───┘└────────────┘     »\n",
       "c0: 2/═══════════════════════════════════════════════════════════════════════»\n",
       "                                                                             »\n",
       "«      ┌────────────────────────────┐\n",
       "«q0_0: ┤ U3(3.4384,0.79627,0.14553) ├\n",
       "«      └┬──────────────────────────┬┘\n",
       "«q0_1: ─┤ U3(4.9937,3.1561,5.6151) ├─\n",
       "«       └──────────────────────────┘ \n",
       "«c0: 2/══════════════════════════════\n",
       "«                                    "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc = QCircuit(2)\n",
    "# Create u4 gate with random parameter\n",
    "qc.u4(0, 1) \n",
    "\n",
    "# Decompose circuit to use base gates\n",
    "qc2 = decompose(qc)\n",
    "qc2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c6af234",
   "metadata": {},
   "source": [
    "Let us assume that we would like to find parameters of the U4 gate implementing exactly ZZ, YY, and XX gate combinations for a given time $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "1afcec75",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "                         ┌─────────┐                   ┌──────────┐┌─────────┐»\n",
       "q1_0: ──■─────────────■──┤ Rx(π/2) ├──■─────────────■──┤ Rx(-π/2) ├┤ Ry(π/2) ├»\n",
       "      ┌─┴─┐┌───────┐┌─┴─┐├─────────┤┌─┴─┐┌───────┐┌─┴─┐├──────────┤├─────────┤»\n",
       "q1_1: ┤ X ├┤ Rz(π) ├┤ X ├┤ Rx(π/2) ├┤ X ├┤ Rz(π) ├┤ X ├┤ Rx(-π/2) ├┤ Ry(π/2) ├»\n",
       "      └───┘└───────┘└───┘└─────────┘└───┘└───────┘└───┘└──────────┘└─────────┘»\n",
       "c1: 2/════════════════════════════════════════════════════════════════════════»\n",
       "                                                                              »\n",
       "«                         ┌──────────┐\n",
       "«q1_0: ──■─────────────■──┤ Ry(-π/2) ├\n",
       "«      ┌─┴─┐┌───────┐┌─┴─┐├──────────┤\n",
       "«q1_1: ┤ X ├┤ Rz(π) ├┤ X ├┤ Ry(-π/2) ├\n",
       "«      └───┘└───────┘└───┘└──────────┘\n",
       "«c1: 2/═══════════════════════════════\n",
       "«                                     "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = π/2\n",
    "\n",
    "qc = QCircuit(2)\n",
    "ZZ(qc, 0, 1, t)\n",
    "YY(qc, 0, 1, t)\n",
    "XX(qc, 0, 1, t)\n",
    "expmat = tomatrix(qc)\n",
    "qc"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16f98856",
   "metadata": {},
   "source": [
    "We have to choose the ansact, in our case this will be a U4 gate and in that case, we are sure that we always can find the correct parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "3610730f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      ┌───────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "q2_0: ┤ U3(1.7624,0.77116,3.9005) ├┤ X ├┤ Rz(5.2775) ├──■────────────────┤ X ├»\n",
       "      └┬──────────────────────────┤└─┬─┘├────────────┤┌─┴─┐┌────────────┐└─┬─┘»\n",
       "q2_1: ─┤ U3(5.0677,5.7902,6.2463) ├──■──┤ Ry(4.4635) ├┤ X ├┤ Ry(2.5877) ├──■──»\n",
       "       └──────────────────────────┘     └────────────┘└───┘└────────────┘     »\n",
       "«      ┌──────────────────────────┐\n",
       "«q2_0: ┤ U3(0.89282,1.8969,3.728) ├\n",
       "«      ├──────────────────────────┤\n",
       "«q2_1: ┤ U3(4.2926,2.4936,2.6366) ├\n",
       "«      └──────────────────────────┘"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "    qr = QuantumRegister(2)\n",
    "    qc = QCircuit(qr)\n",
    "    qc.u4(qr[0], qr[1])\n",
    "\n",
    "    params = getRandParameters(qc)\n",
    "    setparameters!(qc, params)\n",
    "    qc = decompose(qc)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cccc9645",
   "metadata": {},
   "source": [
    "Now we can find the parameter of our ansact that perfectly fit our expected unitary matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "7c6ccd39",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.988143676440324e-8"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "params, _, err, _  = findparam(expmat, qc, debug=false, trystandard=false)\n",
    "err"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "718428f0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "      ┌──────────────────────────┐┌───┐┌──────────┐                ┌───┐»\n",
       "q3_0: ┤ U3(1.4791,0.48683,4.394) ├┤ X ├┤ Rz(3π/2) ├──■─────────────┤ X ├»\n",
       "      └─┬─────────────────────┬──┘└─┬─┘├──────────┤┌─┴─┐┌─────────┐└─┬─┘»\n",
       "q3_1: ──┤ U3(π,5.7747,5.1031) ├─────■──┤ Ry(3π/2) ├┤ X ├┤ Ry(π/2) ├──■──»\n",
       "        └─────────────────────┘        └──────────┘└───┘└─────────┘     »\n",
       "«      ┌───────────────────────────┐\n",
       "«q3_0: ┤ U3(-1.4791,1.8894,4.2257) ├\n",
       "«      └──┬─────────────────────┬──┘\n",
       "«q3_1: ───┤ U3(π,1.9652,2.8646) ├───\n",
       "«         └─────────────────────┘   "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qc"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2282ab55",
   "metadata": {},
   "source": [
    "A description of the optimization method used is available in the notebook [06_Optimization](06_Optimization.ipynb)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5eef8bf0",
   "metadata": {},
   "source": [
    "# 4  The effective implementation of the Trotter step\n",
    "\n",
    "Assume that we would like to implement the Trotter step using the second-order Trotter formula (see _\"Quantum computers as universal quantum simulators: state-of-art and perspectives\"_ https://arxiv.org/abs/1907.03505), we can do this using the circuit in below."
   ]
  },
  {
   "attachments": {
    "Trotter_step.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "c814ae77",
   "metadata": {},
   "source": [
    "![Trotter_step.png](attachment:Trotter_step.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d06179c1",
   "metadata": {},
   "source": [
    "Using Cartan's KAK decomposition we can rewrite this circuit and replace ZZ, YY, and XX by singe U4 gate as in Figure below."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5ee01a7",
   "metadata": {},
   "source": [
    "<img src=\"images/Trotter_u4.jpg\" width=\"400\" height=\"300\">"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce90e021",
   "metadata": {},
   "source": [
    "Assume that we would like to implement the 2 Trotter step using the second-order Trotter formula. Using Cartan's KAK decomposition each step contains 3 U4 gates. But merging the two $U4$ gates on qubits 0 and 1, the two steps of the Trotter formula using KAK decomposition look like in the Figure below. It's worth noting that these two steps of Trotter utilize only 15 CNOT gates (3 CNOT for each of the U4 gates)."
   ]
  },
  {
   "attachments": {
    "Trotter_x2_u4.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "255fe333",
   "metadata": {},
   "source": [
    "![Trotter_x2_u4.png](attachment:Trotter_x2_u4.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "290b4630",
   "metadata": {},
   "source": [
    "Effective implementation of the two steps is important because this is how much we use in the model training process (see notebook [05_trening](05_trening.ipynb)). However, a similar method of connecting two U4 gates can be used for any number of steps. \n",
    "\n",
    "# 5 Simulation\n",
    "First we create helper function which find the parameter for U4 for ZZ, YY, and XX gates and ZZ, YY, XX, ZZ, YY and, XX and Trotter step."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "31f12ccf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "trotter2U4 (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This code com from using OpenScience21.Simulation.Gates module.\n",
    "function findU4paramsZZYYXX(t; debug=false)\n",
    "    qc = QCircuit(2)\n",
    "    ZZ(qc, 0, 1, t)\n",
    "    YY(qc, 0, 1, t)\n",
    "    XX(qc, 0, 1, t)\n",
    "    expmat = tomatrix(qc)\n",
    "\n",
    "\n",
    "    qr = QuantumRegister(2)\n",
    "    qc = QCircuit(qr)\n",
    "    qc.u4(qr[0], qr[1])\n",
    "\n",
    "    params = getRandParameters(qc)\n",
    "    setparameters!(qc, params)\n",
    "    qc = decompose(qc)\n",
    "\n",
    "    params, _, err, _  = findparam(expmat, qc, debug=debug, trystandard=false)\n",
    "\n",
    "    @assert err < 1e-5 \"The error of U gate should be small but it is $err.\"\n",
    "\n",
    "    return params\n",
    "end\n",
    "\n",
    "function findU4paramsZZYYXXx2(t; debug=false)\n",
    "    qc = QCircuit(2)\n",
    "    ZZ(qc, 0, 1, t)\n",
    "    YY(qc, 0, 1, t)\n",
    "    XX(qc, 0, 1, t)\n",
    "    ZZ(qc, 0, 1, t)\n",
    "    YY(qc, 0, 1, t)\n",
    "    XX(qc, 0, 1, t)\n",
    "    expmat = tomatrix(qc)\n",
    "\n",
    "\n",
    "    qr = QuantumRegister(2)\n",
    "    qc = QCircuit(qr)\n",
    "    qc.u4(qr[0], qr[1])\n",
    "\n",
    "    params = getRandParameters(qc)\n",
    "    setparameters!(qc, params)\n",
    "    qc = decompose(qc)\n",
    "\n",
    "    params, _, err, _  = findparam(expmat, qc, debug=debug, trystandard=false)\n",
    "    @assert err < 1e-5 \"The error of U gate should be small but it is $err.\"\n",
    "\n",
    "    return params\n",
    "end\n",
    "\n",
    "function trotter2U4(qc, qubits, t, isFirst, isLast, params, params2, params3)\n",
    "    if isFirst\n",
    "        for i in 1:(length(qubits)-2)\n",
    "            qc.u4(qubits[i], qubits[i+1], params2)\n",
    "        end\n",
    "    end\n",
    "\n",
    "    i = length(qubits) - 1\n",
    "    qc.u4(qubits[i], qubits[i+1], params)\n",
    "\n",
    "    if isLast\n",
    "        for i in 1:(length(qubits)-2)\n",
    "            qc.u4(qubits[i], qubits[i+1], params2)\n",
    "        end\n",
    "    else\n",
    "        for i in 1:(length(qubits)-2)\n",
    "            qc.u4(qubits[i], qubits[i+1], params3)\n",
    "        end\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b4631b93",
   "metadata": {},
   "source": [
    "Next function generate circuit of the simulation algorithm for given Trotter steps number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "ba8a573b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "generate_circuit (generic function with 5 methods)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function generate_circuit(trotter_steps, run_step, t=π, params=nothing, params2=nothing, params3=nothing; init=false, debug=false)\n",
    "    qr = QuantumRegister(7, \"q\")\n",
    "    qc = QCircuit(qr)\n",
    "\n",
    "    # Prepare initial state (remember we are only evolving 3 of the 7 qubits on jakarta qubits (q_5, q_3, q_1) corresponding to the state |110>)\n",
    "    if init\n",
    "        qc.x([3, 5])  # DO NOT MODIFY (|q_5,q_3,q_1> = |110>)\n",
    "    end\n",
    "\n",
    "    if params == nothing\n",
    "        params = findU4paramsZZYYXX(t / trotter_steps, debug=debug)\n",
    "        params2 = findU4paramsZZYYXX(t / (2 * trotter_steps), debug=debug)\n",
    "        params3 = findU4paramsZZYYXXx2(t / (2 * trotter_steps), debug=debug)\n",
    "    end\n",
    "\n",
    "    qubits = [qr[1], qr[3], qr[5]]\n",
    "    for s in 1:run_step\n",
    "        isFirst = s == 1\n",
    "        isLast = s == run_step\n",
    "\n",
    "        trotter2U4(qc, qubits, t / trotter_steps, isFirst, isLast, params, params2, params3)\n",
    "    end\n",
    "\n",
    "    qc = decompose(qc)\n",
    "    return qc, params, params2, params3\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6037e373",
   "metadata": {},
   "source": [
    "Now we can generate the circuit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "8a49727d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "                                                                               »\n",
       " q_0: ─────────────────────────────────────────────────────────────────────────»\n",
       "      ┌────────────────────────┐                              ┌───┐┌──────────┐»\n",
       " q_1: ┤ U3(2π,5.3676,0.032489) ├──────────────────────────────┤ X ├┤ Rz(7π/5) ├»\n",
       "      └────────────────────────┘                              └─┬─┘└──────────┘»\n",
       " q_2: ──────────────────────────────────────────────────────────┼──────────────»\n",
       "                ┌───┐             ┌────────────────────────┐    │  ┌──────────┐»\n",
       " q_3: ──────────┤ X ├─────────────┤ U3(-π,0.70057,-1.7539) ├────■──┤ Ry(7π/5) ├»\n",
       "                └───┘             └────────────────────────┘       └──────────┘»\n",
       " q_4: ─────────────────────────────────────────────────────────────────────────»\n",
       "                ┌───┐           ┌────────────────────────────┐                 »\n",
       " q_5: ──────────┤ X ├───────────┤ U3(3.1416,3.5375,-0.46269) ├─────────────────»\n",
       "                └───┘           └────────────────────────────┘                 »\n",
       " q_6: ─────────────────────────────────────────────────────────────────────────»\n",
       "                                                                               »\n",
       "c2: 3/═════════════════════════════════════════════════════════════════════════»\n",
       "                                                                               »\n",
       "«                                                       »\n",
       "« q_0: ─────────────────────────────────────────────────»\n",
       "«                        ┌───┐ ┌─────────────────────┐  »\n",
       "« q_1: ──■───────────────┤ X ├─┤ U3(0,5.1954,6.6832) ├──»\n",
       "«        │               └─┬─┘ └─────────────────────┘  »\n",
       "« q_2: ──┼─────────────────┼────────────────────────────»\n",
       "«      ┌─┴─┐┌───────────┐  │  ┌────────────────────────┐»\n",
       "« q_3: ┤ X ├┤ Ry(12π/5) ├──■──┤ U3(π,0.96835,0.084917) ├»\n",
       "«      └───┘└───────────┘     └────────────────────────┘»\n",
       "« q_4: ─────────────────────────────────────────────────»\n",
       "«                                                       »\n",
       "« q_5: ─────────────────────────────────────────────────»\n",
       "«                                                       »\n",
       "« q_6: ─────────────────────────────────────────────────»\n",
       "«                                                       »\n",
       "«c2: 3/═════════════════════════════════════════════════»\n",
       "«                                                       »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────┐                                              »\n",
       "« q_1: ──┤ U3(2π,6.524,5.5212) ├──────────────────────────────────────────────»\n",
       "«        └─────────────────────┘                                              »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                         »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                    ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                    └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐  ┌───────────────────────┐     │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├──┤ U3(-π,2.4314,0.33979) ├─────■──»\n",
       "«      └──────────────────────────────┘  └───────────────────────┘        »\n",
       "« q_4: ───────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐   ┌────────────────────────────┐     »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├───┤ U3(3.1416,3.5375,-0.46269) ├─────»\n",
       "«        └─────────────────────────┘   └────────────────────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                         »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════»\n",
       "«                                                                         »\n",
       "«                                                                           »\n",
       "« q_0: ─────────────────────────────────────────────────────────────────────»\n",
       "«      ┌────────────┐                  ┌───┐┌──────────────────────────────┐»\n",
       "« q_1: ┤ Rz(13π/10) ├──■───────────────┤ X ├┤ U3(3.2916e-11,4.531,0.70243) ├»\n",
       "«      └────────────┘  │               └─┬─┘└──────────────────────────────┘»\n",
       "« q_2: ────────────────┼─────────────────┼──────────────────────────────────»\n",
       "«      ┌────────────┐┌─┴─┐┌───────────┐  │      ┌──────────────────────┐    »\n",
       "« q_3: ┤ Ry(13π/10) ├┤ X ├┤ Ry(3π/10) ├──■──────┤ U3(-π,-1.125,4.6372) ├────»\n",
       "«      └────────────┘└───┘└───────────┘         └──────────────────────┘    »\n",
       "« q_4: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_5: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_6: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "«c2: 3/═════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────┐                                              »\n",
       "« q_1: ──┤ U3(2π,6.524,5.5212) ├──────────────────────────────────────────────»\n",
       "«        └─────────────────────┘                                              »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                         »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                    ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                    └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐  ┌───────────────────────┐     │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├──┤ U3(-π,2.4314,0.33979) ├─────■──»\n",
       "«      └──────────────────────────────┘  └───────────────────────┘        »\n",
       "« q_4: ───────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐   ┌────────────────────────────┐     »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├───┤ U3(3.1416,3.5375,-0.46269) ├─────»\n",
       "«        └─────────────────────────┘   └────────────────────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                         »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════»\n",
       "«                                                                         »\n",
       "«                                                                           »\n",
       "« q_0: ─────────────────────────────────────────────────────────────────────»\n",
       "«      ┌────────────┐                  ┌───┐┌──────────────────────────────┐»\n",
       "« q_1: ┤ Rz(13π/10) ├──■───────────────┤ X ├┤ U3(3.2916e-11,4.531,0.70243) ├»\n",
       "«      └────────────┘  │               └─┬─┘└──────────────────────────────┘»\n",
       "« q_2: ────────────────┼─────────────────┼──────────────────────────────────»\n",
       "«      ┌────────────┐┌─┴─┐┌───────────┐  │      ┌──────────────────────┐    »\n",
       "« q_3: ┤ Ry(13π/10) ├┤ X ├┤ Ry(3π/10) ├──■──────┤ U3(-π,-1.125,4.6372) ├────»\n",
       "«      └────────────┘└───┘└───────────┘         └──────────────────────┘    »\n",
       "« q_4: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_5: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_6: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "«c2: 3/═════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────┐                                              »\n",
       "« q_1: ──┤ U3(2π,6.524,5.5212) ├──────────────────────────────────────────────»\n",
       "«        └─────────────────────┘                                              »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                         »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                    ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                    └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐  ┌───────────────────────┐     │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├──┤ U3(-π,2.4314,0.33979) ├─────■──»\n",
       "«      └──────────────────────────────┘  └───────────────────────┘        »\n",
       "« q_4: ───────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐   ┌────────────────────────────┐     »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├───┤ U3(3.1416,3.5375,-0.46269) ├─────»\n",
       "«        └─────────────────────────┘   └────────────────────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                         »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════»\n",
       "«                                                                         »\n",
       "«                                                                           »\n",
       "« q_0: ─────────────────────────────────────────────────────────────────────»\n",
       "«      ┌────────────┐                  ┌───┐┌──────────────────────────────┐»\n",
       "« q_1: ┤ Rz(13π/10) ├──■───────────────┤ X ├┤ U3(3.2916e-11,4.531,0.70243) ├»\n",
       "«      └────────────┘  │               └─┬─┘└──────────────────────────────┘»\n",
       "« q_2: ────────────────┼─────────────────┼──────────────────────────────────»\n",
       "«      ┌────────────┐┌─┴─┐┌───────────┐  │      ┌──────────────────────┐    »\n",
       "« q_3: ┤ Ry(13π/10) ├┤ X ├┤ Ry(3π/10) ├──■──────┤ U3(-π,-1.125,4.6372) ├────»\n",
       "«      └────────────┘└───┘└───────────┘         └──────────────────────┘    »\n",
       "« q_4: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_5: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_6: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "«c2: 3/═════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────┐                                              »\n",
       "« q_1: ──┤ U3(2π,6.524,5.5212) ├──────────────────────────────────────────────»\n",
       "«        └─────────────────────┘                                              »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                         »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                    ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                    └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐  ┌───────────────────────┐     │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├──┤ U3(-π,2.4314,0.33979) ├─────■──»\n",
       "«      └──────────────────────────────┘  └───────────────────────┘        »\n",
       "« q_4: ───────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐   ┌────────────────────────────┐     »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├───┤ U3(3.1416,3.5375,-0.46269) ├─────»\n",
       "«        └─────────────────────────┘   └────────────────────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                         »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════»\n",
       "«                                                                         »\n",
       "«                                                                           »\n",
       "« q_0: ─────────────────────────────────────────────────────────────────────»\n",
       "«      ┌────────────┐                  ┌───┐┌──────────────────────────────┐»\n",
       "« q_1: ┤ Rz(13π/10) ├──■───────────────┤ X ├┤ U3(3.2916e-11,4.531,0.70243) ├»\n",
       "«      └────────────┘  │               └─┬─┘└──────────────────────────────┘»\n",
       "« q_2: ────────────────┼─────────────────┼──────────────────────────────────»\n",
       "«      ┌────────────┐┌─┴─┐┌───────────┐  │      ┌──────────────────────┐    »\n",
       "« q_3: ┤ Ry(13π/10) ├┤ X ├┤ Ry(3π/10) ├──■──────┤ U3(-π,-1.125,4.6372) ├────»\n",
       "«      └────────────┘└───┘└───────────┘         └──────────────────────┘    »\n",
       "« q_4: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_5: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_6: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "«c2: 3/═════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────┐                                              »\n",
       "« q_1: ──┤ U3(2π,6.524,5.5212) ├──────────────────────────────────────────────»\n",
       "«        └─────────────────────┘                                              »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                         »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                    ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                    └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐  ┌───────────────────────┐     │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├──┤ U3(-π,2.4314,0.33979) ├─────■──»\n",
       "«      └──────────────────────────────┘  └───────────────────────┘        »\n",
       "« q_4: ───────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐   ┌────────────────────────────┐     »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├───┤ U3(3.1416,3.5375,-0.46269) ├─────»\n",
       "«        └─────────────────────────┘   └────────────────────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                         »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════»\n",
       "«                                                                         »\n",
       "«                                                                           »\n",
       "« q_0: ─────────────────────────────────────────────────────────────────────»\n",
       "«      ┌────────────┐                  ┌───┐┌──────────────────────────────┐»\n",
       "« q_1: ┤ Rz(13π/10) ├──■───────────────┤ X ├┤ U3(3.2916e-11,4.531,0.70243) ├»\n",
       "«      └────────────┘  │               └─┬─┘└──────────────────────────────┘»\n",
       "« q_2: ────────────────┼─────────────────┼──────────────────────────────────»\n",
       "«      ┌────────────┐┌─┴─┐┌───────────┐  │      ┌──────────────────────┐    »\n",
       "« q_3: ┤ Ry(13π/10) ├┤ X ├┤ Ry(3π/10) ├──■──────┤ U3(-π,-1.125,4.6372) ├────»\n",
       "«      └────────────┘└───┘└───────────┘         └──────────────────────┘    »\n",
       "« q_4: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_5: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_6: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "«c2: 3/═════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────┐                                              »\n",
       "« q_1: ──┤ U3(2π,6.524,5.5212) ├──────────────────────────────────────────────»\n",
       "«        └─────────────────────┘                                              »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                         »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                    ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                    └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐  ┌───────────────────────┐     │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├──┤ U3(-π,2.4314,0.33979) ├─────■──»\n",
       "«      └──────────────────────────────┘  └───────────────────────┘        »\n",
       "« q_4: ───────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐   ┌────────────────────────────┐     »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├───┤ U3(3.1416,3.5375,-0.46269) ├─────»\n",
       "«        └─────────────────────────┘   └────────────────────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                         »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════»\n",
       "«                                                                         »\n",
       "«                                                                           »\n",
       "« q_0: ─────────────────────────────────────────────────────────────────────»\n",
       "«      ┌────────────┐                  ┌───┐┌──────────────────────────────┐»\n",
       "« q_1: ┤ Rz(13π/10) ├──■───────────────┤ X ├┤ U3(3.2916e-11,4.531,0.70243) ├»\n",
       "«      └────────────┘  │               └─┬─┘└──────────────────────────────┘»\n",
       "« q_2: ────────────────┼─────────────────┼──────────────────────────────────»\n",
       "«      ┌────────────┐┌─┴─┐┌───────────┐  │      ┌──────────────────────┐    »\n",
       "« q_3: ┤ Ry(13π/10) ├┤ X ├┤ Ry(3π/10) ├──■──────┤ U3(-π,-1.125,4.6372) ├────»\n",
       "«      └────────────┘└───┘└───────────┘         └──────────────────────┘    »\n",
       "« q_4: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_5: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_6: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "«c2: 3/═════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────┐                                              »\n",
       "« q_1: ──┤ U3(2π,6.524,5.5212) ├──────────────────────────────────────────────»\n",
       "«        └─────────────────────┘                                              »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                         »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                    ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                    └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐  ┌───────────────────────┐     │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├──┤ U3(-π,2.4314,0.33979) ├─────■──»\n",
       "«      └──────────────────────────────┘  └───────────────────────┘        »\n",
       "« q_4: ───────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐   ┌────────────────────────────┐     »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├───┤ U3(3.1416,3.5375,-0.46269) ├─────»\n",
       "«        └─────────────────────────┘   └────────────────────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                         »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════»\n",
       "«                                                                         »\n",
       "«                                                                           »\n",
       "« q_0: ─────────────────────────────────────────────────────────────────────»\n",
       "«      ┌────────────┐                  ┌───┐┌──────────────────────────────┐»\n",
       "« q_1: ┤ Rz(13π/10) ├──■───────────────┤ X ├┤ U3(3.2916e-11,4.531,0.70243) ├»\n",
       "«      └────────────┘  │               └─┬─┘└──────────────────────────────┘»\n",
       "« q_2: ────────────────┼─────────────────┼──────────────────────────────────»\n",
       "«      ┌────────────┐┌─┴─┐┌───────────┐  │      ┌──────────────────────┐    »\n",
       "« q_3: ┤ Ry(13π/10) ├┤ X ├┤ Ry(3π/10) ├──■──────┤ U3(-π,-1.125,4.6372) ├────»\n",
       "«      └────────────┘└───┘└───────────┘         └──────────────────────┘    »\n",
       "« q_4: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_5: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_6: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "«c2: 3/═════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────┐                                              »\n",
       "« q_1: ──┤ U3(2π,6.524,5.5212) ├──────────────────────────────────────────────»\n",
       "«        └─────────────────────┘                                              »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                         »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                    ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                    └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐  ┌───────────────────────┐     │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├──┤ U3(-π,2.4314,0.33979) ├─────■──»\n",
       "«      └──────────────────────────────┘  └───────────────────────┘        »\n",
       "« q_4: ───────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐   ┌────────────────────────────┐     »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├───┤ U3(3.1416,3.5375,-0.46269) ├─────»\n",
       "«        └─────────────────────────┘   └────────────────────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                         »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════»\n",
       "«                                                                         »\n",
       "«                                                                           »\n",
       "« q_0: ─────────────────────────────────────────────────────────────────────»\n",
       "«      ┌────────────┐                  ┌───┐┌──────────────────────────────┐»\n",
       "« q_1: ┤ Rz(13π/10) ├──■───────────────┤ X ├┤ U3(3.2916e-11,4.531,0.70243) ├»\n",
       "«      └────────────┘  │               └─┬─┘└──────────────────────────────┘»\n",
       "« q_2: ────────────────┼─────────────────┼──────────────────────────────────»\n",
       "«      ┌────────────┐┌─┴─┐┌───────────┐  │      ┌──────────────────────┐    »\n",
       "« q_3: ┤ Ry(13π/10) ├┤ X ├┤ Ry(3π/10) ├──■──────┤ U3(-π,-1.125,4.6372) ├────»\n",
       "«      └────────────┘└───┘└───────────┘         └──────────────────────┘    »\n",
       "« q_4: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_5: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_6: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "«c2: 3/═════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────┐                                              »\n",
       "« q_1: ──┤ U3(2π,6.524,5.5212) ├──────────────────────────────────────────────»\n",
       "«        └─────────────────────┘                                              »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                         »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                    ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                    └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐  ┌───────────────────────┐     │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├──┤ U3(-π,2.4314,0.33979) ├─────■──»\n",
       "«      └──────────────────────────────┘  └───────────────────────┘        »\n",
       "« q_4: ───────────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐   ┌────────────────────────────┐     »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├───┤ U3(3.1416,3.5375,-0.46269) ├─────»\n",
       "«        └─────────────────────────┘   └────────────────────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────»\n",
       "«                                                                         »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════»\n",
       "«                                                                         »\n",
       "«                                                                           »\n",
       "« q_0: ─────────────────────────────────────────────────────────────────────»\n",
       "«      ┌────────────┐                  ┌───┐┌──────────────────────────────┐»\n",
       "« q_1: ┤ Rz(13π/10) ├──■───────────────┤ X ├┤ U3(3.2916e-11,4.531,0.70243) ├»\n",
       "«      └────────────┘  │               └─┬─┘└──────────────────────────────┘»\n",
       "« q_2: ────────────────┼─────────────────┼──────────────────────────────────»\n",
       "«      ┌────────────┐┌─┴─┐┌───────────┐  │      ┌──────────────────────┐    »\n",
       "« q_3: ┤ Ry(13π/10) ├┤ X ├┤ Ry(3π/10) ├──■──────┤ U3(-π,-1.125,4.6372) ├────»\n",
       "«      └────────────┘└───┘└───────────┘         └──────────────────────┘    »\n",
       "« q_4: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_5: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "« q_6: ─────────────────────────────────────────────────────────────────────»\n",
       "«                                                                           »\n",
       "«c2: 3/═════════════════════════════════════════════════════════════════════»\n",
       "«                                                                           »\n",
       "«                                                                             »\n",
       "« q_0: ───────────────────────────────────────────────────────────────────────»\n",
       "«       ┌────────────────────────┐                                            »\n",
       "« q_1: ─┤ U3(2π,5.3676,0.032489) ├────────────────────────────────────────────»\n",
       "«       └────────────────────────┘                                            »\n",
       "« q_2: ───────────────────────────────────────────────────────────────────────»\n",
       "«      ┌──────────────────────────┐┌───┐┌────────────┐                   ┌───┐»\n",
       "« q_3: ┤ U3(6.2832,2.3456,1.5083) ├┤ X ├┤ Rz(13π/10) ├──■────────────────┤ X ├»\n",
       "«      └──────────────────────────┘└─┬─┘└────────────┘  │                └─┬─┘»\n",
       "« q_4: ──────────────────────────────┼──────────────────┼──────────────────┼──»\n",
       "«                                    │  ┌────────────┐┌─┴─┐┌────────────┐  │  »\n",
       "« q_5: ──────────────────────────────■──┤ Ry(13π/10) ├┤ X ├┤ Ry(7.2257) ├──■──»\n",
       "«                                       └────────────┘└───┘└────────────┘     »\n",
       "« q_6: ───────────────────────────────────────────────────────────────────────»\n",
       "«                                                                             »\n",
       "«c2: 3/═══════════════════════════════════════════════════════════════════════»\n",
       "«                                                                             »\n",
       "«                                                                     »\n",
       "« q_0: ───────────────────────────────────────────────────────────────»\n",
       "«                                                                ┌───┐»\n",
       "« q_1: ──────────────────────────────────────────────────────────┤ X ├»\n",
       "«                                                                └─┬─┘»\n",
       "« q_2: ────────────────────────────────────────────────────────────┼──»\n",
       "«      ┌──────────────────────────────┐┌────────────────────────┐  │  »\n",
       "« q_3: ┤ U3(2.7748e-06,2.0573,5.0844) ├┤ U3(-π,0.70057,-1.7539) ├──■──»\n",
       "«      └──────────────────────────────┘└────────────────────────┘     »\n",
       "« q_4: ───────────────────────────────────────────────────────────────»\n",
       "«        ┌─────────────────────────┐              ┌─┐                 »\n",
       "« q_5: ──┤ U3(3.1416,2.819,6.6726) ├──────────────┤M├─────────────────»\n",
       "«        └─────────────────────────┘              └╥┘                 »\n",
       "« q_6: ────────────────────────────────────────────╫──────────────────»\n",
       "«                                                  ║                  »\n",
       "«c2: 3/════════════════════════════════════════════╩══════════════════»\n",
       "«                                                  2                  »\n",
       "«                                                                         \n",
       "« q_0: ───────────────────────────────────────────────────────────────────\n",
       "«      ┌──────────┐                  ┌───┐ ┌─────────────────────┐  ┌─┐   \n",
       "« q_1: ┤ Rz(7π/5) ├──■───────────────┤ X ├─┤ U3(0,5.1954,6.6832) ├──┤M├───\n",
       "«      └──────────┘  │               └─┬─┘ └─────────────────────┘  └╥┘   \n",
       "« q_2: ──────────────┼─────────────────┼─────────────────────────────╫────\n",
       "«      ┌──────────┐┌─┴─┐┌───────────┐  │  ┌────────────────────────┐ ║ ┌─┐\n",
       "« q_3: ┤ Ry(7π/5) ├┤ X ├┤ Ry(12π/5) ├──■──┤ U3(π,0.96835,0.084917) ├─╫─┤M├\n",
       "«      └──────────┘└───┘└───────────┘     └────────────────────────┘ ║ └╥┘\n",
       "« q_4: ──────────────────────────────────────────────────────────────╫──╫─\n",
       "«                                                                    ║  ║ \n",
       "« q_5: ──────────────────────────────────────────────────────────────╫──╫─\n",
       "«                                                                    ║  ║ \n",
       "« q_6: ──────────────────────────────────────────────────────────────╫──╫─\n",
       "«                                                                    ║  ║ \n",
       "«c2: 3/══════════════════════════════════════════════════════════════╩══╩═\n",
       "«                                                                    0  1 "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = π\n",
    "trotter_steps = 10\n",
    "qc_full, _, _, _ = generate_circuit(trotter_steps, trotter_steps, t, init=true)\n",
    "\n",
    "# Add measurment\n",
    "cr = ClassicalRegister(3)\n",
    "setClassicalRegister!(qc_full, cr)\n",
    "qc_full.measure([1, 3, 5], [0, 1, 2])\n",
    "qc_full"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "82faa8d9",
   "metadata": {},
   "source": [
    "Next helper function is used to compare the simulation results with  exact values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "1df31558",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "check_simulation_err (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using QuantumCircuits.QCircuits.Math\n",
    "using QuantumCircuits.QCircuits.Gates: Xmatrix, Ymatrix, Zmatrix\n",
    "\n",
    "XXs = kron(kron(eye(2), Xmatrix), Xmatrix) + kron(kron(Xmatrix, Xmatrix), eye(2))\n",
    "YYs = kron(kron(eye(2), Ymatrix), Ymatrix) + kron(kron(Ymatrix, Ymatrix), eye(2))\n",
    "ZZs = kron(kron(eye(2), Zmatrix), Zmatrix) + kron(kron(Zmatrix, Zmatrix), eye(2))\n",
    "Hs = XXs + YYs + ZZs\n",
    "\n",
    "U_heis3(t) = exp(-im * Hs * t)\n",
    "\n",
    "const backend = QuantumSimulator()\n",
    "\n",
    "function check_simulation_err(qc, t)\n",
    "    sym_full = execute(backend, qc)\n",
    "    exp_full = U_heis3(t) * ket\"110\"\n",
    "    exp_full = abs.(exp_full) .^ 2\n",
    "    return sum(abs.(sym_full - exp_full))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "980a025e",
   "metadata": {},
   "source": [
    "And, now we can check the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "8596435e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10278310866155166"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "check_simulation_err(qc_full, t)"
   ]
  }
 ],
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