docs/source/quilt_waveforms.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_formats = ['svg']\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#  Pulses and waveforms\n",
    "\n",
    "\n",
    "A waveform is simply a time varying signal. When a pulse operation occurs, a digital to analog converter produces a signal by combinding the following data:\n",
    "\n",
    "* a time varying complex function $ u(t) $ which describes the baseband waveform\n",
    "* a *frequency* $f$, in Hertz,\n",
    "* a *phase* $\\theta$, in radians,\n",
    "* a unitless *scale* $\\alpha$.\n",
    "\n",
    "The signal generated from this has the mathematical form\n",
    "\\begin{equation}\n",
    "x(t) = \\text{Re}[\\alpha u(t)e^{i (\\theta + 2 \\pi f t)}],\n",
    "\\end{equation}\n",
    "where $\\text{Re}$ denotes the real part of a complex number. \n",
    "\n",
    "In accordance with the usual conventions, we will refer to the real part of $u(t)$ as the **in-phase** component, and the imaginary part as the **quadrature** component. A general complex number $z = x + iy$ will sometimes be referred to as an **IQ-value**, with $I = x$ and $Q = y$.\n",
    "\n",
    "\n",
    "Quil-T provides the programmer with pulse and capture-level control, both by allowing for custom baseband waveforms, as well as allowing for direct control over the run-time frequency, phase, and scale.\n",
    "\n",
    "\n",
    "## Waveform Templates and References\n",
    "\n",
    "There are two ways to specify a Quil-T waveform:\n",
    "- by using a pre-existing template, which has a shape dictated by certain parameters\n",
    "- by referencing a custom waveform definition.\n",
    "\n",
    "Consider the following Quil-T code:\n",
    "```\n",
    "PULSE 0 \"rf\" flat(duration: 1e-6, iq: 0.5 + 0.5*i)\n",
    "```\n",
    "This denotes a pulse operation, on frame `0 \"rf\"`, with a total duration of one microsecond and with baseband waveform given by the `flat` template, (corresponding to $u(t) = 1/2 + i/2$ for the duration of the signal). The resulting signal produced depends also on the phase, frequency, and scaling factor associated with frame `0 \"rf\"`. These may be set explicitly, as in\n",
    "```\n",
    "SET-SCALE 0 \"rf\" 1.0\n",
    "SET-FREQUENCY 0 \"rf\" 5e9\n",
    "SET-PHASE 0 \"rf\" 0.0\n",
    "```\n",
    "\n",
    "On the other hand, a custom waveform may be defined and used. This is done using the `DEFWAVEFORM` form,\n",
    "```\n",
    "DEFWAVEFORM my_waveform:\n",
    "    0.01, 0.01+0.01*i, ...\n",
    "```\n",
    "the body of which consists of a list of complex numbers. Such a custom waveform may be referenced directly by name, as in\n",
    "```\n",
    "PULSE 0 \"rf\" my_waveform.\n",
    "```\n",
    "\n",
    "The precise meaning of this depends on the *sample rate* of the associated frame: this indicates the number of samples per second consumed by the underlying digital-to-analog converter. For example, suppose that the definition of `0 \"rf\"` looked like this\n",
    "```\n",
    "DEFFRAME 0 \"rf\":\n",
    "    SAMPLE-RATE: 1000000000.0\n",
    "    INITIAL-FREQUENCY: 4807541957.13474\n",
    "    DIRECTION: \"tx\"\n",
    "```\n",
    "and `my_waveform` has a definition consisting of complex numbers $z_1, \\ldots, z_N$. Then, letting $r=10^9$ denote the sample rate, the resulting pulse has total duration $\\left \\lceil{\\frac{N}{r}}\\right \\rceil$, corresponding to the baseband waveform $ u(t) = z_{\\left \\lfloor{tr}\\right \\rfloor}. $ Here $\\left \\lceil{x}\\right \\rceil $ and $\\left \\lfloor{x}\\right \\rfloor$ denote the ceiling and floor of $x$, respectively. \n",
    "\n",
    "As before, this baseband waveform is combined with the frame's scale, frequency, and phase during digital to analog conversion.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Catalog of Template Waveforms\n",
    "\n",
    "Rigetti provides a number of templates by default. These include\n",
    "\n",
    "- `flat`, corresponding to simple rectangular waveforms\n",
    "- `gaussian`, for a Gaussian waveform\n",
    "- `drag_gaussian`, for a Gaussian waveform modified by the Derivative Removal by Adiabatic Gate (DRAG) technique\n",
    "- `hrm_gaussian`, for a DRAG Gaussian waveform with second-order corrections\n",
    "- `erf_square`, for a flat waveform with smooth edges derived from the Gaussian error function\n",
    "- `boxcar_kernel`, for a flat waveform which is normalized to integrate to 1\n",
    "\n",
    "Each of these waveforms has a corresponding definition in `pyquil.quiltwaveforms`. In addition to providing documentation on the meaning of each of the waveform parameters, this module also contains routines for generating samples for each template.\n",
    "\n",
    "Below we look at each individidually, discussing the meaning of various parameters and plotting the real part of the waveform envelope."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pyquil.quilatom import TemplateWaveform\n",
    "\n",
    "def plot_waveform(wf: TemplateWaveform, sample_rate: float):\n",
    "    \"\"\" Plot a template waveform by sampling at the specified sample rate. \"\"\"\n",
    "    samples = wf.samples(sample_rate)\n",
    "    times = np.arange(len(samples))/sample_rate\n",
    "\n",
    "    print(wf)\n",
    "    plt.plot(times, samples.real)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### flat\n",
    "\n",
    "A flat waveform is simple: it represents a constant signal for a certain duration. There are two required parameters:\n",
    "\n",
    "* `duration`: the length of the waveform, in seconds\n",
    "* `iq`: a complex number"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "flat(duration: 1e-6, iq: 1)\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"400.785625pt\" height=\"309.86825pt\" viewBox=\"0 0 400.785625 309.86825\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2023-09-15T14:32:51.996913</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.7.2, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 309.86825 \n",
       "L 400.785625 309.86825 \n",
       "L 400.785625 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 36.465625 273.312 \n",
       "L 393.585625 273.312 \n",
       "L 393.585625 7.2 \n",
       "L 36.465625 7.2 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path id=\"m68e64daaf9\" d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m68e64daaf9\" x=\"52.698352\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(44.74679 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
       "Q 1547 4250 1301 3770 \n",
       "Q 1056 3291 1056 2328 \n",
       "Q 1056 1369 1301 889 \n",
       "Q 1547 409 2034 409 \n",
       "Q 2525 409 2770 889 \n",
       "Q 3016 1369 3016 2328 \n",
       "Q 3016 3291 2770 3770 \n",
       "Q 2525 4250 2034 4250 \n",
       "z\n",
       "M 2034 4750 \n",
       "Q 2819 4750 3233 4129 \n",
       "Q 3647 3509 3647 2328 \n",
       "Q 3647 1150 3233 529 \n",
       "Q 2819 -91 2034 -91 \n",
       "Q 1250 -91 836 529 \n",
       "Q 422 1150 422 2328 \n",
       "Q 422 3509 836 4129 \n",
       "Q 1250 4750 2034 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-2e\" d=\"M 684 794 \n",
       "L 1344 794 \n",
       "L 1344 0 \n",
       "L 684 0 \n",
       "L 684 794 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m68e64daaf9\" x=\"117.694257\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(109.742695 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-32\" d=\"M 1228 531 \n",
       "L 3431 531 \n",
       "L 3431 0 \n",
       "L 469 0 \n",
       "L 469 531 \n",
       "Q 828 903 1448 1529 \n",
       "Q 2069 2156 2228 2338 \n",
       "Q 2531 2678 2651 2914 \n",
       "Q 2772 3150 2772 3378 \n",
       "Q 2772 3750 2511 3984 \n",
       "Q 2250 4219 1831 4219 \n",
       "Q 1534 4219 1204 4116 \n",
       "Q 875 4013 500 3803 \n",
       "L 500 4441 \n",
       "Q 881 4594 1212 4672 \n",
       "Q 1544 4750 1819 4750 \n",
       "Q 2544 4750 2975 4387 \n",
       "Q 3406 4025 3406 3419 \n",
       "Q 3406 3131 3298 2873 \n",
       "Q 3191 2616 2906 2266 \n",
       "Q 2828 2175 2409 1742 \n",
       "Q 1991 1309 1228 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m68e64daaf9\" x=\"182.690162\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(174.7386 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
       "L 825 1625 \n",
       "L 2419 1625 \n",
       "L 2419 4116 \n",
       "z\n",
       "M 2253 4666 \n",
       "L 3047 4666 \n",
       "L 3047 1625 \n",
       "L 3713 1625 \n",
       "L 3713 1100 \n",
       "L 3047 1100 \n",
       "L 3047 0 \n",
       "L 2419 0 \n",
       "L 2419 1100 \n",
       "L 313 1100 \n",
       "L 313 1709 \n",
       "L 2253 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m68e64daaf9\" x=\"247.686067\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(239.734505 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
       "Q 1688 2584 1439 2293 \n",
       "Q 1191 2003 1191 1497 \n",
       "Q 1191 994 1439 701 \n",
       "Q 1688 409 2113 409 \n",
       "Q 2538 409 2786 701 \n",
       "Q 3034 994 3034 1497 \n",
       "Q 3034 2003 2786 2293 \n",
       "Q 2538 2584 2113 2584 \n",
       "z\n",
       "M 3366 4563 \n",
       "L 3366 3988 \n",
       "Q 3128 4100 2886 4159 \n",
       "Q 2644 4219 2406 4219 \n",
       "Q 1781 4219 1451 3797 \n",
       "Q 1122 3375 1075 2522 \n",
       "Q 1259 2794 1537 2939 \n",
       "Q 1816 3084 2150 3084 \n",
       "Q 2853 3084 3261 2657 \n",
       "Q 3669 2231 3669 1497 \n",
       "Q 3669 778 3244 343 \n",
       "Q 2819 -91 2113 -91 \n",
       "Q 1303 -91 875 529 \n",
       "Q 447 1150 447 2328 \n",
       "Q 447 3434 972 4092 \n",
       "Q 1497 4750 2381 4750 \n",
       "Q 2619 4750 2861 4703 \n",
       "Q 3103 4656 3366 4563 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m68e64daaf9\" x=\"312.681972\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(304.73041 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \n",
       "Q 1584 2216 1326 1975 \n",
       "Q 1069 1734 1069 1313 \n",
       "Q 1069 891 1326 650 \n",
       "Q 1584 409 2034 409 \n",
       "Q 2484 409 2743 651 \n",
       "Q 3003 894 3003 1313 \n",
       "Q 3003 1734 2745 1975 \n",
       "Q 2488 2216 2034 2216 \n",
       "z\n",
       "M 1403 2484 \n",
       "Q 997 2584 770 2862 \n",
       "Q 544 3141 544 3541 \n",
       "Q 544 4100 942 4425 \n",
       "Q 1341 4750 2034 4750 \n",
       "Q 2731 4750 3128 4425 \n",
       "Q 3525 4100 3525 3541 \n",
       "Q 3525 3141 3298 2862 \n",
       "Q 3072 2584 2669 2484 \n",
       "Q 3125 2378 3379 2068 \n",
       "Q 3634 1759 3634 1313 \n",
       "Q 3634 634 3220 271 \n",
       "Q 2806 -91 2034 -91 \n",
       "Q 1263 -91 848 271 \n",
       "Q 434 634 434 1313 \n",
       "Q 434 1759 690 2068 \n",
       "Q 947 2378 1403 2484 \n",
       "z\n",
       "M 1172 3481 \n",
       "Q 1172 3119 1398 2916 \n",
       "Q 1625 2713 2034 2713 \n",
       "Q 2441 2713 2670 2916 \n",
       "Q 2900 3119 2900 3481 \n",
       "Q 2900 3844 2670 4047 \n",
       "Q 2441 4250 2034 4250 \n",
       "Q 1625 4250 1398 4047 \n",
       "Q 1172 3844 1172 3481 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m68e64daaf9\" x=\"377.677877\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(369.726315 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-31\" d=\"M 794 531 \n",
       "L 1825 531 \n",
       "L 1825 4091 \n",
       "L 703 3866 \n",
       "L 703 4441 \n",
       "L 1819 4666 \n",
       "L 2450 4666 \n",
       "L 2450 531 \n",
       "L 3481 531 \n",
       "L 3481 0 \n",
       "L 794 0 \n",
       "L 794 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- 1e−6 -->\n",
       "     <g transform=\"translate(366.327813 300.588563) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-65\" d=\"M 3597 1894 \n",
       "L 3597 1613 \n",
       "L 953 1613 \n",
       "Q 991 1019 1311 708 \n",
       "Q 1631 397 2203 397 \n",
       "Q 2534 397 2845 478 \n",
       "Q 3156 559 3463 722 \n",
       "L 3463 178 \n",
       "Q 3153 47 2828 -22 \n",
       "Q 2503 -91 2169 -91 \n",
       "Q 1331 -91 842 396 \n",
       "Q 353 884 353 1716 \n",
       "Q 353 2575 817 3079 \n",
       "Q 1281 3584 2069 3584 \n",
       "Q 2775 3584 3186 3129 \n",
       "Q 3597 2675 3597 1894 \n",
       "z\n",
       "M 3022 2063 \n",
       "Q 3016 2534 2758 2815 \n",
       "Q 2500 3097 2075 3097 \n",
       "Q 1594 3097 1305 2825 \n",
       "Q 1016 2553 972 2059 \n",
       "L 3022 2063 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2212\" d=\"M 678 2272 \n",
       "L 4684 2272 \n",
       "L 4684 1741 \n",
       "L 678 1741 \n",
       "L 678 2272 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-65\" x=\"63.623047\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2212\" x=\"125.146484\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-36\" x=\"208.935547\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <defs>\n",
       "       <path id=\"m15e75d3297\" d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m15e75d3297\" x=\"36.465625\" y=\"237.024\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 0.96 -->\n",
       "      <g transform=\"translate(7.2 240.823219) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-39\" d=\"M 703 97 \n",
       "L 703 672 \n",
       "Q 941 559 1184 500 \n",
       "Q 1428 441 1663 441 \n",
       "Q 2288 441 2617 861 \n",
       "Q 2947 1281 2994 2138 \n",
       "Q 2813 1869 2534 1725 \n",
       "Q 2256 1581 1919 1581 \n",
       "Q 1219 1581 811 2004 \n",
       "Q 403 2428 403 3163 \n",
       "Q 403 3881 828 4315 \n",
       "Q 1253 4750 1959 4750 \n",
       "Q 2769 4750 3195 4129 \n",
       "Q 3622 3509 3622 2328 \n",
       "Q 3622 1225 3098 567 \n",
       "Q 2575 -91 1691 -91 \n",
       "Q 1453 -91 1209 -44 \n",
       "Q 966 3 703 97 \n",
       "z\n",
       "M 1959 2075 \n",
       "Q 2384 2075 2632 2365 \n",
       "Q 2881 2656 2881 3163 \n",
       "Q 2881 3666 2632 3958 \n",
       "Q 2384 4250 1959 4250 \n",
       "Q 1534 4250 1286 3958 \n",
       "Q 1038 3666 1038 3163 \n",
       "Q 1038 2656 1286 2365 \n",
       "Q 1534 2075 1959 2075 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m15e75d3297\" x=\"36.465625\" y=\"188.64\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0.98 -->\n",
       "      <g transform=\"translate(7.2 192.439219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m15e75d3297\" x=\"36.465625\" y=\"140.256\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 1.00 -->\n",
       "      <g transform=\"translate(7.2 144.055219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m15e75d3297\" x=\"36.465625\" y=\"91.872\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 1.02 -->\n",
       "      <g transform=\"translate(7.2 95.671219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m15e75d3297\" x=\"36.465625\" y=\"43.488\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 1.04 -->\n",
       "      <g transform=\"translate(7.2 47.287219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_12\">\n",
       "    <path d=\"M 52.698352 140.256 \n",
       "L 377.352898 140.256 \n",
       "L 377.352898 140.256 \n",
       "\" clip-path=\"url(#pd8e2c5c775)\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 36.465625 273.312 \n",
       "L 36.465625 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 393.585625 273.312 \n",
       "L 393.585625 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 36.465625 273.312 \n",
       "L 393.585625 273.312 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 36.465625 7.2 \n",
       "L 393.585625 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pd8e2c5c775\">\n",
       "   <rect x=\"36.465625\" y=\"7.2\" width=\"357.12\" height=\"266.112\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pyquil.quiltwaveforms import FlatWaveform\n",
    "\n",
    "plot_waveform(FlatWaveform(duration=1e-6, iq=1.0), sample_rate=1e9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Scale, Phase, Detuning\n",
    "\n",
    "In addition to the parameters specific to each template waveform, there are also a few generic parameters. One of these we have met: each template waveform has a required `duration` argument, indicating the length of the waveform in seconds.\n",
    "\n",
    "The other arguments are *optional*, and are used to modulate the basic shape. These are\n",
    "\n",
    "* scale $\\alpha$, which has the effect of scaling the baseband $u(t) \\mapsto \\alpha u(t)$\n",
    "* phase $\\theta$, in radians, which has the effect of a phase shift on the baseband $u(t) \\mapsto e^{i \\theta} u(t)$\n",
    "* detuning $f_d$, in Hertz, which has the effect of $u(t) \\mapsto e^{2 \\pi i f_d} u(t)$\n",
    "\n",
    "These may be provided as arguments to any template waveform, e.g.\n",
    "```\n",
    "PULSE 0 \"rf\" flat(duration: 1e-8, iq: 1.0, scale: 0.3, phase: 1.570796, detuning: 1e8) \n",
    "```\n",
    "\n",
    "Below we consider this by way of the pyQuil bindings."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "flat(detuning: 10000000, duration: 1e-6, iq: 1)\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"409.165312pt\" height=\"309.86825pt\" viewBox=\"0 0 409.165312 309.86825\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2023-09-15T14:32:52.057655</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.7.2, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 309.86825 \n",
       "L 409.165312 309.86825 \n",
       "L 409.165312 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 44.845313 273.312 \n",
       "L 401.965312 273.312 \n",
       "L 401.965312 7.2 \n",
       "L 44.845313 7.2 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path id=\"m57bcf01be9\" d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m57bcf01be9\" x=\"61.07804\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(53.126477 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
       "Q 1547 4250 1301 3770 \n",
       "Q 1056 3291 1056 2328 \n",
       "Q 1056 1369 1301 889 \n",
       "Q 1547 409 2034 409 \n",
       "Q 2525 409 2770 889 \n",
       "Q 3016 1369 3016 2328 \n",
       "Q 3016 3291 2770 3770 \n",
       "Q 2525 4250 2034 4250 \n",
       "z\n",
       "M 2034 4750 \n",
       "Q 2819 4750 3233 4129 \n",
       "Q 3647 3509 3647 2328 \n",
       "Q 3647 1150 3233 529 \n",
       "Q 2819 -91 2034 -91 \n",
       "Q 1250 -91 836 529 \n",
       "Q 422 1150 422 2328 \n",
       "Q 422 3509 836 4129 \n",
       "Q 1250 4750 2034 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-2e\" d=\"M 684 794 \n",
       "L 1344 794 \n",
       "L 1344 0 \n",
       "L 684 0 \n",
       "L 684 794 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m57bcf01be9\" x=\"126.073945\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(118.122382 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-32\" d=\"M 1228 531 \n",
       "L 3431 531 \n",
       "L 3431 0 \n",
       "L 469 0 \n",
       "L 469 531 \n",
       "Q 828 903 1448 1529 \n",
       "Q 2069 2156 2228 2338 \n",
       "Q 2531 2678 2651 2914 \n",
       "Q 2772 3150 2772 3378 \n",
       "Q 2772 3750 2511 3984 \n",
       "Q 2250 4219 1831 4219 \n",
       "Q 1534 4219 1204 4116 \n",
       "Q 875 4013 500 3803 \n",
       "L 500 4441 \n",
       "Q 881 4594 1212 4672 \n",
       "Q 1544 4750 1819 4750 \n",
       "Q 2544 4750 2975 4387 \n",
       "Q 3406 4025 3406 3419 \n",
       "Q 3406 3131 3298 2873 \n",
       "Q 3191 2616 2906 2266 \n",
       "Q 2828 2175 2409 1742 \n",
       "Q 1991 1309 1228 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m57bcf01be9\" x=\"191.06985\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(183.118287 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
       "L 825 1625 \n",
       "L 2419 1625 \n",
       "L 2419 4116 \n",
       "z\n",
       "M 2253 4666 \n",
       "L 3047 4666 \n",
       "L 3047 1625 \n",
       "L 3713 1625 \n",
       "L 3713 1100 \n",
       "L 3047 1100 \n",
       "L 3047 0 \n",
       "L 2419 0 \n",
       "L 2419 1100 \n",
       "L 313 1100 \n",
       "L 313 1709 \n",
       "L 2253 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m57bcf01be9\" x=\"256.065755\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(248.114192 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
       "Q 1688 2584 1439 2293 \n",
       "Q 1191 2003 1191 1497 \n",
       "Q 1191 994 1439 701 \n",
       "Q 1688 409 2113 409 \n",
       "Q 2538 409 2786 701 \n",
       "Q 3034 994 3034 1497 \n",
       "Q 3034 2003 2786 2293 \n",
       "Q 2538 2584 2113 2584 \n",
       "z\n",
       "M 3366 4563 \n",
       "L 3366 3988 \n",
       "Q 3128 4100 2886 4159 \n",
       "Q 2644 4219 2406 4219 \n",
       "Q 1781 4219 1451 3797 \n",
       "Q 1122 3375 1075 2522 \n",
       "Q 1259 2794 1537 2939 \n",
       "Q 1816 3084 2150 3084 \n",
       "Q 2853 3084 3261 2657 \n",
       "Q 3669 2231 3669 1497 \n",
       "Q 3669 778 3244 343 \n",
       "Q 2819 -91 2113 -91 \n",
       "Q 1303 -91 875 529 \n",
       "Q 447 1150 447 2328 \n",
       "Q 447 3434 972 4092 \n",
       "Q 1497 4750 2381 4750 \n",
       "Q 2619 4750 2861 4703 \n",
       "Q 3103 4656 3366 4563 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m57bcf01be9\" x=\"321.06166\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(313.110097 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \n",
       "Q 1584 2216 1326 1975 \n",
       "Q 1069 1734 1069 1313 \n",
       "Q 1069 891 1326 650 \n",
       "Q 1584 409 2034 409 \n",
       "Q 2484 409 2743 651 \n",
       "Q 3003 894 3003 1313 \n",
       "Q 3003 1734 2745 1975 \n",
       "Q 2488 2216 2034 2216 \n",
       "z\n",
       "M 1403 2484 \n",
       "Q 997 2584 770 2862 \n",
       "Q 544 3141 544 3541 \n",
       "Q 544 4100 942 4425 \n",
       "Q 1341 4750 2034 4750 \n",
       "Q 2731 4750 3128 4425 \n",
       "Q 3525 4100 3525 3541 \n",
       "Q 3525 3141 3298 2862 \n",
       "Q 3072 2584 2669 2484 \n",
       "Q 3125 2378 3379 2068 \n",
       "Q 3634 1759 3634 1313 \n",
       "Q 3634 634 3220 271 \n",
       "Q 2806 -91 2034 -91 \n",
       "Q 1263 -91 848 271 \n",
       "Q 434 634 434 1313 \n",
       "Q 434 1759 690 2068 \n",
       "Q 947 2378 1403 2484 \n",
       "z\n",
       "M 1172 3481 \n",
       "Q 1172 3119 1398 2916 \n",
       "Q 1625 2713 2034 2713 \n",
       "Q 2441 2713 2670 2916 \n",
       "Q 2900 3119 2900 3481 \n",
       "Q 2900 3844 2670 4047 \n",
       "Q 2441 4250 2034 4250 \n",
       "Q 1625 4250 1398 4047 \n",
       "Q 1172 3844 1172 3481 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m57bcf01be9\" x=\"386.057565\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(378.106002 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-31\" d=\"M 794 531 \n",
       "L 1825 531 \n",
       "L 1825 4091 \n",
       "L 703 3866 \n",
       "L 703 4441 \n",
       "L 1819 4666 \n",
       "L 2450 4666 \n",
       "L 2450 531 \n",
       "L 3481 531 \n",
       "L 3481 0 \n",
       "L 794 0 \n",
       "L 794 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- 1e−6 -->\n",
       "     <g transform=\"translate(374.7075 300.588562) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-65\" d=\"M 3597 1894 \n",
       "L 3597 1613 \n",
       "L 953 1613 \n",
       "Q 991 1019 1311 708 \n",
       "Q 1631 397 2203 397 \n",
       "Q 2534 397 2845 478 \n",
       "Q 3156 559 3463 722 \n",
       "L 3463 178 \n",
       "Q 3153 47 2828 -22 \n",
       "Q 2503 -91 2169 -91 \n",
       "Q 1331 -91 842 396 \n",
       "Q 353 884 353 1716 \n",
       "Q 353 2575 817 3079 \n",
       "Q 1281 3584 2069 3584 \n",
       "Q 2775 3584 3186 3129 \n",
       "Q 3597 2675 3597 1894 \n",
       "z\n",
       "M 3022 2063 \n",
       "Q 3016 2534 2758 2815 \n",
       "Q 2500 3097 2075 3097 \n",
       "Q 1594 3097 1305 2825 \n",
       "Q 1016 2553 972 2059 \n",
       "L 3022 2063 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2212\" d=\"M 678 2272 \n",
       "L 4684 2272 \n",
       "L 4684 1741 \n",
       "L 678 1741 \n",
       "L 678 2272 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-65\" x=\"63.623047\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2212\" x=\"125.146484\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-36\" x=\"208.935547\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <defs>\n",
       "       <path id=\"mad6efb098e\" d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#mad6efb098e\" x=\"44.845313\" y=\"261.216\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- −1.00 -->\n",
       "      <g transform=\"translate(7.2 265.015219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-2212\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-31\" x=\"83.789062\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"147.412109\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"179.199219\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"242.822266\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mad6efb098e\" x=\"44.845313\" y=\"230.976\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- −0.75 -->\n",
       "      <g transform=\"translate(7.2 234.775219) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-37\" d=\"M 525 4666 \n",
       "L 3525 4666 \n",
       "L 3525 4397 \n",
       "L 1831 0 \n",
       "L 1172 0 \n",
       "L 2766 4134 \n",
       "L 525 4134 \n",
       "L 525 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-35\" d=\"M 691 4666 \n",
       "L 3169 4666 \n",
       "L 3169 4134 \n",
       "L 1269 4134 \n",
       "L 1269 2991 \n",
       "Q 1406 3038 1543 3061 \n",
       "Q 1681 3084 1819 3084 \n",
       "Q 2600 3084 3056 2656 \n",
       "Q 3513 2228 3513 1497 \n",
       "Q 3513 744 3044 326 \n",
       "Q 2575 -91 1722 -91 \n",
       "Q 1428 -91 1123 -41 \n",
       "Q 819 9 494 109 \n",
       "L 494 744 \n",
       "Q 775 591 1075 516 \n",
       "Q 1375 441 1709 441 \n",
       "Q 2250 441 2565 725 \n",
       "Q 2881 1009 2881 1497 \n",
       "Q 2881 1984 2565 2268 \n",
       "Q 2250 2553 1709 2553 \n",
       "Q 1456 2553 1204 2497 \n",
       "Q 953 2441 691 2322 \n",
       "L 691 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-2212\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"83.789062\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"147.412109\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-37\" x=\"179.199219\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" x=\"242.822266\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mad6efb098e\" x=\"44.845313\" y=\"200.736\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- −0.50 -->\n",
       "      <g transform=\"translate(7.2 204.535219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-2212\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"83.789062\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"147.412109\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" x=\"179.199219\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"242.822266\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mad6efb098e\" x=\"44.845313\" y=\"170.496\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- −0.25 -->\n",
       "      <g transform=\"translate(7.2 174.295219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-2212\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"83.789062\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"147.412109\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"179.199219\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" x=\"242.822266\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mad6efb098e\" x=\"44.845313\" y=\"140.256\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.00 -->\n",
       "      <g transform=\"translate(15.579688 144.055219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mad6efb098e\" x=\"44.845313\" y=\"110.016\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 0.25 -->\n",
       "      <g transform=\"translate(15.579688 113.815219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_7\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mad6efb098e\" x=\"44.845313\" y=\"79.776\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_14\">\n",
       "      <!-- 0.50 -->\n",
       "      <g transform=\"translate(15.579688 83.575219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_8\">\n",
       "     <g id=\"line2d_14\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mad6efb098e\" x=\"44.845313\" y=\"49.536\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_15\">\n",
       "      <!-- 0.75 -->\n",
       "      <g transform=\"translate(15.579688 53.335219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-37\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-35\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_9\">\n",
       "     <g id=\"line2d_15\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mad6efb098e\" x=\"44.845313\" y=\"19.296\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_16\">\n",
       "      <!-- 1.00 -->\n",
       "      <g transform=\"translate(15.579688 23.095219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"159.033203\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_16\">\n",
       "    <path d=\"M 61.07804 19.296 \n",
       "L 61.403019 19.534687 \n",
       "L 61.727999 20.249806 \n",
       "L 62.052978 21.438534 \n",
       "L 62.702937 25.216204 \n",
       "L 63.352896 30.80812 \n",
       "L 64.002855 38.126094 \n",
       "L 64.977794 52.079955 \n",
       "L 65.952733 69.157496 \n",
       "L 67.252651 95.727654 \n",
       "L 69.527507 147.851141 \n",
       "L 71.802364 198.528924 \n",
       "L 73.102282 223.058818 \n",
       "L 74.077221 238.114696 \n",
       "L 75.052159 249.70388 \n",
       "L 75.702118 255.295796 \n",
       "L 76.352077 259.073466 \n",
       "L 76.677057 260.262194 \n",
       "L 77.002036 260.977313 \n",
       "L 77.327016 261.216 \n",
       "L 77.651996 260.977313 \n",
       "L 77.976975 260.262194 \n",
       "L 78.301955 259.073466 \n",
       "L 78.951914 255.295796 \n",
       "L 79.601873 249.70388 \n",
       "L 80.251832 242.385906 \n",
       "L 81.22677 228.432045 \n",
       "L 82.201709 211.354504 \n",
       "L 83.501627 184.784346 \n",
       "L 85.776484 132.660859 \n",
       "L 88.05134 81.983076 \n",
       "L 89.351258 57.453182 \n",
       "L 90.326197 42.397304 \n",
       "L 91.301136 30.80812 \n",
       "L 91.951095 25.216204 \n",
       "L 92.601054 21.438534 \n",
       "L 92.926033 20.249806 \n",
       "L 93.251013 19.534687 \n",
       "L 93.575992 19.296 \n",
       "L 93.900972 19.534687 \n",
       "L 94.225951 20.249806 \n",
       "L 94.550931 21.438534 \n",
       "L 95.20089 25.216204 \n",
       "L 95.850849 30.80812 \n",
       "L 96.500808 38.126094 \n",
       "L 97.475747 52.079955 \n",
       "L 98.450685 69.157496 \n",
       "L 99.750603 95.727654 \n",
       "L 102.02546 147.851141 \n",
       "L 104.300317 198.528924 \n",
       "L 105.600235 223.058818 \n",
       "L 106.575173 238.114696 \n",
       "L 107.550112 249.70388 \n",
       "L 108.200071 255.295796 \n",
       "L 108.85003 259.073466 \n",
       "L 109.175009 260.262194 \n",
       "L 109.499989 260.977313 \n",
       "L 109.824969 261.216 \n",
       "L 110.149948 260.977313 \n",
       "L 110.474928 260.262194 \n",
       "L 110.799907 259.073466 \n",
       "L 111.449866 255.295796 \n",
       "L 112.099825 249.70388 \n",
       "L 112.749784 242.385906 \n",
       "L 113.724723 228.432045 \n",
       "L 114.699661 211.354504 \n",
       "L 115.999579 184.784346 \n",
       "L 118.274436 132.660859 \n",
       "L 120.549293 81.983076 \n",
       "L 121.849211 57.453182 \n",
       "L 122.82415 42.397304 \n",
       "L 123.799088 30.80812 \n",
       "L 124.449047 25.216204 \n",
       "L 125.099006 21.438534 \n",
       "L 125.423986 20.249806 \n",
       "L 125.748965 19.534687 \n",
       "L 126.073945 19.296 \n",
       "L 126.398924 19.534687 \n",
       "L 126.723904 20.249806 \n",
       "L 127.048883 21.438534 \n",
       "L 127.698842 25.216204 \n",
       "L 128.348801 30.80812 \n",
       "L 128.99876 38.126094 \n",
       "L 129.973699 52.079955 \n",
       "L 130.948638 69.157496 \n",
       "L 132.248556 95.727654 \n",
       "L 134.523412 147.851141 \n",
       "L 136.798269 198.528924 \n",
       "L 138.098187 223.058818 \n",
       "L 139.073126 238.114696 \n",
       "L 140.048064 249.70388 \n",
       "L 140.698023 255.295796 \n",
       "L 141.347982 259.073466 \n",
       "L 141.672962 260.262194 \n",
       "L 141.997941 260.977313 \n",
       "L 142.322921 261.216 \n",
       "L 142.647901 260.977313 \n",
       "L 142.97288 260.262194 \n",
       "L 143.29786 259.073466 \n",
       "L 143.947819 255.295796 \n",
       "L 144.597778 249.70388 \n",
       "L 145.247737 242.385906 \n",
       "L 146.222675 228.432045 \n",
       "L 147.197614 211.354504 \n",
       "L 148.497532 184.784346 \n",
       "L 150.772389 132.660859 \n",
       "L 153.047245 81.983076 \n",
       "L 154.347163 57.453182 \n",
       "L 155.322102 42.397304 \n",
       "L 156.297041 30.80812 \n",
       "L 156.947 25.216204 \n",
       "L 157.596959 21.438534 \n",
       "L 157.921938 20.249806 \n",
       "L 158.246918 19.534687 \n",
       "L 158.571897 19.296 \n",
       "L 158.896877 19.534687 \n",
       "L 159.221856 20.249806 \n",
       "L 159.546836 21.438534 \n",
       "L 160.196795 25.216204 \n",
       "L 160.846754 30.80812 \n",
       "L 161.496713 38.126094 \n",
       "L 162.471652 52.079955 \n",
       "L 163.44659 69.157496 \n",
       "L 164.746508 95.727654 \n",
       "L 167.021365 147.851141 \n",
       "L 169.296222 198.528924 \n",
       "L 170.59614 223.058818 \n",
       "L 171.571078 238.114696 \n",
       "L 172.546017 249.70388 \n",
       "L 173.195976 255.295796 \n",
       "L 173.845935 259.073466 \n",
       "L 174.170914 260.262194 \n",
       "L 174.495894 260.977313 \n",
       "L 174.820874 261.216 \n",
       "L 175.145853 260.977313 \n",
       "L 175.470833 260.262194 \n",
       "L 175.795812 259.073466 \n",
       "L 176.445771 255.295796 \n",
       "L 177.09573 249.70388 \n",
       "L 177.745689 242.385906 \n",
       "L 178.720628 228.432045 \n",
       "L 179.695566 211.354504 \n",
       "L 180.995484 184.784346 \n",
       "L 183.270341 132.660859 \n",
       "L 185.545198 81.983076 \n",
       "L 186.845116 57.453182 \n",
       "L 187.820055 42.397304 \n",
       "L 188.794993 30.80812 \n",
       "L 189.444952 25.216204 \n",
       "L 190.094911 21.438534 \n",
       "L 190.419891 20.249806 \n",
       "L 190.74487 19.534687 \n",
       "L 191.06985 19.296 \n",
       "L 191.394829 19.534687 \n",
       "L 191.719809 20.249806 \n",
       "L 192.044788 21.438534 \n",
       "L 192.694747 25.216204 \n",
       "L 193.344706 30.80812 \n",
       "L 193.994665 38.126094 \n",
       "L 194.969604 52.079955 \n",
       "L 195.944543 69.157496 \n",
       "L 197.244461 95.727654 \n",
       "L 199.519317 147.851141 \n",
       "L 201.794174 198.528924 \n",
       "L 203.094092 223.058818 \n",
       "L 204.069031 238.114696 \n",
       "L 205.043969 249.70388 \n",
       "L 205.693928 255.295796 \n",
       "L 206.343887 259.073466 \n",
       "L 206.668867 260.262194 \n",
       "L 206.993846 260.977313 \n",
       "L 207.318826 261.216 \n",
       "L 207.643806 260.977313 \n",
       "L 207.968785 260.262194 \n",
       "L 208.293765 259.073466 \n",
       "L 208.943724 255.295796 \n",
       "L 209.593683 249.70388 \n",
       "L 210.243642 242.385906 \n",
       "L 211.21858 228.432045 \n",
       "L 212.193519 211.354504 \n",
       "L 213.493437 184.784346 \n",
       "L 215.768294 132.660859 \n",
       "L 218.04315 81.983076 \n",
       "L 219.343068 57.453182 \n",
       "L 220.318007 42.397304 \n",
       "L 221.292946 30.80812 \n",
       "L 221.942905 25.216204 \n",
       "L 222.592864 21.438534 \n",
       "L 222.917843 20.249806 \n",
       "L 223.242823 19.534687 \n",
       "L 223.567802 19.296 \n",
       "L 223.892782 19.534687 \n",
       "L 224.217761 20.249806 \n",
       "L 224.542741 21.438534 \n",
       "L 225.1927 25.216204 \n",
       "L 225.842659 30.80812 \n",
       "L 226.492618 38.126094 \n",
       "L 227.467557 52.079955 \n",
       "L 228.442495 69.157496 \n",
       "L 229.742413 95.727654 \n",
       "L 232.01727 147.851141 \n",
       "L 234.292127 198.528924 \n",
       "L 235.592045 223.058818 \n",
       "L 236.566983 238.114696 \n",
       "L 237.541922 249.70388 \n",
       "L 238.191881 255.295796 \n",
       "L 238.84184 259.073466 \n",
       "L 239.166819 260.262194 \n",
       "L 239.491799 260.977313 \n",
       "L 239.816779 261.216 \n",
       "L 240.141758 260.977313 \n",
       "L 240.466738 260.262194 \n",
       "L 240.791717 259.073466 \n",
       "L 241.441676 255.295796 \n",
       "L 242.091635 249.70388 \n",
       "L 242.741594 242.385906 \n",
       "L 243.716533 228.432045 \n",
       "L 244.691471 211.354504 \n",
       "L 245.991389 184.784346 \n",
       "L 248.266246 132.660859 \n",
       "L 250.541103 81.983076 \n",
       "L 251.841021 57.453182 \n",
       "L 252.81596 42.397304 \n",
       "L 253.790898 30.80812 \n",
       "L 254.440857 25.216204 \n",
       "L 255.090816 21.438534 \n",
       "L 255.415796 20.249806 \n",
       "L 255.740775 19.534687 \n",
       "L 256.065755 19.296 \n",
       "L 256.390734 19.534687 \n",
       "L 256.715714 20.249806 \n",
       "L 257.040693 21.438534 \n",
       "L 257.690652 25.216204 \n",
       "L 258.340611 30.80812 \n",
       "L 258.99057 38.126094 \n",
       "L 259.965509 52.079955 \n",
       "L 260.940448 69.157496 \n",
       "L 262.240366 95.727654 \n",
       "L 264.515222 147.851141 \n",
       "L 266.790079 198.528924 \n",
       "L 268.089997 223.058818 \n",
       "L 269.064936 238.114696 \n",
       "L 270.039874 249.70388 \n",
       "L 270.689833 255.295796 \n",
       "L 271.339792 259.073466 \n",
       "L 271.664772 260.262194 \n",
       "L 271.989751 260.977313 \n",
       "L 272.314731 261.216 \n",
       "L 272.639711 260.977313 \n",
       "L 272.96469 260.262194 \n",
       "L 273.28967 259.073466 \n",
       "L 273.939629 255.295796 \n",
       "L 274.589588 249.70388 \n",
       "L 275.239547 242.385906 \n",
       "L 276.214485 228.432045 \n",
       "L 277.189424 211.354504 \n",
       "L 278.489342 184.784346 \n",
       "L 280.764199 132.660859 \n",
       "L 283.039055 81.983076 \n",
       "L 284.338973 57.453182 \n",
       "L 285.313912 42.397304 \n",
       "L 286.288851 30.80812 \n",
       "L 286.93881 25.216204 \n",
       "L 287.588769 21.438534 \n",
       "L 287.913748 20.249806 \n",
       "L 288.238728 19.534687 \n",
       "L 288.563707 19.296 \n",
       "L 288.888687 19.534687 \n",
       "L 289.213666 20.249806 \n",
       "L 289.538646 21.438534 \n",
       "L 290.188605 25.216204 \n",
       "L 290.838564 30.80812 \n",
       "L 291.488523 38.126094 \n",
       "L 292.463462 52.079955 \n",
       "L 293.4384 69.157496 \n",
       "L 294.738318 95.727654 \n",
       "L 297.013175 147.851141 \n",
       "L 299.288032 198.528924 \n",
       "L 300.58795 223.058818 \n",
       "L 301.562888 238.114696 \n",
       "L 302.537827 249.70388 \n",
       "L 303.187786 255.295796 \n",
       "L 303.837745 259.073466 \n",
       "L 304.162724 260.262194 \n",
       "L 304.487704 260.977313 \n",
       "L 304.812684 261.216 \n",
       "L 305.137663 260.977313 \n",
       "L 305.462643 260.262194 \n",
       "L 305.787622 259.073466 \n",
       "L 306.437581 255.295796 \n",
       "L 307.08754 249.70388 \n",
       "L 307.737499 242.385906 \n",
       "L 308.712438 228.432045 \n",
       "L 309.687376 211.354504 \n",
       "L 310.987294 184.784346 \n",
       "L 313.262151 132.660859 \n",
       "L 315.537008 81.983076 \n",
       "L 316.836926 57.453182 \n",
       "L 317.811865 42.397304 \n",
       "L 318.786803 30.80812 \n",
       "L 319.436762 25.216204 \n",
       "L 320.086721 21.438534 \n",
       "L 320.411701 20.249806 \n",
       "L 320.73668 19.534687 \n",
       "L 321.06166 19.296 \n",
       "L 321.386639 19.534687 \n",
       "L 321.711619 20.249806 \n",
       "L 322.036598 21.438534 \n",
       "L 322.686557 25.216204 \n",
       "L 323.336516 30.80812 \n",
       "L 323.986475 38.126094 \n",
       "L 324.961414 52.079955 \n",
       "L 325.936353 69.157496 \n",
       "L 327.236271 95.727654 \n",
       "L 329.511127 147.851141 \n",
       "L 331.785984 198.528924 \n",
       "L 333.085902 223.058818 \n",
       "L 334.060841 238.114696 \n",
       "L 335.035779 249.70388 \n",
       "L 335.685738 255.295796 \n",
       "L 336.335697 259.073466 \n",
       "L 336.660677 260.262194 \n",
       "L 336.985656 260.977313 \n",
       "L 337.310636 261.216 \n",
       "L 337.635616 260.977313 \n",
       "L 337.960595 260.262194 \n",
       "L 338.285575 259.073466 \n",
       "L 338.935534 255.295796 \n",
       "L 339.585493 249.70388 \n",
       "L 340.235452 242.385906 \n",
       "L 341.21039 228.432045 \n",
       "L 342.185329 211.354504 \n",
       "L 343.485247 184.784346 \n",
       "L 345.760104 132.660859 \n",
       "L 348.03496 81.983076 \n",
       "L 349.334878 57.453182 \n",
       "L 350.309817 42.397304 \n",
       "L 351.284756 30.80812 \n",
       "L 351.934715 25.216204 \n",
       "L 352.584674 21.438534 \n",
       "L 352.909653 20.249806 \n",
       "L 353.234633 19.534687 \n",
       "L 353.559612 19.296 \n",
       "L 353.884592 19.534687 \n",
       "L 354.209571 20.249806 \n",
       "L 354.534551 21.438534 \n",
       "L 355.18451 25.216204 \n",
       "L 355.834469 30.80812 \n",
       "L 356.484428 38.126094 \n",
       "L 357.459367 52.079955 \n",
       "L 358.434305 69.157496 \n",
       "L 359.734223 95.727654 \n",
       "L 362.00908 147.851141 \n",
       "L 364.283937 198.528924 \n",
       "L 365.583855 223.058818 \n",
       "L 366.558793 238.114696 \n",
       "L 367.533732 249.70388 \n",
       "L 368.183691 255.295796 \n",
       "L 368.83365 259.073466 \n",
       "L 369.158629 260.262194 \n",
       "L 369.483609 260.977313 \n",
       "L 369.808589 261.216 \n",
       "L 370.133568 260.977313 \n",
       "L 370.458548 260.262194 \n",
       "L 370.783527 259.073466 \n",
       "L 371.433486 255.295796 \n",
       "L 372.083445 249.70388 \n",
       "L 372.733404 242.385906 \n",
       "L 373.708343 228.432045 \n",
       "L 374.683281 211.354504 \n",
       "L 375.983199 184.784346 \n",
       "L 378.258056 132.660859 \n",
       "L 380.532913 81.983076 \n",
       "L 381.832831 57.453182 \n",
       "L 382.80777 42.397304 \n",
       "L 383.782708 30.80812 \n",
       "L 384.432667 25.216204 \n",
       "L 385.082626 21.438534 \n",
       "L 385.407606 20.249806 \n",
       "L 385.732585 19.534687 \n",
       "L 385.732585 19.534687 \n",
       "\" clip-path=\"url(#pd95c027828)\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 44.845313 273.312 \n",
       "L 44.845313 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 401.965312 273.312 \n",
       "L 401.965312 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 44.845313 273.312 \n",
       "L 401.965312 273.312 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 44.845313 7.2 \n",
       "L 401.965312 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pd95c027828\">\n",
       "   <rect x=\"44.845313\" y=\"7.2\" width=\"357.12\" height=\"266.112\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_waveform(FlatWaveform(duration=1e-6, iq=1.0, detuning=1e7), sample_rate=1e9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### gaussian\n",
    "\n",
    "Several of the template waveforms provided are derived from a standard (unnormalized) Gaussian. Here we have\n",
    "\\begin{equation}\n",
    "u(t) = \\text{exp}\\Big(-\\frac{(t-t_0)^2}{2 \\sigma^2}\\Big),\n",
    "\\end{equation}\n",
    "where $t_0$ denotes the center of the Gaussian, and $\\sigma$ is the usual standard deviation. \n",
    "\n",
    "By Quil-T convention, this is parameterized by the Gaussian's [full width at half maximum](https://en.wikipedia.org/wiki/Full_width_at_half_maximum) (FWHM), which is defined to be\n",
    "\n",
    "\\begin{equation}\n",
    "\\text{FWHM} = 2 \\sqrt{2 \\ln(2)} \\sigma.\n",
    "\\end{equation}\n",
    "\n",
    "As with all Quil-T waveforms, a Quil-T `gaussian` has a finite duration, and thus corresponds to a truncation of a true Gaussian.\n",
    "\n",
    "In short, the parameters are:\n",
    "\n",
    "* `duration`: the duration of the waveform, in seconds. The Gaussian will be truncated to $[0, \\text{duration}]$\n",
    "* `t0`: the center of the Gaussian, in seconds\n",
    "* `fwhm`: the full width half maximum of the Gaussian, in seconds"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gaussian(duration: 1e-6, fwhm: 4e-7, t0: 5e-7)\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"394.423125pt\" height=\"309.86825pt\" viewBox=\"0 0 394.423125 309.86825\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2023-09-15T14:32:52.107436</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.7.2, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 309.86825 \n",
       "L 394.423125 309.86825 \n",
       "L 394.423125 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 387.223125 273.312 \n",
       "L 387.223125 7.2 \n",
       "L 30.103125 7.2 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path id=\"m247b34cf56\" d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m247b34cf56\" x=\"46.335852\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(38.38429 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
       "Q 1547 4250 1301 3770 \n",
       "Q 1056 3291 1056 2328 \n",
       "Q 1056 1369 1301 889 \n",
       "Q 1547 409 2034 409 \n",
       "Q 2525 409 2770 889 \n",
       "Q 3016 1369 3016 2328 \n",
       "Q 3016 3291 2770 3770 \n",
       "Q 2525 4250 2034 4250 \n",
       "z\n",
       "M 2034 4750 \n",
       "Q 2819 4750 3233 4129 \n",
       "Q 3647 3509 3647 2328 \n",
       "Q 3647 1150 3233 529 \n",
       "Q 2819 -91 2034 -91 \n",
       "Q 1250 -91 836 529 \n",
       "Q 422 1150 422 2328 \n",
       "Q 422 3509 836 4129 \n",
       "Q 1250 4750 2034 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-2e\" d=\"M 684 794 \n",
       "L 1344 794 \n",
       "L 1344 0 \n",
       "L 684 0 \n",
       "L 684 794 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m247b34cf56\" x=\"111.331757\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(103.380195 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-32\" d=\"M 1228 531 \n",
       "L 3431 531 \n",
       "L 3431 0 \n",
       "L 469 0 \n",
       "L 469 531 \n",
       "Q 828 903 1448 1529 \n",
       "Q 2069 2156 2228 2338 \n",
       "Q 2531 2678 2651 2914 \n",
       "Q 2772 3150 2772 3378 \n",
       "Q 2772 3750 2511 3984 \n",
       "Q 2250 4219 1831 4219 \n",
       "Q 1534 4219 1204 4116 \n",
       "Q 875 4013 500 3803 \n",
       "L 500 4441 \n",
       "Q 881 4594 1212 4672 \n",
       "Q 1544 4750 1819 4750 \n",
       "Q 2544 4750 2975 4387 \n",
       "Q 3406 4025 3406 3419 \n",
       "Q 3406 3131 3298 2873 \n",
       "Q 3191 2616 2906 2266 \n",
       "Q 2828 2175 2409 1742 \n",
       "Q 1991 1309 1228 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m247b34cf56\" x=\"176.327662\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(168.3761 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
       "L 825 1625 \n",
       "L 2419 1625 \n",
       "L 2419 4116 \n",
       "z\n",
       "M 2253 4666 \n",
       "L 3047 4666 \n",
       "L 3047 1625 \n",
       "L 3713 1625 \n",
       "L 3713 1100 \n",
       "L 3047 1100 \n",
       "L 3047 0 \n",
       "L 2419 0 \n",
       "L 2419 1100 \n",
       "L 313 1100 \n",
       "L 313 1709 \n",
       "L 2253 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m247b34cf56\" x=\"241.323567\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(233.372005 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
       "Q 1688 2584 1439 2293 \n",
       "Q 1191 2003 1191 1497 \n",
       "Q 1191 994 1439 701 \n",
       "Q 1688 409 2113 409 \n",
       "Q 2538 409 2786 701 \n",
       "Q 3034 994 3034 1497 \n",
       "Q 3034 2003 2786 2293 \n",
       "Q 2538 2584 2113 2584 \n",
       "z\n",
       "M 3366 4563 \n",
       "L 3366 3988 \n",
       "Q 3128 4100 2886 4159 \n",
       "Q 2644 4219 2406 4219 \n",
       "Q 1781 4219 1451 3797 \n",
       "Q 1122 3375 1075 2522 \n",
       "Q 1259 2794 1537 2939 \n",
       "Q 1816 3084 2150 3084 \n",
       "Q 2853 3084 3261 2657 \n",
       "Q 3669 2231 3669 1497 \n",
       "Q 3669 778 3244 343 \n",
       "Q 2819 -91 2113 -91 \n",
       "Q 1303 -91 875 529 \n",
       "Q 447 1150 447 2328 \n",
       "Q 447 3434 972 4092 \n",
       "Q 1497 4750 2381 4750 \n",
       "Q 2619 4750 2861 4703 \n",
       "Q 3103 4656 3366 4563 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m247b34cf56\" x=\"306.319472\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(298.36791 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \n",
       "Q 1584 2216 1326 1975 \n",
       "Q 1069 1734 1069 1313 \n",
       "Q 1069 891 1326 650 \n",
       "Q 1584 409 2034 409 \n",
       "Q 2484 409 2743 651 \n",
       "Q 3003 894 3003 1313 \n",
       "Q 3003 1734 2745 1975 \n",
       "Q 2488 2216 2034 2216 \n",
       "z\n",
       "M 1403 2484 \n",
       "Q 997 2584 770 2862 \n",
       "Q 544 3141 544 3541 \n",
       "Q 544 4100 942 4425 \n",
       "Q 1341 4750 2034 4750 \n",
       "Q 2731 4750 3128 4425 \n",
       "Q 3525 4100 3525 3541 \n",
       "Q 3525 3141 3298 2862 \n",
       "Q 3072 2584 2669 2484 \n",
       "Q 3125 2378 3379 2068 \n",
       "Q 3634 1759 3634 1313 \n",
       "Q 3634 634 3220 271 \n",
       "Q 2806 -91 2034 -91 \n",
       "Q 1263 -91 848 271 \n",
       "Q 434 634 434 1313 \n",
       "Q 434 1759 690 2068 \n",
       "Q 947 2378 1403 2484 \n",
       "z\n",
       "M 1172 3481 \n",
       "Q 1172 3119 1398 2916 \n",
       "Q 1625 2713 2034 2713 \n",
       "Q 2441 2713 2670 2916 \n",
       "Q 2900 3119 2900 3481 \n",
       "Q 2900 3844 2670 4047 \n",
       "Q 2441 4250 2034 4250 \n",
       "Q 1625 4250 1398 4047 \n",
       "Q 1172 3844 1172 3481 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m247b34cf56\" x=\"371.315377\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(363.363815 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-31\" d=\"M 794 531 \n",
       "L 1825 531 \n",
       "L 1825 4091 \n",
       "L 703 3866 \n",
       "L 703 4441 \n",
       "L 1819 4666 \n",
       "L 2450 4666 \n",
       "L 2450 531 \n",
       "L 3481 531 \n",
       "L 3481 0 \n",
       "L 794 0 \n",
       "L 794 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- 1e−6 -->\n",
       "     <g transform=\"translate(359.965312 300.588562) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-65\" d=\"M 3597 1894 \n",
       "L 3597 1613 \n",
       "L 953 1613 \n",
       "Q 991 1019 1311 708 \n",
       "Q 1631 397 2203 397 \n",
       "Q 2534 397 2845 478 \n",
       "Q 3156 559 3463 722 \n",
       "L 3463 178 \n",
       "Q 3153 47 2828 -22 \n",
       "Q 2503 -91 2169 -91 \n",
       "Q 1331 -91 842 396 \n",
       "Q 353 884 353 1716 \n",
       "Q 353 2575 817 3079 \n",
       "Q 1281 3584 2069 3584 \n",
       "Q 2775 3584 3186 3129 \n",
       "Q 3597 2675 3597 1894 \n",
       "z\n",
       "M 3022 2063 \n",
       "Q 3016 2534 2758 2815 \n",
       "Q 2500 3097 2075 3097 \n",
       "Q 1594 3097 1305 2825 \n",
       "Q 1016 2553 972 2059 \n",
       "L 3022 2063 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2212\" d=\"M 678 2272 \n",
       "L 4684 2272 \n",
       "L 4684 1741 \n",
       "L 678 1741 \n",
       "L 678 2272 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-65\" x=\"63.623047\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2212\" x=\"125.146484\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-36\" x=\"208.935547\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <defs>\n",
       "       <path id=\"m9aabff2a8b\" d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m9aabff2a8b\" x=\"30.103125\" y=\"264.436908\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(7.2 268.236127) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m9aabff2a8b\" x=\"30.103125\" y=\"215.408726\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(7.2 219.207945) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m9aabff2a8b\" x=\"30.103125\" y=\"166.380545\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(7.2 170.179764) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m9aabff2a8b\" x=\"30.103125\" y=\"117.352363\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(7.2 121.151582) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m9aabff2a8b\" x=\"30.103125\" y=\"68.324182\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(7.2 72.1234) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m9aabff2a8b\" x=\"30.103125\" y=\"19.296\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(7.2 23.095219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_13\">\n",
       "    <path d=\"M 46.335852 261.216 \n",
       "L 51.535525 260.205706 \n",
       "L 56.085238 259.103834 \n",
       "L 60.309972 257.865318 \n",
       "L 64.534706 256.386469 \n",
       "L 68.43446 254.777951 \n",
       "L 72.009235 253.072564 \n",
       "L 75.58401 251.122029 \n",
       "L 79.158784 248.901993 \n",
       "L 82.40858 246.629046 \n",
       "L 85.658375 244.094163 \n",
       "L 88.90817 241.278849 \n",
       "L 92.157965 238.165121 \n",
       "L 95.407761 234.735849 \n",
       "L 98.657556 230.975124 \n",
       "L 101.907351 226.868647 \n",
       "L 105.157146 222.404143 \n",
       "L 108.406942 217.571795 \n",
       "L 111.656737 212.364672 \n",
       "L 115.231512 206.199774 \n",
       "L 118.806286 199.577568 \n",
       "L 122.381061 192.504627 \n",
       "L 126.280815 184.290417 \n",
       "L 130.505549 174.835938 \n",
       "L 135.055263 164.062863 \n",
       "L 139.929955 151.930182 \n",
       "L 145.779587 136.741092 \n",
       "L 154.879014 112.369471 \n",
       "L 163.653461 89.044506 \n",
       "L 168.853133 75.829511 \n",
       "L 173.077867 65.665361 \n",
       "L 176.652642 57.586722 \n",
       "L 179.902437 50.737005 \n",
       "L 182.827253 45.029649 \n",
       "L 185.752068 39.801359 \n",
       "L 188.351905 35.589452 \n",
       "L 190.951741 31.815116 \n",
       "L 193.226598 28.890513 \n",
       "L 195.501454 26.333833 \n",
       "L 197.451331 24.444918 \n",
       "L 199.401209 22.842635 \n",
       "L 201.351086 21.532907 \n",
       "L 203.300963 20.520594 \n",
       "L 204.92586 19.906945 \n",
       "L 206.550758 19.504062 \n",
       "L 208.175656 19.312991 \n",
       "L 209.800553 19.334229 \n",
       "L 211.425451 19.567719 \n",
       "L 213.050349 20.012856 \n",
       "L 214.675246 20.668485 \n",
       "L 216.300144 21.532907 \n",
       "L 218.250021 22.842635 \n",
       "L 220.199898 24.444918 \n",
       "L 222.149775 26.333833 \n",
       "L 224.424632 28.890513 \n",
       "L 226.699489 31.815116 \n",
       "L 228.974345 35.093212 \n",
       "L 231.574182 39.251826 \n",
       "L 234.174018 43.82491 \n",
       "L 237.098833 49.429517 \n",
       "L 240.348629 56.176838 \n",
       "L 243.923403 64.157894 \n",
       "L 247.823158 73.431795 \n",
       "L 252.372871 84.846381 \n",
       "L 258.222503 100.173486 \n",
       "L 278.371233 153.579033 \n",
       "L 283.570906 166.419408 \n",
       "L 288.120619 177.066826 \n",
       "L 292.345353 186.391468 \n",
       "L 296.245107 194.477681 \n",
       "L 300.144861 202.038278 \n",
       "L 303.719636 208.494617 \n",
       "L 307.294411 214.492805 \n",
       "L 310.869186 220.034456 \n",
       "L 314.118981 224.681825 \n",
       "L 317.368776 228.965924 \n",
       "L 320.618571 232.897855 \n",
       "L 323.868367 236.490955 \n",
       "L 327.118162 239.760381 \n",
       "L 330.367957 242.7227 \n",
       "L 333.617752 245.395511 \n",
       "L 336.867548 247.797093 \n",
       "L 340.442322 250.147815 \n",
       "L 344.017097 252.217797 \n",
       "L 347.591872 254.031638 \n",
       "L 351.491626 255.746388 \n",
       "L 355.391381 257.214688 \n",
       "L 359.616114 258.559959 \n",
       "L 364.165828 259.760608 \n",
       "L 369.040521 260.803704 \n",
       "L 370.990398 261.159756 \n",
       "L 370.990398 261.159756 \n",
       "\" clip-path=\"url(#pb3422c3a43)\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 30.103125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 387.223125 273.312 \n",
       "L 387.223125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 387.223125 273.312 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 30.103125 7.2 \n",
       "L 387.223125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pb3422c3a43\">\n",
       "   <rect x=\"30.103125\" y=\"7.2\" width=\"357.12\" height=\"266.112\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pyquil.quiltwaveforms import GaussianWaveform\n",
    "\n",
    "plot_waveform(GaussianWaveform(duration=1e-6, t0=5e-7, fwhm=4e-7), sample_rate=1e9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### drag_gaussian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `drag_gaussian` waveform extends the basic Gaussian with an additional correction factor (cf. https://arxiv.org/abs/1809.04919 and references therein). The shape is given by\n",
    "\\begin{equation}\n",
    "u(t) = \\Big(1 + i\\frac{\\alpha}{2 \\pi \\eta \\sigma^2}\\Big)\\text{exp}\\Big(-\\frac{(t-t_0)^2}{2 \\sigma^2}\\Big),\n",
    "\\end{equation}\n",
    "where $\\eta$ is the anharmonicity constant, in Hz, and $\\alpha$ is a dimensionless shape parameter.\n",
    "\n",
    "As before, rather than providing $\\sigma$ explicity, `drag_gaussian` takes a full-width half max parameter. \n",
    "\n",
    "In summary, the required arguments to `drag_gaussian` are:\n",
    "\n",
    "* `duration`: the duration of the waveform, in seconds. The Gaussian will be truncated to $[0, \\text{duration}]$\n",
    "* `t0`: the center of the Gaussian, in seconds\n",
    "* `fwhm`: the full width half maximum of the Gaussian, in seconds\n",
    "* `anh`: the anharmonicity constant $\\eta$, in Hertz\n",
    "* `alpha`: dimensionless shape parameter $\\alpha$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "drag_gaussian(alpha: 1, anh: 1.1, duration: 1e-6, fwhm: 4e-7, t0: 5e-7)\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"394.423125pt\" height=\"309.86825pt\" viewBox=\"0 0 394.423125 309.86825\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2023-09-15T14:32:52.165426</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.7.2, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 309.86825 \n",
       "L 394.423125 309.86825 \n",
       "L 394.423125 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 387.223125 273.312 \n",
       "L 387.223125 7.2 \n",
       "L 30.103125 7.2 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path id=\"m8c5c6bc59d\" d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m8c5c6bc59d\" x=\"46.335852\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(38.38429 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
       "Q 1547 4250 1301 3770 \n",
       "Q 1056 3291 1056 2328 \n",
       "Q 1056 1369 1301 889 \n",
       "Q 1547 409 2034 409 \n",
       "Q 2525 409 2770 889 \n",
       "Q 3016 1369 3016 2328 \n",
       "Q 3016 3291 2770 3770 \n",
       "Q 2525 4250 2034 4250 \n",
       "z\n",
       "M 2034 4750 \n",
       "Q 2819 4750 3233 4129 \n",
       "Q 3647 3509 3647 2328 \n",
       "Q 3647 1150 3233 529 \n",
       "Q 2819 -91 2034 -91 \n",
       "Q 1250 -91 836 529 \n",
       "Q 422 1150 422 2328 \n",
       "Q 422 3509 836 4129 \n",
       "Q 1250 4750 2034 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-2e\" d=\"M 684 794 \n",
       "L 1344 794 \n",
       "L 1344 0 \n",
       "L 684 0 \n",
       "L 684 794 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m8c5c6bc59d\" x=\"111.331757\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(103.380195 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-32\" d=\"M 1228 531 \n",
       "L 3431 531 \n",
       "L 3431 0 \n",
       "L 469 0 \n",
       "L 469 531 \n",
       "Q 828 903 1448 1529 \n",
       "Q 2069 2156 2228 2338 \n",
       "Q 2531 2678 2651 2914 \n",
       "Q 2772 3150 2772 3378 \n",
       "Q 2772 3750 2511 3984 \n",
       "Q 2250 4219 1831 4219 \n",
       "Q 1534 4219 1204 4116 \n",
       "Q 875 4013 500 3803 \n",
       "L 500 4441 \n",
       "Q 881 4594 1212 4672 \n",
       "Q 1544 4750 1819 4750 \n",
       "Q 2544 4750 2975 4387 \n",
       "Q 3406 4025 3406 3419 \n",
       "Q 3406 3131 3298 2873 \n",
       "Q 3191 2616 2906 2266 \n",
       "Q 2828 2175 2409 1742 \n",
       "Q 1991 1309 1228 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m8c5c6bc59d\" x=\"176.327662\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(168.3761 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
       "L 825 1625 \n",
       "L 2419 1625 \n",
       "L 2419 4116 \n",
       "z\n",
       "M 2253 4666 \n",
       "L 3047 4666 \n",
       "L 3047 1625 \n",
       "L 3713 1625 \n",
       "L 3713 1100 \n",
       "L 3047 1100 \n",
       "L 3047 0 \n",
       "L 2419 0 \n",
       "L 2419 1100 \n",
       "L 313 1100 \n",
       "L 313 1709 \n",
       "L 2253 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m8c5c6bc59d\" x=\"241.323567\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(233.372005 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
       "Q 1688 2584 1439 2293 \n",
       "Q 1191 2003 1191 1497 \n",
       "Q 1191 994 1439 701 \n",
       "Q 1688 409 2113 409 \n",
       "Q 2538 409 2786 701 \n",
       "Q 3034 994 3034 1497 \n",
       "Q 3034 2003 2786 2293 \n",
       "Q 2538 2584 2113 2584 \n",
       "z\n",
       "M 3366 4563 \n",
       "L 3366 3988 \n",
       "Q 3128 4100 2886 4159 \n",
       "Q 2644 4219 2406 4219 \n",
       "Q 1781 4219 1451 3797 \n",
       "Q 1122 3375 1075 2522 \n",
       "Q 1259 2794 1537 2939 \n",
       "Q 1816 3084 2150 3084 \n",
       "Q 2853 3084 3261 2657 \n",
       "Q 3669 2231 3669 1497 \n",
       "Q 3669 778 3244 343 \n",
       "Q 2819 -91 2113 -91 \n",
       "Q 1303 -91 875 529 \n",
       "Q 447 1150 447 2328 \n",
       "Q 447 3434 972 4092 \n",
       "Q 1497 4750 2381 4750 \n",
       "Q 2619 4750 2861 4703 \n",
       "Q 3103 4656 3366 4563 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m8c5c6bc59d\" x=\"306.319472\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(298.36791 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \n",
       "Q 1584 2216 1326 1975 \n",
       "Q 1069 1734 1069 1313 \n",
       "Q 1069 891 1326 650 \n",
       "Q 1584 409 2034 409 \n",
       "Q 2484 409 2743 651 \n",
       "Q 3003 894 3003 1313 \n",
       "Q 3003 1734 2745 1975 \n",
       "Q 2488 2216 2034 2216 \n",
       "z\n",
       "M 1403 2484 \n",
       "Q 997 2584 770 2862 \n",
       "Q 544 3141 544 3541 \n",
       "Q 544 4100 942 4425 \n",
       "Q 1341 4750 2034 4750 \n",
       "Q 2731 4750 3128 4425 \n",
       "Q 3525 4100 3525 3541 \n",
       "Q 3525 3141 3298 2862 \n",
       "Q 3072 2584 2669 2484 \n",
       "Q 3125 2378 3379 2068 \n",
       "Q 3634 1759 3634 1313 \n",
       "Q 3634 634 3220 271 \n",
       "Q 2806 -91 2034 -91 \n",
       "Q 1263 -91 848 271 \n",
       "Q 434 634 434 1313 \n",
       "Q 434 1759 690 2068 \n",
       "Q 947 2378 1403 2484 \n",
       "z\n",
       "M 1172 3481 \n",
       "Q 1172 3119 1398 2916 \n",
       "Q 1625 2713 2034 2713 \n",
       "Q 2441 2713 2670 2916 \n",
       "Q 2900 3119 2900 3481 \n",
       "Q 2900 3844 2670 4047 \n",
       "Q 2441 4250 2034 4250 \n",
       "Q 1625 4250 1398 4047 \n",
       "Q 1172 3844 1172 3481 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m8c5c6bc59d\" x=\"371.315377\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(363.363815 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-31\" d=\"M 794 531 \n",
       "L 1825 531 \n",
       "L 1825 4091 \n",
       "L 703 3866 \n",
       "L 703 4441 \n",
       "L 1819 4666 \n",
       "L 2450 4666 \n",
       "L 2450 531 \n",
       "L 3481 531 \n",
       "L 3481 0 \n",
       "L 794 0 \n",
       "L 794 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- 1e−6 -->\n",
       "     <g transform=\"translate(359.965312 300.588562) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-65\" d=\"M 3597 1894 \n",
       "L 3597 1613 \n",
       "L 953 1613 \n",
       "Q 991 1019 1311 708 \n",
       "Q 1631 397 2203 397 \n",
       "Q 2534 397 2845 478 \n",
       "Q 3156 559 3463 722 \n",
       "L 3463 178 \n",
       "Q 3153 47 2828 -22 \n",
       "Q 2503 -91 2169 -91 \n",
       "Q 1331 -91 842 396 \n",
       "Q 353 884 353 1716 \n",
       "Q 353 2575 817 3079 \n",
       "Q 1281 3584 2069 3584 \n",
       "Q 2775 3584 3186 3129 \n",
       "Q 3597 2675 3597 1894 \n",
       "z\n",
       "M 3022 2063 \n",
       "Q 3016 2534 2758 2815 \n",
       "Q 2500 3097 2075 3097 \n",
       "Q 1594 3097 1305 2825 \n",
       "Q 1016 2553 972 2059 \n",
       "L 3022 2063 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2212\" d=\"M 678 2272 \n",
       "L 4684 2272 \n",
       "L 4684 1741 \n",
       "L 678 1741 \n",
       "L 678 2272 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-65\" x=\"63.623047\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2212\" x=\"125.146484\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-36\" x=\"208.935547\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <defs>\n",
       "       <path id=\"m702dbb169d\" d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m702dbb169d\" x=\"30.103125\" y=\"264.436908\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(7.2 268.236127) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m702dbb169d\" x=\"30.103125\" y=\"215.408726\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(7.2 219.207945) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m702dbb169d\" x=\"30.103125\" y=\"166.380545\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(7.2 170.179764) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m702dbb169d\" x=\"30.103125\" y=\"117.352363\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(7.2 121.151582) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m702dbb169d\" x=\"30.103125\" y=\"68.324182\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(7.2 72.1234) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m702dbb169d\" x=\"30.103125\" y=\"19.296\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(7.2 23.095219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_13\">\n",
       "    <path d=\"M 46.335852 261.216 \n",
       "L 51.535525 260.205706 \n",
       "L 56.085238 259.103834 \n",
       "L 60.309972 257.865318 \n",
       "L 64.534706 256.386469 \n",
       "L 68.43446 254.777951 \n",
       "L 72.009235 253.072564 \n",
       "L 75.58401 251.122029 \n",
       "L 79.158784 248.901993 \n",
       "L 82.40858 246.629046 \n",
       "L 85.658375 244.094163 \n",
       "L 88.90817 241.278849 \n",
       "L 92.157965 238.165121 \n",
       "L 95.407761 234.735849 \n",
       "L 98.657556 230.975124 \n",
       "L 101.907351 226.868647 \n",
       "L 105.157146 222.404143 \n",
       "L 108.406942 217.571795 \n",
       "L 111.656737 212.364672 \n",
       "L 115.231512 206.199774 \n",
       "L 118.806286 199.577568 \n",
       "L 122.381061 192.504627 \n",
       "L 126.280815 184.290417 \n",
       "L 130.505549 174.835938 \n",
       "L 135.055263 164.062863 \n",
       "L 139.929955 151.930182 \n",
       "L 145.779587 136.741092 \n",
       "L 154.879014 112.369471 \n",
       "L 163.653461 89.044506 \n",
       "L 168.853133 75.829511 \n",
       "L 173.077867 65.665361 \n",
       "L 176.652642 57.586722 \n",
       "L 179.902437 50.737005 \n",
       "L 182.827253 45.029649 \n",
       "L 185.752068 39.801359 \n",
       "L 188.351905 35.589452 \n",
       "L 190.951741 31.815116 \n",
       "L 193.226598 28.890513 \n",
       "L 195.501454 26.333833 \n",
       "L 197.451331 24.444918 \n",
       "L 199.401209 22.842635 \n",
       "L 201.351086 21.532907 \n",
       "L 203.300963 20.520594 \n",
       "L 204.92586 19.906945 \n",
       "L 206.550758 19.504062 \n",
       "L 208.175656 19.312991 \n",
       "L 209.800553 19.334229 \n",
       "L 211.425451 19.567719 \n",
       "L 213.050349 20.012856 \n",
       "L 214.675246 20.668485 \n",
       "L 216.300144 21.532907 \n",
       "L 218.250021 22.842635 \n",
       "L 220.199898 24.444918 \n",
       "L 222.149775 26.333833 \n",
       "L 224.424632 28.890513 \n",
       "L 226.699489 31.815116 \n",
       "L 228.974345 35.093212 \n",
       "L 231.574182 39.251826 \n",
       "L 234.174018 43.82491 \n",
       "L 237.098833 49.429517 \n",
       "L 240.348629 56.176838 \n",
       "L 243.923403 64.157894 \n",
       "L 247.823158 73.431795 \n",
       "L 252.372871 84.846381 \n",
       "L 258.222503 100.173486 \n",
       "L 278.371233 153.579033 \n",
       "L 283.570906 166.419408 \n",
       "L 288.120619 177.066826 \n",
       "L 292.345353 186.391468 \n",
       "L 296.245107 194.477681 \n",
       "L 300.144861 202.038278 \n",
       "L 303.719636 208.494617 \n",
       "L 307.294411 214.492805 \n",
       "L 310.869186 220.034456 \n",
       "L 314.118981 224.681825 \n",
       "L 317.368776 228.965924 \n",
       "L 320.618571 232.897855 \n",
       "L 323.868367 236.490955 \n",
       "L 327.118162 239.760381 \n",
       "L 330.367957 242.7227 \n",
       "L 333.617752 245.395511 \n",
       "L 336.867548 247.797093 \n",
       "L 340.442322 250.147815 \n",
       "L 344.017097 252.217797 \n",
       "L 347.591872 254.031638 \n",
       "L 351.491626 255.746388 \n",
       "L 355.391381 257.214688 \n",
       "L 359.616114 258.559959 \n",
       "L 364.165828 259.760608 \n",
       "L 369.040521 260.803704 \n",
       "L 370.990398 261.159756 \n",
       "L 370.990398 261.159756 \n",
       "\" clip-path=\"url(#pae6f227ece)\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 30.103125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 387.223125 273.312 \n",
       "L 387.223125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 387.223125 273.312 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 30.103125 7.2 \n",
       "L 387.223125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pae6f227ece\">\n",
       "   <rect x=\"30.103125\" y=\"7.2\" width=\"357.12\" height=\"266.112\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pyquil.quiltwaveforms import DragGaussianWaveform\n",
    "\n",
    "plot_waveform(\n",
    "    DragGaussianWaveform(duration=1e-6, t0=5e-7, fwhm=4e-7, anh=1.1, alpha=1.0),\n",
    "    sample_rate=1e9\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, we only plotted the *real part* of the waveform above. The imaginary part is relevant when the baseband waveform is converted to a passband waveform, as we can demonstrate below via the optional `detuning` argument."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "drag_gaussian(alpha: 1, anh: 1.1, detuning: 10000000, duration: 1e-6, fwhm: 4e-7, t0: 5e-7)\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"425.074687pt\" height=\"309.86825pt\" viewBox=\"0 0 425.074687 309.86825\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2023-09-15T14:32:52.235035</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.7.2, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 309.86825 \n",
       "L 425.074687 309.86825 \n",
       "L 425.074687 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 60.754688 273.312 \n",
       "L 417.874688 273.312 \n",
       "L 417.874688 7.2 \n",
       "L 60.754688 7.2 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path id=\"m432b39d8cf\" d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m432b39d8cf\" x=\"76.987415\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(69.035852 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
       "Q 1547 4250 1301 3770 \n",
       "Q 1056 3291 1056 2328 \n",
       "Q 1056 1369 1301 889 \n",
       "Q 1547 409 2034 409 \n",
       "Q 2525 409 2770 889 \n",
       "Q 3016 1369 3016 2328 \n",
       "Q 3016 3291 2770 3770 \n",
       "Q 2525 4250 2034 4250 \n",
       "z\n",
       "M 2034 4750 \n",
       "Q 2819 4750 3233 4129 \n",
       "Q 3647 3509 3647 2328 \n",
       "Q 3647 1150 3233 529 \n",
       "Q 2819 -91 2034 -91 \n",
       "Q 1250 -91 836 529 \n",
       "Q 422 1150 422 2328 \n",
       "Q 422 3509 836 4129 \n",
       "Q 1250 4750 2034 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-2e\" d=\"M 684 794 \n",
       "L 1344 794 \n",
       "L 1344 0 \n",
       "L 684 0 \n",
       "L 684 794 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m432b39d8cf\" x=\"141.98332\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(134.031757 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-32\" d=\"M 1228 531 \n",
       "L 3431 531 \n",
       "L 3431 0 \n",
       "L 469 0 \n",
       "L 469 531 \n",
       "Q 828 903 1448 1529 \n",
       "Q 2069 2156 2228 2338 \n",
       "Q 2531 2678 2651 2914 \n",
       "Q 2772 3150 2772 3378 \n",
       "Q 2772 3750 2511 3984 \n",
       "Q 2250 4219 1831 4219 \n",
       "Q 1534 4219 1204 4116 \n",
       "Q 875 4013 500 3803 \n",
       "L 500 4441 \n",
       "Q 881 4594 1212 4672 \n",
       "Q 1544 4750 1819 4750 \n",
       "Q 2544 4750 2975 4387 \n",
       "Q 3406 4025 3406 3419 \n",
       "Q 3406 3131 3298 2873 \n",
       "Q 3191 2616 2906 2266 \n",
       "Q 2828 2175 2409 1742 \n",
       "Q 1991 1309 1228 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m432b39d8cf\" x=\"206.979225\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(199.027662 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
       "L 825 1625 \n",
       "L 2419 1625 \n",
       "L 2419 4116 \n",
       "z\n",
       "M 2253 4666 \n",
       "L 3047 4666 \n",
       "L 3047 1625 \n",
       "L 3713 1625 \n",
       "L 3713 1100 \n",
       "L 3047 1100 \n",
       "L 3047 0 \n",
       "L 2419 0 \n",
       "L 2419 1100 \n",
       "L 313 1100 \n",
       "L 313 1709 \n",
       "L 2253 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m432b39d8cf\" x=\"271.97513\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(264.023567 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
       "Q 1688 2584 1439 2293 \n",
       "Q 1191 2003 1191 1497 \n",
       "Q 1191 994 1439 701 \n",
       "Q 1688 409 2113 409 \n",
       "Q 2538 409 2786 701 \n",
       "Q 3034 994 3034 1497 \n",
       "Q 3034 2003 2786 2293 \n",
       "Q 2538 2584 2113 2584 \n",
       "z\n",
       "M 3366 4563 \n",
       "L 3366 3988 \n",
       "Q 3128 4100 2886 4159 \n",
       "Q 2644 4219 2406 4219 \n",
       "Q 1781 4219 1451 3797 \n",
       "Q 1122 3375 1075 2522 \n",
       "Q 1259 2794 1537 2939 \n",
       "Q 1816 3084 2150 3084 \n",
       "Q 2853 3084 3261 2657 \n",
       "Q 3669 2231 3669 1497 \n",
       "Q 3669 778 3244 343 \n",
       "Q 2819 -91 2113 -91 \n",
       "Q 1303 -91 875 529 \n",
       "Q 447 1150 447 2328 \n",
       "Q 447 3434 972 4092 \n",
       "Q 1497 4750 2381 4750 \n",
       "Q 2619 4750 2861 4703 \n",
       "Q 3103 4656 3366 4563 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m432b39d8cf\" x=\"336.971035\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(329.019472 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \n",
       "Q 1584 2216 1326 1975 \n",
       "Q 1069 1734 1069 1313 \n",
       "Q 1069 891 1326 650 \n",
       "Q 1584 409 2034 409 \n",
       "Q 2484 409 2743 651 \n",
       "Q 3003 894 3003 1313 \n",
       "Q 3003 1734 2745 1975 \n",
       "Q 2488 2216 2034 2216 \n",
       "z\n",
       "M 1403 2484 \n",
       "Q 997 2584 770 2862 \n",
       "Q 544 3141 544 3541 \n",
       "Q 544 4100 942 4425 \n",
       "Q 1341 4750 2034 4750 \n",
       "Q 2731 4750 3128 4425 \n",
       "Q 3525 4100 3525 3541 \n",
       "Q 3525 3141 3298 2862 \n",
       "Q 3072 2584 2669 2484 \n",
       "Q 3125 2378 3379 2068 \n",
       "Q 3634 1759 3634 1313 \n",
       "Q 3634 634 3220 271 \n",
       "Q 2806 -91 2034 -91 \n",
       "Q 1263 -91 848 271 \n",
       "Q 434 634 434 1313 \n",
       "Q 434 1759 690 2068 \n",
       "Q 947 2378 1403 2484 \n",
       "z\n",
       "M 1172 3481 \n",
       "Q 1172 3119 1398 2916 \n",
       "Q 1625 2713 2034 2713 \n",
       "Q 2441 2713 2670 2916 \n",
       "Q 2900 3119 2900 3481 \n",
       "Q 2900 3844 2670 4047 \n",
       "Q 2441 4250 2034 4250 \n",
       "Q 1625 4250 1398 4047 \n",
       "Q 1172 3844 1172 3481 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m432b39d8cf\" x=\"401.96694\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(394.015377 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-31\" d=\"M 794 531 \n",
       "L 1825 531 \n",
       "L 1825 4091 \n",
       "L 703 3866 \n",
       "L 703 4441 \n",
       "L 1819 4666 \n",
       "L 2450 4666 \n",
       "L 2450 531 \n",
       "L 3481 531 \n",
       "L 3481 0 \n",
       "L 794 0 \n",
       "L 794 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- 1e−6 -->\n",
       "     <g transform=\"translate(390.616875 300.588562) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-65\" d=\"M 3597 1894 \n",
       "L 3597 1613 \n",
       "L 953 1613 \n",
       "Q 991 1019 1311 708 \n",
       "Q 1631 397 2203 397 \n",
       "Q 2534 397 2845 478 \n",
       "Q 3156 559 3463 722 \n",
       "L 3463 178 \n",
       "Q 3153 47 2828 -22 \n",
       "Q 2503 -91 2169 -91 \n",
       "Q 1331 -91 842 396 \n",
       "Q 353 884 353 1716 \n",
       "Q 353 2575 817 3079 \n",
       "Q 1281 3584 2069 3584 \n",
       "Q 2775 3584 3186 3129 \n",
       "Q 3597 2675 3597 1894 \n",
       "z\n",
       "M 3022 2063 \n",
       "Q 3016 2534 2758 2815 \n",
       "Q 2500 3097 2075 3097 \n",
       "Q 1594 3097 1305 2825 \n",
       "Q 1016 2553 972 2059 \n",
       "L 3022 2063 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2212\" d=\"M 678 2272 \n",
       "L 4684 2272 \n",
       "L 4684 1741 \n",
       "L 678 1741 \n",
       "L 678 2272 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-65\" x=\"63.623047\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2212\" x=\"125.146484\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-36\" x=\"208.935547\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <defs>\n",
       "       <path id=\"md8df0c3dbc\" d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#md8df0c3dbc\" x=\"60.754688\" y=\"242.030794\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- −400000 -->\n",
       "      <g transform=\"translate(7.2 245.830013) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-2212\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"83.789062\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"147.412109\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"211.035156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"274.658203\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"338.28125\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"401.904297\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#md8df0c3dbc\" x=\"60.754688\" y=\"193.407269\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- −200000 -->\n",
       "      <g transform=\"translate(7.2 197.206488) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-2212\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"83.789062\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"147.412109\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"211.035156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"274.658203\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"338.28125\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"401.904297\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#md8df0c3dbc\" x=\"60.754688\" y=\"144.783744\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 0 -->\n",
       "      <g transform=\"translate(47.392188 148.582963) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#md8df0c3dbc\" x=\"60.754688\" y=\"96.16022\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 200000 -->\n",
       "      <g transform=\"translate(15.579687 99.959438) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-32\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"127.246094\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"190.869141\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"254.492188\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"318.115234\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#md8df0c3dbc\" x=\"60.754688\" y=\"47.536695\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 400000 -->\n",
       "      <g transform=\"translate(15.579687 51.335914) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-34\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"127.246094\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"190.869141\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"254.492188\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"318.115234\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_12\">\n",
       "    <path d=\"M 76.987415 144.783741 \n",
       "L 78.937292 141.553706 \n",
       "L 81.862108 136.664898 \n",
       "L 83.162026 134.889149 \n",
       "L 84.136964 133.864506 \n",
       "L 85.111903 133.176326 \n",
       "L 85.761862 132.933321 \n",
       "L 86.411821 132.880174 \n",
       "L 87.06178 133.028837 \n",
       "L 87.711739 133.38836 \n",
       "L 88.361698 133.964559 \n",
       "L 89.336637 135.239132 \n",
       "L 90.311575 136.99722 \n",
       "L 91.286514 139.210294 \n",
       "L 92.586432 142.780649 \n",
       "L 94.21133 147.985304 \n",
       "L 98.111084 160.943187 \n",
       "L 99.411002 164.322839 \n",
       "L 100.385941 166.218804 \n",
       "L 101.0359 167.111659 \n",
       "L 101.685859 167.671234 \n",
       "L 102.335818 167.871123 \n",
       "L 102.985777 167.689922 \n",
       "L 103.635736 167.111856 \n",
       "L 104.285695 166.127338 \n",
       "L 104.935654 164.733437 \n",
       "L 105.910592 161.88595 \n",
       "L 106.885531 158.17304 \n",
       "L 108.185449 152.024022 \n",
       "L 109.810347 142.848201 \n",
       "L 114.36006 115.679682 \n",
       "L 115.334999 111.172788 \n",
       "L 116.309937 107.646532 \n",
       "L 116.959896 105.940517 \n",
       "L 117.609855 104.811177 \n",
       "L 117.934835 104.476666 \n",
       "L 118.259814 104.301929 \n",
       "L 118.584794 104.290984 \n",
       "L 118.909773 104.447229 \n",
       "L 119.234753 104.773411 \n",
       "L 119.884712 105.943184 \n",
       "L 120.534671 107.808412 \n",
       "L 121.18463 110.365486 \n",
       "L 122.159569 115.460541 \n",
       "L 123.134507 121.968939 \n",
       "L 124.434425 132.548774 \n",
       "L 126.059323 148.020567 \n",
       "L 129.959077 186.632876 \n",
       "L 131.258995 196.857269 \n",
       "L 132.233934 202.754044 \n",
       "L 132.883893 205.667507 \n",
       "L 133.533852 207.675331 \n",
       "L 133.858832 208.319088 \n",
       "L 134.183811 208.713553 \n",
       "L 134.508791 208.853013 \n",
       "L 134.83377 208.732755 \n",
       "L 135.15875 208.349094 \n",
       "L 135.483729 207.699408 \n",
       "L 136.133688 205.596963 \n",
       "L 136.783647 202.426686 \n",
       "L 137.433606 198.208193 \n",
       "L 138.408545 190.003661 \n",
       "L 139.383484 179.73734 \n",
       "L 140.683402 163.364795 \n",
       "L 142.633279 135.011204 \n",
       "L 145.558095 91.9626 \n",
       "L 146.858013 75.987767 \n",
       "L 147.832951 66.372842 \n",
       "L 148.80789 59.283244 \n",
       "L 149.457849 56.142104 \n",
       "L 150.107808 54.369426 \n",
       "L 150.432787 54.016558 \n",
       "L 150.757767 54.02735 \n",
       "L 151.082746 54.405974 \n",
       "L 151.407726 55.155089 \n",
       "L 152.057685 57.767712 \n",
       "L 152.707644 61.855327 \n",
       "L 153.357603 67.382695 \n",
       "L 154.332542 78.234051 \n",
       "L 155.30748 91.87491 \n",
       "L 156.607398 113.652322 \n",
       "L 158.557276 151.293044 \n",
       "L 161.482091 208.168758 \n",
       "L 162.782009 229.229457 \n",
       "L 163.756948 241.943445 \n",
       "L 164.731887 251.40934 \n",
       "L 165.381846 255.699745 \n",
       "L 166.031805 258.259946 \n",
       "L 166.356784 258.869332 \n",
       "L 166.681764 259.023561 \n",
       "L 167.006743 258.718976 \n",
       "L 167.331723 257.953846 \n",
       "L 167.656702 256.728381 \n",
       "L 168.306661 252.907067 \n",
       "L 168.95662 247.295803 \n",
       "L 169.606579 239.966304 \n",
       "L 170.581518 225.9809 \n",
       "L 171.556457 208.827551 \n",
       "L 172.856375 182.066211 \n",
       "L 175.131231 129.446945 \n",
       "L 177.406088 78.478026 \n",
       "L 178.706006 54.122373 \n",
       "L 179.680945 39.44774 \n",
       "L 180.655883 28.492884 \n",
       "L 181.305842 23.473669 \n",
       "L 181.955801 20.389482 \n",
       "L 182.280781 19.591739 \n",
       "L 182.60576 19.296 \n",
       "L 182.93074 19.503919 \n",
       "L 183.255719 20.214993 \n",
       "L 183.580699 21.426566 \n",
       "L 184.230658 25.329863 \n",
       "L 184.880617 31.150029 \n",
       "L 185.530576 38.790374 \n",
       "L 186.505515 53.375862 \n",
       "L 187.480453 71.21081 \n",
       "L 188.780371 98.862813 \n",
       "L 193.980044 216.214004 \n",
       "L 194.954982 232.730243 \n",
       "L 195.929921 245.830728 \n",
       "L 196.57988 252.442785 \n",
       "L 197.229839 257.245297 \n",
       "L 197.879798 260.178501 \n",
       "L 198.204778 260.93455 \n",
       "L 198.529757 261.216 \n",
       "L 198.854737 261.024795 \n",
       "L 199.179716 260.364955 \n",
       "L 199.504696 259.242546 \n",
       "L 200.154655 255.644275 \n",
       "L 200.804614 250.317755 \n",
       "L 201.454573 243.380118 \n",
       "L 202.429511 230.274477 \n",
       "L 203.72943 208.696577 \n",
       "L 205.679307 170.925158 \n",
       "L 208.279143 119.974421 \n",
       "L 209.579061 98.122536 \n",
       "L 210.554 84.49344 \n",
       "L 211.528938 73.718329 \n",
       "L 212.178897 68.271885 \n",
       "L 212.828856 64.279989 \n",
       "L 213.478815 61.765945 \n",
       "L 213.803795 61.062546 \n",
       "L 214.128774 60.724397 \n",
       "L 214.453754 60.746474 \n",
       "L 214.778733 61.122088 \n",
       "L 215.103713 61.842934 \n",
       "L 215.753672 64.279416 \n",
       "L 216.403631 67.95973 \n",
       "L 217.37857 75.550581 \n",
       "L 218.353508 85.202016 \n",
       "L 219.653426 100.34185 \n",
       "L 223.87816 151.831029 \n",
       "L 225.178078 163.878844 \n",
       "L 226.153017 170.797328 \n",
       "L 226.802976 174.301844 \n",
       "L 227.452935 176.894414 \n",
       "L 228.102894 178.578382 \n",
       "L 228.427873 179.086679 \n",
       "L 228.752853 179.378903 \n",
       "L 229.077832 179.461407 \n",
       "L 229.402812 179.341734 \n",
       "L 229.727792 179.02855 \n",
       "L 230.377751 177.86152 \n",
       "L 231.02771 176.04941 \n",
       "L 232.002648 172.348077 \n",
       "L 233.302566 166.185673 \n",
       "L 235.902402 153.313573 \n",
       "L 236.877341 149.476771 \n",
       "L 237.85228 146.666311 \n",
       "L 238.502239 145.468709 \n",
       "L 239.152198 144.860048 \n",
       "L 239.477177 144.783501 \n",
       "L 239.802157 144.860048 \n",
       "L 240.127136 145.08907 \n",
       "L 240.777095 145.995884 \n",
       "L 241.427054 147.474523 \n",
       "L 242.077013 149.476771 \n",
       "L 243.051952 153.313573 \n",
       "L 244.35187 159.520123 \n",
       "L 246.951706 172.348077 \n",
       "L 247.926645 176.04941 \n",
       "L 248.576604 177.86152 \n",
       "L 249.226563 179.02855 \n",
       "L 249.551543 179.341734 \n",
       "L 249.876522 179.461407 \n",
       "L 250.201502 179.378903 \n",
       "L 250.526481 179.086679 \n",
       "L 250.851461 178.578382 \n",
       "L 251.50142 176.894414 \n",
       "L 252.151379 174.301844 \n",
       "L 252.801338 170.797328 \n",
       "L 253.776276 163.878844 \n",
       "L 254.751215 155.117276 \n",
       "L 256.051133 141.052084 \n",
       "L 258.00101 116.816843 \n",
       "L 260.275867 88.785702 \n",
       "L 261.575785 75.550581 \n",
       "L 262.550724 67.95973 \n",
       "L 263.200683 64.279416 \n",
       "L 263.850642 61.842934 \n",
       "L 264.175621 61.122088 \n",
       "L 264.500601 60.746474 \n",
       "L 264.82558 60.724397 \n",
       "L 265.15056 61.062546 \n",
       "L 265.475539 61.765945 \n",
       "L 266.125498 64.279989 \n",
       "L 266.775457 68.271885 \n",
       "L 267.425416 73.718329 \n",
       "L 268.400355 84.49344 \n",
       "L 269.375294 98.122536 \n",
       "L 270.675212 119.974421 \n",
       "L 272.625089 157.797455 \n",
       "L 275.224925 208.696577 \n",
       "L 276.524843 230.274477 \n",
       "L 277.499782 243.380118 \n",
       "L 278.47472 253.190387 \n",
       "L 279.124679 257.665641 \n",
       "L 279.774638 260.364955 \n",
       "L 280.099618 261.024795 \n",
       "L 280.424597 261.216 \n",
       "L 280.749577 260.93455 \n",
       "L 281.074556 260.178501 \n",
       "L 281.399536 258.948 \n",
       "L 282.049495 255.074745 \n",
       "L 282.699454 249.357929 \n",
       "L 283.349413 241.873731 \n",
       "L 284.324352 227.578355 \n",
       "L 285.29929 210.046353 \n",
       "L 286.599208 182.729453 \n",
       "L 289.199045 121.487612 \n",
       "L 291.148922 77.764186 \n",
       "L 292.44884 53.375862 \n",
       "L 293.423778 38.790374 \n",
       "L 294.398717 28.005626 \n",
       "L 295.048676 23.133833 \n",
       "L 295.698635 20.214993 \n",
       "L 296.023615 19.503919 \n",
       "L 296.348594 19.296 \n",
       "L 296.673574 19.591739 \n",
       "L 296.998553 20.389482 \n",
       "L 297.323533 21.685432 \n",
       "L 297.973492 25.746176 \n",
       "L 298.623451 31.701712 \n",
       "L 299.27341 39.44774 \n",
       "L 300.248348 54.122373 \n",
       "L 301.548267 78.478026 \n",
       "L 303.173164 114.24536 \n",
       "L 307.072918 202.504622 \n",
       "L 308.372837 225.9809 \n",
       "L 309.347775 239.966304 \n",
       "L 310.322714 250.321414 \n",
       "L 310.972673 255.044748 \n",
       "L 311.622632 257.953846 \n",
       "L 311.947611 258.718976 \n",
       "L 312.272591 259.023561 \n",
       "L 312.59757 258.869332 \n",
       "L 312.92255 258.259946 \n",
       "L 313.247529 257.200948 \n",
       "L 313.897488 253.76555 \n",
       "L 314.547448 248.643788 \n",
       "L 315.197407 241.943445 \n",
       "L 316.172345 229.229457 \n",
       "L 317.472263 208.168758 \n",
       "L 319.097161 177.401626 \n",
       "L 322.671936 107.880402 \n",
       "L 323.971854 87.041955 \n",
       "L 324.946792 74.289276 \n",
       "L 325.921731 64.442236 \n",
       "L 326.57169 59.628752 \n",
       "L 327.221649 56.275817 \n",
       "L 327.871608 54.405974 \n",
       "L 328.196588 54.02735 \n",
       "L 328.521567 54.016558 \n",
       "L 328.846547 54.369426 \n",
       "L 329.171526 55.080303 \n",
       "L 329.821485 57.546355 \n",
       "L 330.471444 61.341684 \n",
       "L 331.121403 66.372842 \n",
       "L 332.096342 75.987767 \n",
       "L 333.39626 91.9626 \n",
       "L 335.021158 115.279364 \n",
       "L 338.595932 167.704968 \n",
       "L 339.89585 183.372049 \n",
       "L 340.870789 192.979678 \n",
       "L 341.845728 200.446515 \n",
       "L 342.495687 204.144512 \n",
       "L 343.145646 206.782169 \n",
       "L 343.795605 208.349094 \n",
       "L 344.120584 208.732755 \n",
       "L 344.445564 208.853013 \n",
       "L 344.770543 208.713553 \n",
       "L 345.095523 208.319088 \n",
       "L 345.745482 206.788945 \n",
       "L 346.395441 204.319455 \n",
       "L 347.0454 200.981287 \n",
       "L 348.020339 194.529334 \n",
       "L 349.320257 183.740959 \n",
       "L 350.945154 167.96418 \n",
       "L 354.519929 132.548774 \n",
       "L 355.819847 121.968939 \n",
       "L 356.794786 115.460541 \n",
       "L 357.769724 110.365486 \n",
       "L 358.419683 107.808412 \n",
       "L 359.069642 105.943184 \n",
       "L 359.719602 104.773411 \n",
       "L 360.044581 104.447229 \n",
       "L 360.369561 104.290984 \n",
       "L 360.69454 104.301929 \n",
       "L 361.01952 104.476666 \n",
       "L 361.669479 105.300851 \n",
       "L 362.319438 106.724474 \n",
       "L 362.969397 108.700044 \n",
       "L 363.944335 112.576791 \n",
       "L 365.244253 119.119095 \n",
       "L 366.869151 128.72491 \n",
       "L 370.443926 150.302232 \n",
       "L 371.743844 156.756246 \n",
       "L 372.718783 160.741178 \n",
       "L 373.693721 163.883966 \n",
       "L 374.66866 166.127338 \n",
       "L 375.318619 167.111856 \n",
       "L 375.968578 167.689922 \n",
       "L 376.618537 167.871123 \n",
       "L 377.268496 167.671234 \n",
       "L 377.918455 167.111659 \n",
       "L 378.568414 166.218804 \n",
       "L 379.543353 164.322839 \n",
       "L 380.518291 161.865017 \n",
       "L 381.818209 157.940447 \n",
       "L 384.418045 149.089417 \n",
       "L 386.367923 142.780649 \n",
       "L 387.667841 139.210294 \n",
       "L 388.642779 136.99722 \n",
       "L 389.617718 135.239132 \n",
       "L 390.592656 133.964559 \n",
       "L 391.242615 133.38836 \n",
       "L 391.892574 133.028837 \n",
       "L 392.542534 132.880174 \n",
       "L 393.192493 132.933321 \n",
       "L 393.842452 133.176326 \n",
       "L 394.81739 133.864506 \n",
       "L 395.792329 134.889149 \n",
       "L 397.092247 136.664898 \n",
       "L 399.042124 139.865272 \n",
       "L 401.64196 144.273101 \n",
       "L 401.64196 144.273101 \n",
       "\" clip-path=\"url(#p742dd30804)\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 60.754688 273.312 \n",
       "L 60.754688 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 417.874688 273.312 \n",
       "L 417.874688 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 60.754688 273.312 \n",
       "L 417.874688 273.312 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 60.754688 7.2 \n",
       "L 417.874688 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p742dd30804\">\n",
       "   <rect x=\"60.754688\" y=\"7.2\" width=\"357.12\" height=\"266.112\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pyquil.quiltwaveforms import DragGaussianWaveform\n",
    "\n",
    "plot_waveform(\n",
    "    DragGaussianWaveform(duration=1e-6, t0=5e-7, fwhm=4e-7, anh=1.1, alpha=1.0, detuning=1e7),\n",
    "    sample_rate=1e9\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### hrm_gaussian\n",
    "\n",
    "The `hrm_gaussian` waveform is a variant of `drag_gaussian` which incorporates a higher order term. The shape is given by\n",
    "\\begin{equation}\n",
    "u(t) = \\Big(1 - H_2 \\frac{(t-t_0)^2}{2 \\sigma^2} + i\\frac{\\alpha (t-t_0)}{2 \\pi \\eta \\sigma^2} \\big(1 - H_2 \\big(\\frac{(t-t_0)^2}{2\\sigma^2} - 1\\big)\\Big)\\text{exp}\\Big(-\\frac{(t-t_0)^2}{2 \\sigma^2}\\Big),\n",
    "\\end{equation}\n",
    "where $\\alpha$ is a dimensionless DRAG parameter, $\\eta$ is the anharmonicity constant, and $H_2$ is a second order correction coefficient (cf. [1]). Note that when $H_2$ equals 0 this reduces to an ordinary `drag_gaussian`.\n",
    "\n",
    "The required arguments to `hrm_gaussian` are:\n",
    "\n",
    "* `duration`: the duration of the waveform, in seconds. The Gaussian will be truncated to $[0, \\text{duration}]$\n",
    "* `t0`: the center of the Gaussian, in seconds\n",
    "* `fwhm`: the full width half maximum of the Gaussian, in seconds\n",
    "* `anh`: the anharmonicity constant $\\eta$, in Hertz\n",
    "* `alpha`: dimensionless shape parameter $\\alpha$\n",
    "* `second_order_hrm_coeff`: the constant $H_2$.\n",
    "\n",
    "[1] Warren, W. S. (1984). Effects of arbitrary laser or NMR pulse shapes on population inversion and coherence. The Journal of Chemical Physics, 81(12), 5437–5448. doi:10.1063/1.447644 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hrm_gaussian(alpha: 1, anh: 1.1, duration: 1e-6, fwhm: 4e-7, second_order_hrm_coeff: 0.5, t0: 5e-7)\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"394.423125pt\" height=\"309.86825pt\" viewBox=\"0 0 394.423125 309.86825\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2023-09-15T14:32:52.298030</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.7.2, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 309.86825 \n",
       "L 394.423125 309.86825 \n",
       "L 394.423125 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 387.223125 273.312 \n",
       "L 387.223125 7.2 \n",
       "L 30.103125 7.2 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path id=\"mdec00cb84a\" d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#mdec00cb84a\" x=\"46.335852\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(38.38429 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
       "Q 1547 4250 1301 3770 \n",
       "Q 1056 3291 1056 2328 \n",
       "Q 1056 1369 1301 889 \n",
       "Q 1547 409 2034 409 \n",
       "Q 2525 409 2770 889 \n",
       "Q 3016 1369 3016 2328 \n",
       "Q 3016 3291 2770 3770 \n",
       "Q 2525 4250 2034 4250 \n",
       "z\n",
       "M 2034 4750 \n",
       "Q 2819 4750 3233 4129 \n",
       "Q 3647 3509 3647 2328 \n",
       "Q 3647 1150 3233 529 \n",
       "Q 2819 -91 2034 -91 \n",
       "Q 1250 -91 836 529 \n",
       "Q 422 1150 422 2328 \n",
       "Q 422 3509 836 4129 \n",
       "Q 1250 4750 2034 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-2e\" d=\"M 684 794 \n",
       "L 1344 794 \n",
       "L 1344 0 \n",
       "L 684 0 \n",
       "L 684 794 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mdec00cb84a\" x=\"111.331757\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(103.380195 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-32\" d=\"M 1228 531 \n",
       "L 3431 531 \n",
       "L 3431 0 \n",
       "L 469 0 \n",
       "L 469 531 \n",
       "Q 828 903 1448 1529 \n",
       "Q 2069 2156 2228 2338 \n",
       "Q 2531 2678 2651 2914 \n",
       "Q 2772 3150 2772 3378 \n",
       "Q 2772 3750 2511 3984 \n",
       "Q 2250 4219 1831 4219 \n",
       "Q 1534 4219 1204 4116 \n",
       "Q 875 4013 500 3803 \n",
       "L 500 4441 \n",
       "Q 881 4594 1212 4672 \n",
       "Q 1544 4750 1819 4750 \n",
       "Q 2544 4750 2975 4387 \n",
       "Q 3406 4025 3406 3419 \n",
       "Q 3406 3131 3298 2873 \n",
       "Q 3191 2616 2906 2266 \n",
       "Q 2828 2175 2409 1742 \n",
       "Q 1991 1309 1228 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mdec00cb84a\" x=\"176.327662\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(168.3761 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
       "L 825 1625 \n",
       "L 2419 1625 \n",
       "L 2419 4116 \n",
       "z\n",
       "M 2253 4666 \n",
       "L 3047 4666 \n",
       "L 3047 1625 \n",
       "L 3713 1625 \n",
       "L 3713 1100 \n",
       "L 3047 1100 \n",
       "L 3047 0 \n",
       "L 2419 0 \n",
       "L 2419 1100 \n",
       "L 313 1100 \n",
       "L 313 1709 \n",
       "L 2253 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mdec00cb84a\" x=\"241.323567\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(233.372005 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
       "Q 1688 2584 1439 2293 \n",
       "Q 1191 2003 1191 1497 \n",
       "Q 1191 994 1439 701 \n",
       "Q 1688 409 2113 409 \n",
       "Q 2538 409 2786 701 \n",
       "Q 3034 994 3034 1497 \n",
       "Q 3034 2003 2786 2293 \n",
       "Q 2538 2584 2113 2584 \n",
       "z\n",
       "M 3366 4563 \n",
       "L 3366 3988 \n",
       "Q 3128 4100 2886 4159 \n",
       "Q 2644 4219 2406 4219 \n",
       "Q 1781 4219 1451 3797 \n",
       "Q 1122 3375 1075 2522 \n",
       "Q 1259 2794 1537 2939 \n",
       "Q 1816 3084 2150 3084 \n",
       "Q 2853 3084 3261 2657 \n",
       "Q 3669 2231 3669 1497 \n",
       "Q 3669 778 3244 343 \n",
       "Q 2819 -91 2113 -91 \n",
       "Q 1303 -91 875 529 \n",
       "Q 447 1150 447 2328 \n",
       "Q 447 3434 972 4092 \n",
       "Q 1497 4750 2381 4750 \n",
       "Q 2619 4750 2861 4703 \n",
       "Q 3103 4656 3366 4563 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mdec00cb84a\" x=\"306.319472\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(298.36791 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \n",
       "Q 1584 2216 1326 1975 \n",
       "Q 1069 1734 1069 1313 \n",
       "Q 1069 891 1326 650 \n",
       "Q 1584 409 2034 409 \n",
       "Q 2484 409 2743 651 \n",
       "Q 3003 894 3003 1313 \n",
       "Q 3003 1734 2745 1975 \n",
       "Q 2488 2216 2034 2216 \n",
       "z\n",
       "M 1403 2484 \n",
       "Q 997 2584 770 2862 \n",
       "Q 544 3141 544 3541 \n",
       "Q 544 4100 942 4425 \n",
       "Q 1341 4750 2034 4750 \n",
       "Q 2731 4750 3128 4425 \n",
       "Q 3525 4100 3525 3541 \n",
       "Q 3525 3141 3298 2862 \n",
       "Q 3072 2584 2669 2484 \n",
       "Q 3125 2378 3379 2068 \n",
       "Q 3634 1759 3634 1313 \n",
       "Q 3634 634 3220 271 \n",
       "Q 2806 -91 2034 -91 \n",
       "Q 1263 -91 848 271 \n",
       "Q 434 634 434 1313 \n",
       "Q 434 1759 690 2068 \n",
       "Q 947 2378 1403 2484 \n",
       "z\n",
       "M 1172 3481 \n",
       "Q 1172 3119 1398 2916 \n",
       "Q 1625 2713 2034 2713 \n",
       "Q 2441 2713 2670 2916 \n",
       "Q 2900 3119 2900 3481 \n",
       "Q 2900 3844 2670 4047 \n",
       "Q 2441 4250 2034 4250 \n",
       "Q 1625 4250 1398 4047 \n",
       "Q 1172 3844 1172 3481 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mdec00cb84a\" x=\"371.315377\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(363.363815 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-31\" d=\"M 794 531 \n",
       "L 1825 531 \n",
       "L 1825 4091 \n",
       "L 703 3866 \n",
       "L 703 4441 \n",
       "L 1819 4666 \n",
       "L 2450 4666 \n",
       "L 2450 531 \n",
       "L 3481 531 \n",
       "L 3481 0 \n",
       "L 794 0 \n",
       "L 794 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- 1e−6 -->\n",
       "     <g transform=\"translate(359.965312 300.588562) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-65\" d=\"M 3597 1894 \n",
       "L 3597 1613 \n",
       "L 953 1613 \n",
       "Q 991 1019 1311 708 \n",
       "Q 1631 397 2203 397 \n",
       "Q 2534 397 2845 478 \n",
       "Q 3156 559 3463 722 \n",
       "L 3463 178 \n",
       "Q 3153 47 2828 -22 \n",
       "Q 2503 -91 2169 -91 \n",
       "Q 1331 -91 842 396 \n",
       "Q 353 884 353 1716 \n",
       "Q 353 2575 817 3079 \n",
       "Q 1281 3584 2069 3584 \n",
       "Q 2775 3584 3186 3129 \n",
       "Q 3597 2675 3597 1894 \n",
       "z\n",
       "M 3022 2063 \n",
       "Q 3016 2534 2758 2815 \n",
       "Q 2500 3097 2075 3097 \n",
       "Q 1594 3097 1305 2825 \n",
       "Q 1016 2553 972 2059 \n",
       "L 3022 2063 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2212\" d=\"M 678 2272 \n",
       "L 4684 2272 \n",
       "L 4684 1741 \n",
       "L 678 1741 \n",
       "L 678 2272 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-65\" x=\"63.623047\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2212\" x=\"125.146484\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-36\" x=\"208.935547\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <defs>\n",
       "       <path id=\"mbd51bac4fa\" d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbd51bac4fa\" x=\"30.103125\" y=\"255.340034\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(7.2 259.139253) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbd51bac4fa\" x=\"30.103125\" y=\"208.131227\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(7.2 211.930446) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbd51bac4fa\" x=\"30.103125\" y=\"160.92242\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(7.2 164.721639) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbd51bac4fa\" x=\"30.103125\" y=\"113.713614\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(7.2 117.512832) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbd51bac4fa\" x=\"30.103125\" y=\"66.504807\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(7.2 70.304026) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mbd51bac4fa\" x=\"30.103125\" y=\"19.296\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(7.2 23.095219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_13\">\n",
       "    <path d=\"M 46.335852 258.956511 \n",
       "L 63.559767 260.751618 \n",
       "L 69.409399 261.123215 \n",
       "L 74.284091 261.213318 \n",
       "L 78.508825 261.061358 \n",
       "L 82.0836 260.717075 \n",
       "L 85.333395 260.194869 \n",
       "L 88.583191 259.438077 \n",
       "L 91.508006 258.526496 \n",
       "L 94.432822 257.368107 \n",
       "L 97.032658 256.108462 \n",
       "L 99.632494 254.611527 \n",
       "L 102.232331 252.857307 \n",
       "L 104.832167 250.826091 \n",
       "L 107.432003 248.498735 \n",
       "L 110.031839 245.85696 \n",
       "L 112.631675 242.883678 \n",
       "L 115.231512 239.563324 \n",
       "L 117.831348 235.882198 \n",
       "L 120.431184 231.828814 \n",
       "L 123.03102 227.39425 \n",
       "L 125.955836 221.942364 \n",
       "L 128.880652 215.996857 \n",
       "L 131.805467 209.559789 \n",
       "L 135.055263 201.841247 \n",
       "L 138.305058 193.547174 \n",
       "L 141.879833 183.797012 \n",
       "L 145.779587 172.478093 \n",
       "L 150.004321 159.522618 \n",
       "L 155.203993 142.8116 \n",
       "L 163.653461 114.728001 \n",
       "L 171.12799 90.185634 \n",
       "L 175.677703 76.003864 \n",
       "L 179.252478 65.521973 \n",
       "L 182.502273 56.639581 \n",
       "L 185.427089 49.267012 \n",
       "L 188.026925 43.272259 \n",
       "L 190.626761 37.855379 \n",
       "L 192.901618 33.625715 \n",
       "L 195.176475 29.901243 \n",
       "L 197.126352 27.12966 \n",
       "L 199.076229 24.760906 \n",
       "L 200.701127 23.103123 \n",
       "L 202.326024 21.738891 \n",
       "L 203.950922 20.672904 \n",
       "L 205.57582 19.908841 \n",
       "L 206.875738 19.516786 \n",
       "L 208.175656 19.320541 \n",
       "L 209.475574 19.320541 \n",
       "L 210.775492 19.516786 \n",
       "L 212.07541 19.908841 \n",
       "L 213.375328 20.495838 \n",
       "L 215.000226 21.501701 \n",
       "L 216.625123 22.806628 \n",
       "L 218.250021 24.406124 \n",
       "L 219.874919 26.294701 \n",
       "L 221.824796 28.933421 \n",
       "L 223.774673 31.966145 \n",
       "L 225.72455 35.378114 \n",
       "L 227.999407 39.815988 \n",
       "L 230.274263 44.718654 \n",
       "L 232.8741 50.850229 \n",
       "L 235.798915 58.361082 \n",
       "L 239.048711 67.37645 \n",
       "L 242.623485 77.978258 \n",
       "L 246.848219 91.227007 \n",
       "L 252.697851 110.397454 \n",
       "L 266.022011 154.37824 \n",
       "L 270.896704 169.547588 \n",
       "L 275.121438 181.957801 \n",
       "L 278.696213 191.821885 \n",
       "L 282.270987 201.037302 \n",
       "L 285.520783 208.814512 \n",
       "L 288.770578 215.996857 \n",
       "L 291.695394 221.942364 \n",
       "L 294.620209 227.39425 \n",
       "L 297.545025 232.356176 \n",
       "L 300.144861 236.362484 \n",
       "L 302.744697 239.997837 \n",
       "L 305.344534 243.274016 \n",
       "L 307.94437 246.204974 \n",
       "L 310.544206 248.806484 \n",
       "L 313.144042 251.095803 \n",
       "L 315.743878 253.091333 \n",
       "L 318.343715 254.812306 \n",
       "L 320.943551 256.278483 \n",
       "L 323.543387 257.509877 \n",
       "L 326.468203 258.639461 \n",
       "L 329.393019 259.525331 \n",
       "L 332.642814 260.257064 \n",
       "L 335.892609 260.75773 \n",
       "L 339.467384 261.082125 \n",
       "L 343.367138 261.213318 \n",
       "L 347.916851 261.136439 \n",
       "L 353.441503 260.803539 \n",
       "L 360.916032 260.10219 \n",
       "L 370.990398 258.992349 \n",
       "L 370.990398 258.992349 \n",
       "\" clip-path=\"url(#p5427faf227)\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 30.103125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 387.223125 273.312 \n",
       "L 387.223125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 387.223125 273.312 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 30.103125 7.2 \n",
       "L 387.223125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p5427faf227\">\n",
       "   <rect x=\"30.103125\" y=\"7.2\" width=\"357.12\" height=\"266.112\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pyquil.quiltwaveforms import HrmGaussianWaveform\n",
    "\n",
    "plot_waveform(\n",
    "    HrmGaussianWaveform(duration=1e-6, t0=5e-7, fwhm=4e-7, anh=1.1, alpha=1.0, second_order_hrm_coeff=0.5),\n",
    "    sample_rate=1e9\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### erf_square\n",
    "\n",
    "The `erf_square` waveform is a variant of `flat_waveform` with the boundary discontinuities smoothed via the [error function](https://en.wikipedia.org/wiki/Error_function) (erf), and additional zero-padding.\n",
    "\n",
    "The required arguments are:\n",
    "\n",
    "* `duration`: the duration of the nonzero part of the waveform, in seconds\n",
    "* `risetime`: width of each of the rise and fall sections of the pulse, in seconds\n",
    "* `pad_left`: amount of zero-padding to add to the left of the pulse, in seconds\n",
    "* `pad_right`: amount of zero-padding to add to the right of the pulse, in seconds\n",
    "\n",
    "**NOTE**: The total duration of the waveform is `duration + pad_left + pad_right`; the total duration of the *support* (nonzero entries) is `duration`.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "erf_square(duration: 1e-6, pad_left: 1e-7, pad_right: 1e-7, risetime: 1e-7)\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"394.423125pt\" height=\"309.86825pt\" viewBox=\"0 0 394.423125 309.86825\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2023-09-15T14:32:52.363442</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.7.2, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 309.86825 \n",
       "L 394.423125 309.86825 \n",
       "L 394.423125 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 387.223125 273.312 \n",
       "L 387.223125 7.2 \n",
       "L 30.103125 7.2 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path id=\"mf85f878502\" d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#mf85f878502\" x=\"46.335852\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(38.38429 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
       "Q 1547 4250 1301 3770 \n",
       "Q 1056 3291 1056 2328 \n",
       "Q 1056 1369 1301 889 \n",
       "Q 1547 409 2034 409 \n",
       "Q 2525 409 2770 889 \n",
       "Q 3016 1369 3016 2328 \n",
       "Q 3016 3291 2770 3770 \n",
       "Q 2525 4250 2034 4250 \n",
       "z\n",
       "M 2034 4750 \n",
       "Q 2819 4750 3233 4129 \n",
       "Q 3647 3509 3647 2328 \n",
       "Q 3647 1150 3233 529 \n",
       "Q 2819 -91 2034 -91 \n",
       "Q 1250 -91 836 529 \n",
       "Q 422 1150 422 2328 \n",
       "Q 422 3509 836 4129 \n",
       "Q 1250 4750 2034 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-2e\" d=\"M 684 794 \n",
       "L 1344 794 \n",
       "L 1344 0 \n",
       "L 684 0 \n",
       "L 684 794 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mf85f878502\" x=\"100.490072\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(92.538509 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-32\" d=\"M 1228 531 \n",
       "L 3431 531 \n",
       "L 3431 0 \n",
       "L 469 0 \n",
       "L 469 531 \n",
       "Q 828 903 1448 1529 \n",
       "Q 2069 2156 2228 2338 \n",
       "Q 2531 2678 2651 2914 \n",
       "Q 2772 3150 2772 3378 \n",
       "Q 2772 3750 2511 3984 \n",
       "Q 2250 4219 1831 4219 \n",
       "Q 1534 4219 1204 4116 \n",
       "Q 875 4013 500 3803 \n",
       "L 500 4441 \n",
       "Q 881 4594 1212 4672 \n",
       "Q 1544 4750 1819 4750 \n",
       "Q 2544 4750 2975 4387 \n",
       "Q 3406 4025 3406 3419 \n",
       "Q 3406 3131 3298 2873 \n",
       "Q 3191 2616 2906 2266 \n",
       "Q 2828 2175 2409 1742 \n",
       "Q 1991 1309 1228 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mf85f878502\" x=\"154.644291\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(146.692729 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
       "L 825 1625 \n",
       "L 2419 1625 \n",
       "L 2419 4116 \n",
       "z\n",
       "M 2253 4666 \n",
       "L 3047 4666 \n",
       "L 3047 1625 \n",
       "L 3713 1625 \n",
       "L 3713 1100 \n",
       "L 3047 1100 \n",
       "L 3047 0 \n",
       "L 2419 0 \n",
       "L 2419 1100 \n",
       "L 313 1100 \n",
       "L 313 1709 \n",
       "L 2253 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mf85f878502\" x=\"208.798511\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(200.846948 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
       "Q 1688 2584 1439 2293 \n",
       "Q 1191 2003 1191 1497 \n",
       "Q 1191 994 1439 701 \n",
       "Q 1688 409 2113 409 \n",
       "Q 2538 409 2786 701 \n",
       "Q 3034 994 3034 1497 \n",
       "Q 3034 2003 2786 2293 \n",
       "Q 2538 2584 2113 2584 \n",
       "z\n",
       "M 3366 4563 \n",
       "L 3366 3988 \n",
       "Q 3128 4100 2886 4159 \n",
       "Q 2644 4219 2406 4219 \n",
       "Q 1781 4219 1451 3797 \n",
       "Q 1122 3375 1075 2522 \n",
       "Q 1259 2794 1537 2939 \n",
       "Q 1816 3084 2150 3084 \n",
       "Q 2853 3084 3261 2657 \n",
       "Q 3669 2231 3669 1497 \n",
       "Q 3669 778 3244 343 \n",
       "Q 2819 -91 2113 -91 \n",
       "Q 1303 -91 875 529 \n",
       "Q 447 1150 447 2328 \n",
       "Q 447 3434 972 4092 \n",
       "Q 1497 4750 2381 4750 \n",
       "Q 2619 4750 2861 4703 \n",
       "Q 3103 4656 3366 4563 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mf85f878502\" x=\"262.95273\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(255.001167 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \n",
       "Q 1584 2216 1326 1975 \n",
       "Q 1069 1734 1069 1313 \n",
       "Q 1069 891 1326 650 \n",
       "Q 1584 409 2034 409 \n",
       "Q 2484 409 2743 651 \n",
       "Q 3003 894 3003 1313 \n",
       "Q 3003 1734 2745 1975 \n",
       "Q 2488 2216 2034 2216 \n",
       "z\n",
       "M 1403 2484 \n",
       "Q 997 2584 770 2862 \n",
       "Q 544 3141 544 3541 \n",
       "Q 544 4100 942 4425 \n",
       "Q 1341 4750 2034 4750 \n",
       "Q 2731 4750 3128 4425 \n",
       "Q 3525 4100 3525 3541 \n",
       "Q 3525 3141 3298 2862 \n",
       "Q 3072 2584 2669 2484 \n",
       "Q 3125 2378 3379 2068 \n",
       "Q 3634 1759 3634 1313 \n",
       "Q 3634 634 3220 271 \n",
       "Q 2806 -91 2034 -91 \n",
       "Q 1263 -91 848 271 \n",
       "Q 434 634 434 1313 \n",
       "Q 434 1759 690 2068 \n",
       "Q 947 2378 1403 2484 \n",
       "z\n",
       "M 1172 3481 \n",
       "Q 1172 3119 1398 2916 \n",
       "Q 1625 2713 2034 2713 \n",
       "Q 2441 2713 2670 2916 \n",
       "Q 2900 3119 2900 3481 \n",
       "Q 2900 3844 2670 4047 \n",
       "Q 2441 4250 2034 4250 \n",
       "Q 1625 4250 1398 4047 \n",
       "Q 1172 3844 1172 3481 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mf85f878502\" x=\"317.106949\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(309.155387 287.910437) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-31\" d=\"M 794 531 \n",
       "L 1825 531 \n",
       "L 1825 4091 \n",
       "L 703 3866 \n",
       "L 703 4441 \n",
       "L 1819 4666 \n",
       "L 2450 4666 \n",
       "L 2450 531 \n",
       "L 3481 531 \n",
       "L 3481 0 \n",
       "L 794 0 \n",
       "L 794 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_7\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#mf85f878502\" x=\"371.261169\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_7\">\n",
       "      <!-- 1.2 -->\n",
       "      <g transform=\"translate(363.309606 287.910437) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_8\">\n",
       "     <!-- 1e−6 -->\n",
       "     <g transform=\"translate(359.965312 300.588562) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-65\" d=\"M 3597 1894 \n",
       "L 3597 1613 \n",
       "L 953 1613 \n",
       "Q 991 1019 1311 708 \n",
       "Q 1631 397 2203 397 \n",
       "Q 2534 397 2845 478 \n",
       "Q 3156 559 3463 722 \n",
       "L 3463 178 \n",
       "Q 3153 47 2828 -22 \n",
       "Q 2503 -91 2169 -91 \n",
       "Q 1331 -91 842 396 \n",
       "Q 353 884 353 1716 \n",
       "Q 353 2575 817 3079 \n",
       "Q 1281 3584 2069 3584 \n",
       "Q 2775 3584 3186 3129 \n",
       "Q 3597 2675 3597 1894 \n",
       "z\n",
       "M 3022 2063 \n",
       "Q 3016 2534 2758 2815 \n",
       "Q 2500 3097 2075 3097 \n",
       "Q 1594 3097 1305 2825 \n",
       "Q 1016 2553 972 2059 \n",
       "L 3022 2063 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2212\" d=\"M 678 2272 \n",
       "L 4684 2272 \n",
       "L 4684 1741 \n",
       "L 678 1741 \n",
       "L 678 2272 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-65\" x=\"63.623047\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2212\" x=\"125.146484\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-36\" x=\"208.935547\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <defs>\n",
       "       <path id=\"m2bfafa33ab\" d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m2bfafa33ab\" x=\"30.103125\" y=\"261.216\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(7.2 265.015219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m2bfafa33ab\" x=\"30.103125\" y=\"212.832\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(7.2 216.631219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m2bfafa33ab\" x=\"30.103125\" y=\"164.448\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(7.2 168.247219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m2bfafa33ab\" x=\"30.103125\" y=\"116.064\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(7.2 119.863219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m2bfafa33ab\" x=\"30.103125\" y=\"67.68\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_13\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(7.2 71.479219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m2bfafa33ab\" x=\"30.103125\" y=\"19.296\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_14\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(7.2 23.095219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_14\">\n",
       "    <path d=\"M 46.335852 261.216 \n",
       "L 73.412962 261.111033 \n",
       "L 74.766817 260.886414 \n",
       "L 75.579131 260.592834 \n",
       "L 76.391444 260.080356 \n",
       "L 77.203757 259.220669 \n",
       "L 77.7453 258.368553 \n",
       "L 78.286842 257.216922 \n",
       "L 78.828384 255.687831 \n",
       "L 79.640697 252.491562 \n",
       "L 80.453011 247.918787 \n",
       "L 81.265324 241.631918 \n",
       "L 82.077637 233.325631 \n",
       "L 82.88995 222.779394 \n",
       "L 83.973035 205.118004 \n",
       "L 85.056119 183.674536 \n",
       "L 86.680746 146.679379 \n",
       "L 89.117686 91.164122 \n",
       "L 90.20077 70.600465 \n",
       "L 91.283854 53.959857 \n",
       "L 92.096168 44.177259 \n",
       "L 92.908481 36.573944 \n",
       "L 93.720794 30.894971 \n",
       "L 94.533108 26.81878 \n",
       "L 95.345421 24.007155 \n",
       "L 96.157734 22.143447 \n",
       "L 96.970047 20.956267 \n",
       "L 97.782361 20.229539 \n",
       "L 98.594674 19.802029 \n",
       "L 99.677758 19.507154 \n",
       "L 101.302385 19.34626 \n",
       "L 105.09318 19.29698 \n",
       "L 220.170897 19.296 \n",
       "L 317.106949 19.400967 \n",
       "L 318.460805 19.625586 \n",
       "L 319.273118 19.919166 \n",
       "L 320.085431 20.431644 \n",
       "L 320.897745 21.291331 \n",
       "L 321.439287 22.143447 \n",
       "L 321.980829 23.295078 \n",
       "L 322.522371 24.824169 \n",
       "L 323.334685 28.020438 \n",
       "L 324.146998 32.593213 \n",
       "L 324.959311 38.880082 \n",
       "L 325.771625 47.186369 \n",
       "L 326.583938 57.732606 \n",
       "L 327.667022 75.393996 \n",
       "L 328.750107 96.837464 \n",
       "L 330.374733 133.832621 \n",
       "L 332.811673 189.347878 \n",
       "L 333.894757 209.911535 \n",
       "L 334.977842 226.552143 \n",
       "L 335.790155 236.334741 \n",
       "L 336.602468 243.938056 \n",
       "L 337.414782 249.617029 \n",
       "L 338.227095 253.69322 \n",
       "L 339.039408 256.504845 \n",
       "L 339.851722 258.368553 \n",
       "L 340.664035 259.555733 \n",
       "L 341.476348 260.282461 \n",
       "L 342.288661 260.709971 \n",
       "L 343.371746 261.004846 \n",
       "L 345.267144 261.216 \n",
       "L 370.990398 261.216 \n",
       "L 370.990398 261.216 \n",
       "\" clip-path=\"url(#pbfcd07f618)\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 30.103125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 387.223125 273.312 \n",
       "L 387.223125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 30.103125 273.312 \n",
       "L 387.223125 273.312 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 30.103125 7.2 \n",
       "L 387.223125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"pbfcd07f618\">\n",
       "   <rect x=\"30.103125\" y=\"7.2\" width=\"357.12\" height=\"266.112\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pyquil.quiltwaveforms import ErfSquareWaveform\n",
    "\n",
    "plot_waveform(\n",
    "    ErfSquareWaveform(duration=1e-6, risetime=1e-7, pad_left=1e-7, pad_right=1e-7),\n",
    "    sample_rate=1e9\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Capture and Kernels\n",
    "\n",
    "In Quil-T, waveforms are used in two places:\n",
    "\n",
    "* in a `PULSE` operation, to specify what signal is generated\n",
    "* in a `CAPTURE` operation, to specify how to resolve a signal into a number\n",
    "\n",
    "The `CAPTURE` operation involves reading in a signal on a suitable signal line, and integrating it with respect to a *kernel*. \n",
    "\n",
    "Mathematically, there is a real-valued signal $ s(t), $ corresponding to a reading on the signal line. This is combined with a complex-valued \"baseband\" kernel $k(t)$ to get a resulting **IQ value** $z$ by\n",
    "\\begin{equation}\n",
    "z = \\int_{t_\\text{min}}^{t_\\text{max}} k(t) s(t) e^{-2 \\pi i f t} \\, dt, \n",
    "\\end{equation}\n",
    "where $f$ denotes the frequency of the associated capture frame.\n",
    "\n",
    "\n",
    "In Quil-T, this integrating kernel is specified by a waveform, with the usual convention that is is scaled to satisfy $\\int k(t) \\, dt = 1.$ The most common example is the `boxcar_kernel`, which corresponds to a flat pulse scaled to satisfy this condition.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### boxcar_kernel\n",
    "\n",
    "Because of the normalization condition, the `boxcar_kernel` requires only a `duration` argument for its construction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "boxcar_kernel(duration: 1e-6)\n"
     ]
    },
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"419.873125pt\" height=\"309.86825pt\" viewBox=\"0 0 419.873125 309.86825\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2023-09-15T14:32:52.436673</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.7.2, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 309.86825 \n",
       "L 419.873125 309.86825 \n",
       "L 419.873125 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 55.553125 273.312 \n",
       "L 412.673125 273.312 \n",
       "L 412.673125 7.2 \n",
       "L 55.553125 7.2 \n",
       "z\n",
       "\" style=\"fill: #ffffff\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <defs>\n",
       "       <path id=\"m4b2ea9684b\" d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m4b2ea9684b\" x=\"71.785852\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- 0.0 -->\n",
       "      <g transform=\"translate(63.83429 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \n",
       "Q 1547 4250 1301 3770 \n",
       "Q 1056 3291 1056 2328 \n",
       "Q 1056 1369 1301 889 \n",
       "Q 1547 409 2034 409 \n",
       "Q 2525 409 2770 889 \n",
       "Q 3016 1369 3016 2328 \n",
       "Q 3016 3291 2770 3770 \n",
       "Q 2525 4250 2034 4250 \n",
       "z\n",
       "M 2034 4750 \n",
       "Q 2819 4750 3233 4129 \n",
       "Q 3647 3509 3647 2328 \n",
       "Q 3647 1150 3233 529 \n",
       "Q 2819 -91 2034 -91 \n",
       "Q 1250 -91 836 529 \n",
       "Q 422 1150 422 2328 \n",
       "Q 422 3509 836 4129 \n",
       "Q 1250 4750 2034 4750 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "        <path id=\"DejaVuSans-2e\" d=\"M 684 794 \n",
       "L 1344 794 \n",
       "L 1344 0 \n",
       "L 684 0 \n",
       "L 684 794 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_2\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m4b2ea9684b\" x=\"136.781757\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- 0.2 -->\n",
       "      <g transform=\"translate(128.830195 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-32\" d=\"M 1228 531 \n",
       "L 3431 531 \n",
       "L 3431 0 \n",
       "L 469 0 \n",
       "L 469 531 \n",
       "Q 828 903 1448 1529 \n",
       "Q 2069 2156 2228 2338 \n",
       "Q 2531 2678 2651 2914 \n",
       "Q 2772 3150 2772 3378 \n",
       "Q 2772 3750 2511 3984 \n",
       "Q 2250 4219 1831 4219 \n",
       "Q 1534 4219 1204 4116 \n",
       "Q 875 4013 500 3803 \n",
       "L 500 4441 \n",
       "Q 881 4594 1212 4672 \n",
       "Q 1544 4750 1819 4750 \n",
       "Q 2544 4750 2975 4387 \n",
       "Q 3406 4025 3406 3419 \n",
       "Q 3406 3131 3298 2873 \n",
       "Q 3191 2616 2906 2266 \n",
       "Q 2828 2175 2409 1742 \n",
       "Q 1991 1309 1228 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m4b2ea9684b\" x=\"201.777662\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_3\">\n",
       "      <!-- 0.4 -->\n",
       "      <g transform=\"translate(193.8261 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-34\" d=\"M 2419 4116 \n",
       "L 825 1625 \n",
       "L 2419 1625 \n",
       "L 2419 4116 \n",
       "z\n",
       "M 2253 4666 \n",
       "L 3047 4666 \n",
       "L 3047 1625 \n",
       "L 3713 1625 \n",
       "L 3713 1100 \n",
       "L 3047 1100 \n",
       "L 3047 0 \n",
       "L 2419 0 \n",
       "L 2419 1100 \n",
       "L 313 1100 \n",
       "L 313 1709 \n",
       "L 2253 4666 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m4b2ea9684b\" x=\"266.773567\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 0.6 -->\n",
       "      <g transform=\"translate(258.822005 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-36\" d=\"M 2113 2584 \n",
       "Q 1688 2584 1439 2293 \n",
       "Q 1191 2003 1191 1497 \n",
       "Q 1191 994 1439 701 \n",
       "Q 1688 409 2113 409 \n",
       "Q 2538 409 2786 701 \n",
       "Q 3034 994 3034 1497 \n",
       "Q 3034 2003 2786 2293 \n",
       "Q 2538 2584 2113 2584 \n",
       "z\n",
       "M 3366 4563 \n",
       "L 3366 3988 \n",
       "Q 3128 4100 2886 4159 \n",
       "Q 2644 4219 2406 4219 \n",
       "Q 1781 4219 1451 3797 \n",
       "Q 1122 3375 1075 2522 \n",
       "Q 1259 2794 1537 2939 \n",
       "Q 1816 3084 2150 3084 \n",
       "Q 2853 3084 3261 2657 \n",
       "Q 3669 2231 3669 1497 \n",
       "Q 3669 778 3244 343 \n",
       "Q 2819 -91 2113 -91 \n",
       "Q 1303 -91 875 529 \n",
       "Q 447 1150 447 2328 \n",
       "Q 447 3434 972 4092 \n",
       "Q 1497 4750 2381 4750 \n",
       "Q 2619 4750 2861 4703 \n",
       "Q 3103 4656 3366 4563 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m4b2ea9684b\" x=\"331.769472\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 0.8 -->\n",
       "      <g transform=\"translate(323.81791 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \n",
       "Q 1584 2216 1326 1975 \n",
       "Q 1069 1734 1069 1313 \n",
       "Q 1069 891 1326 650 \n",
       "Q 1584 409 2034 409 \n",
       "Q 2484 409 2743 651 \n",
       "Q 3003 894 3003 1313 \n",
       "Q 3003 1734 2745 1975 \n",
       "Q 2488 2216 2034 2216 \n",
       "z\n",
       "M 1403 2484 \n",
       "Q 997 2584 770 2862 \n",
       "Q 544 3141 544 3541 \n",
       "Q 544 4100 942 4425 \n",
       "Q 1341 4750 2034 4750 \n",
       "Q 2731 4750 3128 4425 \n",
       "Q 3525 4100 3525 3541 \n",
       "Q 3525 3141 3298 2862 \n",
       "Q 3072 2584 2669 2484 \n",
       "Q 3125 2378 3379 2068 \n",
       "Q 3634 1759 3634 1313 \n",
       "Q 3634 634 3220 271 \n",
       "Q 2806 -91 2034 -91 \n",
       "Q 1263 -91 848 271 \n",
       "Q 434 634 434 1313 \n",
       "Q 434 1759 690 2068 \n",
       "Q 947 2378 1403 2484 \n",
       "z\n",
       "M 1172 3481 \n",
       "Q 1172 3119 1398 2916 \n",
       "Q 1625 2713 2034 2713 \n",
       "Q 2441 2713 2670 2916 \n",
       "Q 2900 3119 2900 3481 \n",
       "Q 2900 3844 2670 4047 \n",
       "Q 2441 4250 2034 4250 \n",
       "Q 1625 4250 1398 4047 \n",
       "Q 1172 3844 1172 3481 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m4b2ea9684b\" x=\"396.765377\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 1.0 -->\n",
       "      <g transform=\"translate(388.813815 287.910438) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-31\" d=\"M 794 531 \n",
       "L 1825 531 \n",
       "L 1825 4091 \n",
       "L 703 3866 \n",
       "L 703 4441 \n",
       "L 1819 4666 \n",
       "L 2450 4666 \n",
       "L 2450 531 \n",
       "L 3481 531 \n",
       "L 3481 0 \n",
       "L 794 0 \n",
       "L 794 531 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_7\">\n",
       "     <!-- 1e−6 -->\n",
       "     <g transform=\"translate(385.415313 300.588563) scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path id=\"DejaVuSans-65\" d=\"M 3597 1894 \n",
       "L 3597 1613 \n",
       "L 953 1613 \n",
       "Q 991 1019 1311 708 \n",
       "Q 1631 397 2203 397 \n",
       "Q 2534 397 2845 478 \n",
       "Q 3156 559 3463 722 \n",
       "L 3463 178 \n",
       "Q 3153 47 2828 -22 \n",
       "Q 2503 -91 2169 -91 \n",
       "Q 1331 -91 842 396 \n",
       "Q 353 884 353 1716 \n",
       "Q 353 2575 817 3079 \n",
       "Q 1281 3584 2069 3584 \n",
       "Q 2775 3584 3186 3129 \n",
       "Q 3597 2675 3597 1894 \n",
       "z\n",
       "M 3022 2063 \n",
       "Q 3016 2534 2758 2815 \n",
       "Q 2500 3097 2075 3097 \n",
       "Q 1594 3097 1305 2825 \n",
       "Q 1016 2553 972 2059 \n",
       "L 3022 2063 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       <path id=\"DejaVuSans-2212\" d=\"M 678 2272 \n",
       "L 4684 2272 \n",
       "L 4684 1741 \n",
       "L 678 1741 \n",
       "L 678 2272 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-31\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-65\" x=\"63.623047\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-2212\" x=\"125.146484\"/>\n",
       "      <use xlink:href=\"#DejaVuSans-36\" x=\"208.935547\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <defs>\n",
       "       <path id=\"m40ae30e6f1\" d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use xlink:href=\"#m40ae30e6f1\" x=\"55.553125\" y=\"237.024\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 0.00096 -->\n",
       "      <g transform=\"translate(7.2 240.823219) scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path id=\"DejaVuSans-39\" d=\"M 703 97 \n",
       "L 703 672 \n",
       "Q 941 559 1184 500 \n",
       "Q 1428 441 1663 441 \n",
       "Q 2288 441 2617 861 \n",
       "Q 2947 1281 2994 2138 \n",
       "Q 2813 1869 2534 1725 \n",
       "Q 2256 1581 1919 1581 \n",
       "Q 1219 1581 811 2004 \n",
       "Q 403 2428 403 3163 \n",
       "Q 403 3881 828 4315 \n",
       "Q 1253 4750 1959 4750 \n",
       "Q 2769 4750 3195 4129 \n",
       "Q 3622 3509 3622 2328 \n",
       "Q 3622 1225 3098 567 \n",
       "Q 2575 -91 1691 -91 \n",
       "Q 1453 -91 1209 -44 \n",
       "Q 966 3 703 97 \n",
       "z\n",
       "M 1959 2075 \n",
       "Q 2384 2075 2632 2365 \n",
       "Q 2881 2656 2881 3163 \n",
       "Q 2881 3666 2632 3958 \n",
       "Q 2384 4250 1959 4250 \n",
       "Q 1534 4250 1286 3958 \n",
       "Q 1038 3666 1038 3163 \n",
       "Q 1038 2656 1286 2365 \n",
       "Q 1534 2075 1959 2075 \n",
       "z\n",
       "\" transform=\"scale(0.015625)\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"159.033203\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"222.65625\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"286.279297\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-36\" x=\"349.902344\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m40ae30e6f1\" x=\"55.553125\" y=\"188.64\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 0.00098 -->\n",
       "      <g transform=\"translate(7.2 192.439219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"159.033203\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"222.65625\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-39\" x=\"286.279297\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-38\" x=\"349.902344\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m40ae30e6f1\" x=\"55.553125\" y=\"140.256\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_10\">\n",
       "      <!-- 0.00100 -->\n",
       "      <g transform=\"translate(7.2 144.055219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"159.033203\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-31\" x=\"222.65625\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"286.279297\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"349.902344\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m40ae30e6f1\" x=\"55.553125\" y=\"91.872\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_11\">\n",
       "      <!-- 0.00102 -->\n",
       "      <g transform=\"translate(7.2 95.671219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"159.033203\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-31\" x=\"222.65625\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"286.279297\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-32\" x=\"349.902344\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use xlink:href=\"#m40ae30e6f1\" x=\"55.553125\" y=\"43.488\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_12\">\n",
       "      <!-- 0.00104 -->\n",
       "      <g transform=\"translate(7.2 47.287219) scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-30\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-2e\" x=\"63.623047\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"95.410156\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"159.033203\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-31\" x=\"222.65625\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-30\" x=\"286.279297\"/>\n",
       "       <use xlink:href=\"#DejaVuSans-34\" x=\"349.902344\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_12\">\n",
       "    <path d=\"M 71.785852 140.256 \n",
       "L 396.440398 140.256 \n",
       "L 396.440398 140.256 \n",
       "\" clip-path=\"url(#p4ee37c269e)\" style=\"fill: none; stroke: #1f77b4; stroke-width: 1.5; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 55.553125 273.312 \n",
       "L 55.553125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 412.673125 273.312 \n",
       "L 412.673125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 55.553125 273.312 \n",
       "L 412.673125 273.312 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 55.553125 7.2 \n",
       "L 412.673125 7.2 \n",
       "\" style=\"fill: none; stroke: #000000; stroke-width: 0.8; stroke-linejoin: miter; stroke-linecap: square\"/>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p4ee37c269e\">\n",
       "   <rect x=\"55.553125\" y=\"7.2\" width=\"357.12\" height=\"266.112\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pyquil.quiltwaveforms import BoxcarAveragerKernel\n",
    "\n",
    "plot_waveform(\n",
    "    BoxcarAveragerKernel(duration=1e-6),\n",
    "    sample_rate=1e9\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The samples should sum to (roughly) one."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "assert np.isclose(\n",
    "    np.sum(BoxcarAveragerKernel(duration=1e-6).samples(1e9)),\n",
    "    1.0\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note**: The reference implementations of these waveforms makes use of Python's double precision floating point arithmetic. The *actual* implementation involves a certain amount of hardware dependence. For example, Rigetti's waveform generation hardware currently makes use of 16 bit fixed point arithmetic."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Compile Time versus Run Time\n",
    "\n",
    "When thinking about parameters in Quil-T, there are two times to consider:\n",
    "* the time of compilation, when a Quil-T program is translated to a binary format executable on Rigetti hardware\n",
    "* the time at which the program is run.\n",
    "\n",
    "**All template parameters must be resolved at compile time.** The Quil-T compiler depends on being able to determine the size and contents of waveforms in advane. In other words, the following is not allowed:\n",
    "```\n",
    "DECLARE theta REAL\n",
    "PULSE 0 \"rf\" flat(duration: 1e-8, iq: 1.0, phase: theta)\n",
    "```\n",
    "However, the following program is valid, and does something equivalent\n",
    "```\n",
    "DECLARE theta REAL\n",
    "SHIFT-PHASE 0 \"rf\" theta\n",
    "PULSE 0 \"rf\" flat(duration: 1e-8, iq: 1.0)\n",
    "SHIFT-PHASE 0 \"rf\" -theta\n",
    "```\n",
    "(Why the second `SHIFT-PHASE`? To leave the frame `0 \"rf\"` as it was before we started!)\n",
    "\n",
    "\n",
    "Explicit control over the *run-time* phase, frequency, or scale requires the use of one of the following instructions.\n",
    "\n",
    "* `SET-SCALE`\n",
    "* `SET-PHASE`\n",
    "* `SHIFT-PHASE`\n",
    "* `SET-FREQUENCY`\n",
    "* `SHIFT-FREQUENCY`\n",
    "\n",
    "all of which support run-time parameter arguments."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.14"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}