Qubit Control.ipynb

qubit-control.ipynb

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        "<a href=\"https://colab.research.google.com/gist/0xPinole/5557ea7b22382840ae2ccefa2f28e8c3/qubit-control.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "source": [
        "## Unraveling Qubit Dynamics and Control for Quantum Gates\n",
        "\n",
        "Quantum computing's potential compels us to grasp the intricacies of qubit dynamics and control—a foundational imperative.\n",
        "\n",
        "The energy of a qubit influenced by an electric field can be elegantly modeled with the following operator:\n",
        "\n",
        "$H = \\frac{\\Delta}{2} Ζ + V(t)X_+ + \\bar{V}(t)X_- $\n",
        "\n",
        "\\\\\n",
        "Here, the constant $\\Delta$ possibly signifies the magnitude of a constant field along the $z$ direction.\n",
        "\n",
        "\\\\\n",
        "However, the Schrödinger equation remains elusive to exact solutions for arbitrary control fields. Yet, in this instance, we offer a solution for the simplified scenario when $V(t) = 0$:\n",
        "$H = \\frac{\\Delta}{2} Ζ$\n",
        "\n",
        "\\\\\n",
        "This foundational insight sets the stage for our exploration."
      ],
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      "source": [
        "## Using the Taylor series of the exponential function\n",
        "In order to find the time-evolution of the qubit the Schrödinger equation\n",
        "\n",
        "$i\\frac{d}{dt}|ψ⟩ = H |ψ⟩$\n",
        "\n",
        "The correspondent time-dependent qubit can be computed as\n",
        "\n",
        "$|ψ (t)⟩ = U(t)|ψ (0)⟩$\n",
        "\n",
        "Here, we aim to espress $U(t)=e^{−iHt}$ using the Taylor series.\n",
        "\n",
        "\\\\\n",
        "The exponential function $e^x$ can be represented as a Taylor series: \\\\\n",
        "$e^x = \\sum_{k=0}^\\infty \\frac{x^k}{k!} $\n",
        "\n",
        "\\\\\n",
        "By substituting $x=−iHt$ at $x$ \\\\\n",
        "$e^x = \\sum_{k=0}^\\infty \\frac{(-iHt)^k}{k!} $\n",
        "\n",
        "\\\\\n",
        "At first glance, the series might seem complex due to the alternating signs and the exponentiated operator $-iHt$. Separating this series into odd and even terms simplifies the expression significantly.\n",
        "\n",
        "\\\\\n",
        "Even terms occurs at $2k$ \\\\\n",
        "This yields: $-i^{2k} = (-i * -i)^{k}= (-1)^{k}$\n",
        "\n",
        "\\\\\n",
        "While odds occurs at $2k+1$ \\\\\n",
        "This yields: $-i^{2k+1} = -i(-1)^{k}$\n",
        "\n",
        "\\\\\n",
        "This separation is driven by the properties of the exponential function and the properties of the terms in the series.\n",
        "\n",
        "As a result of this, the expression transforms into \\\\\n",
        "$e^{-iHt} =\n",
        "\\sum_{n=0}^{\\infty} \\frac{(-1)^{k}H^{2k}t^{2k}}{(2k)!} +\n",
        "\\sum_{n=0}^{\\infty} \\frac{-i(-1)^{k}H^{2k+1}t^{2k+1}}{(2k)!}\n",
        "$\n",
        "\n",
        "\\\\\n",
        "This further leads us to the desired outcome \\\\\n",
        "$U(t) = \\sum_{k=0}^\\infty \\frac{(-1)^{k}t^{2k}}{(2k)!}H^{2k} - i \\sum_{k=0}^\\infty \\frac{(-1)^{k}t^{2k+1}}{(2k+1)!}H^{2k+1} $"
      ],
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      "cell_type": "markdown",
      "source": [
        "## Deriving Matrix Forms of Operators\n",
        "In order to understand the derivation process more comprehensively, let's break down the steps\n",
        "\n",
        "The operator HH is initially given by\n",
        "$H = \\frac{\\Delta}{2} Ζ$\n",
        "\n",
        "\\\\\n",
        "Next, we consider $H^{2k}$\n",
        "\n",
        "$H^{2k} = \\frac{\\Delta^{2k}}{2^{2k}}Z^{2k}$\n",
        "\n",
        "Here $Z^{2k} $ is simply $Z$, , which can be attributed to the periodicity of the Pauli Z matrix:\n",
        "\n",
        "\\\\\n",
        "By substituting\n",
        "$H^{2k} = \\frac{\\Delta^{2k}}{2^{2k}}Z = \\frac{\\Delta^{2k}}{2^{2k}}I$\n",
        "\n",
        "\\\\\n",
        "Finally, the matrix form of $H^{2k}$ is determined by considering the properties of the Pauli Z matrix and identity matrix: \\\\\n",
        "$H^{2k} =\n",
        "\\begin{pmatrix}\n",
        "\\frac{\\Delta^{2k}}{2^{2k}} & 0 \\\\\n",
        "0 & -\\frac{\\Delta^{2k}}{2^{2k}}\n",
        "\\end{pmatrix}$\n",
        "\n",
        "\\\\\n",
        "For the case $H^{2k+1}$ \\\\\n",
        "$H^{2k+1} =\n",
        "\\begin{pmatrix}\n",
        "\\frac{\\Delta^{2k+1}}{2^{2k+1}} & 0 \\\\\n",
        "0 & -\\frac{\\Delta^{2k+1}}{2^{2k+1}}\n",
        "\\end{pmatrix}$"
      ],
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      "source": [
        "## Simplifying the Time-Evolution Operator\n",
        "\n",
        "Given $H^{2k}$ and $H^{2k+1}$\n",
        "we can substitute into the time-evolution operator $U(t)$ and demonstrate its simplified form.\n",
        "\n",
        "We start with the original expression $H^{2k}$ and $H^{2k+1}$\n",
        "\n",
        "$H^{2k} =\n",
        "\\begin{pmatrix}\n",
        "\\frac{\\Delta^{2k}}{2^{2k}} & 0 \\\\\n",
        "0 & -\\frac{\\Delta^{2k}}{2^{2k}}\n",
        "\\end{pmatrix} = \\frac{\\Delta^{2k}}{2^{2k}} \\begin{pmatrix} 1 && 0 \\\\ 0 && -1\\end{pmatrix}$\n",
        "\n",
        "\\\\\n",
        "$H^{2k+1} = \\frac{\\Delta^{2k+1}}{2^{2k+1}} \\begin{pmatrix} 1 && 0 \\\\ 0 && -1\\end{pmatrix}$\n",
        "\n",
        "\\\\\n",
        "And the expression for $U(t)$\n",
        "$U(t) = \\sum_{k=0}^\\infty \\frac{(-1)^{k}t^{2k}}{(2k)!}H^{2k} - i \\sum_{k=0}^\\infty \\frac{(-1)^{k}t^{2k+1}}{(2k+1)!}H^{2k+1} $\n",
        "\n",
        "\n",
        "\\\\\n",
        "With the Maclaurin Series of Trigonometric Functions, we can substitute \\\\\n",
        "\n",
        "$sin(x) = \\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n+1)!}x^{2n+1}$ \\\\\n",
        "$cos(x) = \\sum_{n=0}^{\\infty} \\frac{(-1)^n}{(2n)!}x^{2n}$\n",
        "\n",
        "\\\\\n",
        "Then, we substitute the expressions for $H$ into $U(t)$ as \\\\\n",
        "$U(t) = \\sum_{k=0}^\\infty \\frac{(-1)^{k}(t)^{2k}(\\frac{\\Delta}{2})^{2k}}{(2k)!}\\begin{pmatrix} 1 && 0 \\\\ 0 && -1\\end{pmatrix} - i \\sum_{k=0}^\\infty \\frac{(-1)^{k}(t)^{2k+1}(\\frac{\\Delta}{2})^{2k+1}}{(2k+1)!}\\begin{pmatrix} 1 && 0 \\\\ 0 && -1\\end{pmatrix} $\n",
        "\n",
        "\\\\\n",
        "$U(t) = cos(\\frac{\\Delta}{2}t)\\begin{pmatrix} 1 && 0 \\\\ 0 && -1\\end{pmatrix} - i \\; sin(\\frac{\\Delta}{2}t)\\begin{pmatrix} 1 && 0 \\\\ 0 && -1\\end{pmatrix} $\n",
        "\n",
        "\\\\\n",
        "$U(t) =\n",
        "\\begin{pmatrix}\n",
        "cos(\\frac{\\Delta}{2}t) - i \\; sin(\\frac{\\Delta}{2}t) && 0 \\\\\n",
        "0 && -cos(\\frac{\\Delta}{2}t) + i \\; sin(\\frac{\\Delta}{2}t)\n",
        "\\end{pmatrix} $\n",
        "\n"
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      "source": [
        "## Inital Bloch coordinates\n",
        "\n",
        "The time-dependent qubit corresponding to the system can be computed as\n",
        "\n",
        "$|ψ (t)⟩ = U(t)|ψ (0)⟩$\n",
        "\n",
        "\\\\\n",
        "Where the initial state is given by\n",
        "$|ψ (0)⟩ = a_0 |0⟩ + b_0 |1⟩ $ \\\\\n",
        "With \\\\\n",
        "$\\; a_0 = cos θ_0$  \\\\\n",
        "$\\; b_0 = e^{iφ_0} sin θ_0 $\n",
        "\n",
        "\\\\\n",
        "The operator $U(t)$ is expressed as \\\\\n",
        "$ U(t) =\n",
        "\\begin{pmatrix}\n",
        "cos(\\Delta t / 2) - i \\; sin(\\Delta t / 2) && 0 \\\\\n",
        "0 && cos(\\Delta t / 2) + i \\; sin(\\Delta t / 2)\n",
        "\\end{pmatrix}\n",
        "$\n",
        "\n",
        "\\\\\n",
        "And the state $ψ(0)$ is represented by \\\\\n",
        "$ψ(0) =\n",
        "\\begin{pmatrix}\n",
        "a_0 \\\\\n",
        "b_0\n",
        "\\end{pmatrix} =\n",
        "\\begin{pmatrix}\n",
        "cos(θ_0) \\\\\n",
        "e^{iφ_0} sin(θ_0)\n",
        "\\end{pmatrix}\n",
        "$\n",
        "\n",
        "\\\\\n",
        "From the result, the mapping to Bloch coordinates is obtained by the vector solution \\\\\n",
        "$|ψ⟩ ↦ n̂= (⟨X⟩, ⟨Y ⟩, ⟨Z⟩)$\n",
        "\n",
        "$n_x = ⟨X⟩ = 2Re(\\bar{a}b)$ \\\\\n",
        "$n_y = ⟨Y⟩ = 2IM(\\bar{a}b)$ \\\\\n",
        "$n_z = ⟨Z⟩ = |a|^2 - |b|^2$\n",
        "\n",
        "\\\\\n",
        "As initial conditions, we set \\\\\n",
        "$θ := π /4, \\; φ := 0, \\; Δ := 1$"
      ],
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      "source": [
        "import numpy as np\n",
        "\n",
        "# Set initial conditions\n",
        "t = 0\n",
        "delta = 1\n",
        "\n",
        "# Given initial angles\n",
        "theta_0 = np.pi / 4\n",
        "phi_0 = 0\n",
        "\n",
        "# Calculate initial coefficients a_0 and b_0\n",
        "a_0 = np.cos(theta_0)\n",
        "b_0 = np.exp(1j * phi_0) * np.sin(theta_0)\n",
        "\n",
        "# Define the operator U(t)\n",
        "U_t = np.array([[np.cos((delta * t) / 2) - 1j * np.sin((delta * t) / 2),             0],\n",
        "                [0,             np.cos((delta * t) / 2) + 1j * np.sin((delta * t) / 2)]])\n",
        "\n",
        "# Calculate the time-dependent qubit state |ψ(t)⟩\n",
        "psi_t = np.dot(U_t, np.array([a_0, b_0]))\n",
        "\n",
        "# Calculate the Bloch vector components\n",
        "bloch_x = 2 * np.real(np.conj(psi_t[0]) * psi_t[1])\n",
        "bloch_y = 2 * np.imag(np.conj(psi_t[0]) * psi_t[1])\n",
        "bloch_z = np.abs(psi_t[0])**2 - np.abs(psi_t[1])**2\n",
        "\n",
        "print(\"Time-dependent qubit state |ψ(t)⟩:\", psi_t)\n",
        "print(\"Bloch vector components: n_x =\", bloch_x, \", n_y =\", bloch_y, \", n_z =\", bloch_z)"
      ],
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          "base_uri": "https://localhost:8080/"
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      },
      "execution_count": 112,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Time-dependent qubit state |ψ(t)⟩: [0.70710678+0.j 0.70710678+0.j]\n",
            "Bloch vector components: n_x = 1.0 , n_y = 0.0 , n_z = 2.220446049250313e-16\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Dynamics trayectory in the sphere"
      ],
      "metadata": {
        "id": "KWi-5eT3cosU"
      }
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    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "# Set initial conditions\n",
        "delta = 1\n",
        "\n",
        "# Given initial angles\n",
        "theta_0 = np.pi / 4\n",
        "phi_0 = 0\n",
        "\n",
        "# Pre-calculated coefficients a_0 and b_0\n",
        "a_0 = np.cos(theta_0)\n",
        "b_0 = np.exp(1j * phi_0) * np.sin(theta_0)\n",
        "\n",
        "# Anonymous function for U(t)\n",
        "U = lambda t: np.array([[np.cos((delta * t) / 2) - 1j * np.sin((delta * t) / 2), 0],\n",
        "                        [0, np.cos((delta * t) / 2) + 1j * np.sin((delta * t) / 2)]])\n",
        "\n",
        "# Anonymous function for calculating psi\n",
        "psi = lambda U_t: np.dot(U_t, np.array([a_0, b_0]))\n",
        "\n",
        "# Anonymous function for calculating Bloch vector\n",
        "bloch = lambda a,b: [2*np.real(np.conj(a)*b),\n",
        "                     2*np.imag(np.conj(a)*b),\n",
        "                     np.abs(a)**2 - np.abs(b)**2]\n",
        "\n",
        "def Bloch_psi_t(t):\n",
        "  U_t = U(t)\n",
        "  a,b = psi(U_t)\n",
        "  n = bloch(a, b)\n",
        "\n",
        "  return n\n",
        "\n",
        "print(\"Bloch vector components at t: 0: \", Bloch_psi_t(0))"
      ],
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        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "4bZZu_QGjWF9",
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      },
      "execution_count": 113,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Bloch vector components at t: 0:  [1.0, 0.0, 2.220446049250313e-16]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from matplotlib.patches import Circle\n",
        "from matplotlib.animation import FuncAnimation\n",
        "from IPython.display import HTML\n",
        "\n",
        "# Create subplots for X-Y, X-Z, and Y-Z projections\n",
        "fig, axes = plt.subplots(1, 3, figsize=(8, 4))\n",
        "plt.suptitle('Bloch Sphere Projections', fontsize=12)\n",
        "plt.subplots_adjust(wspace=0.5)\n",
        "\n",
        "# Plot of each Bloch sphere view\n",
        "def plot_bloch_sphere(ax, x_label, y_label, n):\n",
        "    ax.clear()\n",
        "    ax.add_patch(Circle((0, 0), 1, fill=False, color='g', alpha=1))\n",
        "    ax.set_xlim([-1.1, 1.1])\n",
        "    ax.set_ylim([-1.1, 1.1])\n",
        "    ax.set_xlabel(x_label)\n",
        "    ax.set_ylabel(y_label)\n",
        "    ax.set_aspect('equal', adjustable='box')\n",
        "    ax.grid(True, linestyle='-', linewidth=0.5, alpha=0.5)\n",
        "    ax.quiver(0, 0, n[0], n[1], angles='xy', scale_units='xy', scale=1, color='k')\n",
        "\n",
        "# Function to update the plot for each animation frame\n",
        "def update(frame):\n",
        "  t = frame / ratio_frames\n",
        "\n",
        "  n_vector = Bloch_psi_t(t)\n",
        "  # X-Y, X-Z and Y-Z plot projection\n",
        "  plot_bloch_sphere(axes[0], \"X: (+-)\", \"Y: (±i)\", [n_vector[0], n_vector[1]])\n",
        "  plot_bloch_sphere(axes[1], \"X: (+-)\", \"Z: (01)\", [n_vector[0], n_vector[2]])\n",
        "  plot_bloch_sphere(axes[2], \"Y: (±i)\", \"Z: (01)\", [n_vector[1], n_vector[2]])\n",
        "\n",
        "# Set the animation parameters\n",
        "ratio_frames = 10  # Adjust as needed\n",
        "animation_duration_s = 10\n",
        "frames_total = animation_duration_s * ratio_frames\n",
        "interval_frame = 1000 / ratio_frames\n",
        "\n",
        "# Create the animation\n",
        "animation = FuncAnimation(fig, update, frames=frames_total, interval=interval_frame)\n",
        "\n",
        "# Set up the HTML video player (colab.research required)\n",
        "video = HTML(animation.to_jshtml())\n",
        "display(video)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 832
        },
        "id": "SjEUrxfQV1CC",
        "outputId": "467e243c-f729-4255-bba9-f253b3580aa8"
      },
      "execution_count": 114,
      "outputs": [
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              "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
              "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
              "  }\n",
              "\n",
              "  /* Define the Animation class */\n",
              "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
              "    this.img_id = img_id;\n",
              "    this.slider_id = slider_id;\n",
              "    this.loop_select_id = loop_select_id;\n",
              "    this.interval = interval;\n",
              "    this.current_frame = 0;\n",
              "    this.direction = 0;\n",
              "    this.timer = null;\n",
              "    this.frames = new Array(frames.length);\n",
              "\n",
              "    for (var i=0; i<frames.length; i++)\n",
              "    {\n",
              "     this.frames[i] = new Image();\n",
              "     this.frames[i].src = frames[i];\n",
              "    }\n",
              "    var slider = document.getElementById(this.slider_id);\n",
              "    slider.max = this.frames.length - 1;\n",
              "    if (isInternetExplorer()) {\n",
              "        // switch from oninput to onchange because IE <= 11 does not conform\n",
              "        // with W3C specification. It ignores oninput and onchange behaves\n",
              "        // like oninput. In contrast, Microsoft Edge behaves correctly.\n",
              "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
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              "  Animation.prototype.set_frame = function(frame){\n",
              "    this.current_frame = frame;\n",
              "    document.getElementById(this.img_id).src =\n",
              "            this.frames[this.current_frame].src;\n",
              "    document.getElementById(this.slider_id).value = this.current_frame;\n",
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              "  {\n",
              "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
              "  }\n",
              "\n",
              "  Animation.prototype.previous_frame = function()\n",
              "  {\n",
              "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
              "  }\n",
              "\n",
              "  Animation.prototype.first_frame = function()\n",
              "  {\n",
              "    this.set_frame(0);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.last_frame = function()\n",
              "  {\n",
              "    this.set_frame(this.frames.length - 1);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.slower = function()\n",
              "  {\n",
              "    this.interval /= 0.7;\n",
              "    if(this.direction > 0){this.play_animation();}\n",
              "    else if(this.direction < 0){this.reverse_animation();}\n",
              "  }\n",
              "\n",
              "  Animation.prototype.faster = function()\n",
              "  {\n",
              "    this.interval *= 0.7;\n",
              "    if(this.direction > 0){this.play_animation();}\n",
              "    else if(this.direction < 0){this.reverse_animation();}\n",
              "  }\n",
              "\n",
              "  Animation.prototype.anim_step_forward = function()\n",
              "  {\n",
              "    this.current_frame += 1;\n",
              "    if(this.current_frame < this.frames.length){\n",
              "      this.set_frame(this.current_frame);\n",
              "    }else{\n",
              "      var loop_state = this.get_loop_state();\n",
              "      if(loop_state == \"loop\"){\n",
              "        this.first_frame();\n",
              "      }else if(loop_state == \"reflect\"){\n",
              "        this.last_frame();\n",
              "        this.reverse_animation();\n",
              "      }else{\n",
              "        this.pause_animation();\n",
              "        this.last_frame();\n",
              "      }\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.anim_step_reverse = function()\n",
              "  {\n",
              "    this.current_frame -= 1;\n",
              "    if(this.current_frame >= 0){\n",
              "      this.set_frame(this.current_frame);\n",
              "    }else{\n",
              "      var loop_state = this.get_loop_state();\n",
              "      if(loop_state == \"loop\"){\n",
              "        this.last_frame();\n",
              "      }else if(loop_state == \"reflect\"){\n",
              "        this.first_frame();\n",
              "        this.play_animation();\n",
              "      }else{\n",
              "        this.pause_animation();\n",
              "        this.first_frame();\n",
              "      }\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.pause_animation = function()\n",
              "  {\n",
              "    this.direction = 0;\n",
              "    if (this.timer){\n",
              "      clearInterval(this.timer);\n",
              "      this.timer = null;\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.play_animation = function()\n",
              "  {\n",
              "    this.pause_animation();\n",
              "    this.direction = 1;\n",
              "    var t = this;\n",
              "    if (!this.timer) this.timer = setInterval(function() {\n",
              "        t.anim_step_forward();\n",
              "    }, this.interval);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.reverse_animation = function()\n",
              "  {\n",
              "    this.pause_animation();\n",
              "    this.direction = -1;\n",
              "    var t = this;\n",
              "    if (!this.timer) this.timer = setInterval(function() {\n",
              "        t.anim_step_reverse();\n",
              "    }, this.interval);\n",
              "  }\n",
              "</script>\n",
              "\n",
              "<style>\n",
              ".animation {\n",
              "    display: inline-block;\n",
              "    text-align: center;\n",
              "}\n",
              "input[type=range].anim-slider {\n",
              "    width: 374px;\n",
              "    margin-left: auto;\n",
              "    margin-right: auto;\n",
              "}\n",
              ".anim-buttons {\n",
              "    margin: 8px 0px;\n",
              "}\n",
              ".anim-buttons button {\n",
              "    padding: 0;\n",
              "    width: 36px;\n",
              "}\n",
              ".anim-state label {\n",
              "    margin-right: 8px;\n",
              "}\n",
              ".anim-state input {\n",
              "    margin: 0;\n",
              "    vertical-align: middle;\n",
              "}\n",
              "</style>\n",
              "\n",
              "<div class=\"animation\">\n",
              "  <img id=\"_anim_imgc8b4f3a7b64c438b89decf1535ed2eab\">\n",
              "  <div class=\"anim-controls\">\n",
              "    <input id=\"_anim_sliderc8b4f3a7b64c438b89decf1535ed2eab\" type=\"range\" class=\"anim-slider\"\n",
              "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
              "           oninput=\"animc8b4f3a7b64c438b89decf1535ed2eab.set_frame(parseInt(this.value));\">\n",
              "    <div class=\"anim-buttons\">\n",
              "      <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"animc8b4f3a7b64c438b89decf1535ed2eab.slower()\">\n",
              "          <i class=\"fa fa-minus\"></i></button>\n",
              "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"animc8b4f3a7b64c438b89decf1535ed2eab.first_frame()\">\n",
              "        <i class=\"fa fa-fast-backward\"></i></button>\n",
              "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"animc8b4f3a7b64c438b89decf1535ed2eab.previous_frame()\">\n",
              "          <i class=\"fa fa-step-backward\"></i></button>\n",
              "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"animc8b4f3a7b64c438b89decf1535ed2eab.reverse_animation()\">\n",
              "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
              "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"animc8b4f3a7b64c438b89decf1535ed2eab.pause_animation()\">\n",
              "          <i class=\"fa fa-pause\"></i></button>\n",
              "      <button title=\"Play\" aria-label=\"Play\" onclick=\"animc8b4f3a7b64c438b89decf1535ed2eab.play_animation()\">\n",
              "          <i class=\"fa fa-play\"></i></button>\n",
              "      <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"animc8b4f3a7b64c438b89decf1535ed2eab.next_frame()\">\n",
              "          <i class=\"fa fa-step-forward\"></i></button>\n",
              "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"animc8b4f3a7b64c438b89decf1535ed2eab.last_frame()\">\n",
              "          <i class=\"fa fa-fast-forward\"></i></button>\n",
              "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"animc8b4f3a7b64c438b89decf1535ed2eab.faster()\">\n",
              "          <i class=\"fa fa-plus\"></i></button>\n",
              "    </div>\n",
              "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_selectc8b4f3a7b64c438b89decf1535ed2eab\"\n",
              "          class=\"anim-state\">\n",
              "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_c8b4f3a7b64c438b89decf1535ed2eab\"\n",
              "             >\n",
              "      <label for=\"_anim_radio1_c8b4f3a7b64c438b89decf1535ed2eab\">Once</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_c8b4f3a7b64c438b89decf1535ed2eab\"\n",
              "             checked>\n",
              "      <label for=\"_anim_radio2_c8b4f3a7b64c438b89decf1535ed2eab\">Loop</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_c8b4f3a7b64c438b89decf1535ed2eab\"\n",
              "             >\n",
              "      <label for=\"_anim_radio3_c8b4f3a7b64c438b89decf1535ed2eab\">Reflect</label>\n",
              "    </form>\n",
              "  </div>\n",
              "</div>\n",
              "\n",
              "\n",
              "<script language=\"javascript\">\n",
              "  /* Instantiate the Animation class. */\n",
              "  /* The IDs given should match those used in the template above. */\n",
              "  (function() {\n",
              "    var img_id = \"_anim_imgc8b4f3a7b64c438b89decf1535ed2eab\";\n",
              "    var slider_id = \"_anim_sliderc8b4f3a7b64c438b89decf1535ed2eab\";\n",
              "    var loop_select_id = \"_anim_loop_selectc8b4f3a7b64c438b89decf1535ed2eab\";\n",
              "    var frames = new Array(100);\n",
              "    \n",
              "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAGQCAYAAABWJQQ0AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
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