Basic_Circuit_Identities.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "colab_type": "text",
        "id": "view-in-github"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/MonitSharma/Learn-Quantum-Computing-with-Qiskit/blob/main/Basic_Circuit_Identities.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3bsZWDvWTKTq"
      },
      "source": [
        "# Basic Circuit Identities\n",
        "\n",
        "In the realm of quantum computing, our objective is to construct valuable quantum circuits utilizing fundamental building blocks. However, there may be instances where we lack certain desired basic building blocks. In this section, we will explore methods for converting between basic gates and utilizing them to construct relatively straightforward yet slightly more intricate gates.\n",
        "\n",
        "The techniques covered in this chapter draw upon the pioneering work of Barenco and collaborators, initially introduced in their influential 1995 paper. These techniques offer valuable insights into transforming gates and assembling them to expand our repertoire of quantum circuit components. By harnessing these methods, we can navigate the constraints posed by limited gate availability and continue to develop sophisticated quantum circuits."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "id": "AOONanl2TFTz"
      },
      "outputs": [],
      "source": [
        "from qiskit import QuantumCircuit\n",
        "from qiskit.circuit import Gate\n",
        "from math import pi\n",
        "qc = QuantumCircuit(2)\n",
        "c = 0\n",
        "t = 1"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sz7C7KUpTeXJ"
      },
      "source": [
        "##  Making a Controlled-Z from a CNOT\n",
        "\n",
        "The Controlled Z gate is a well used two qubit gate. Just as CNOT applies an $X$ gate to its target qubit whenever the control is $|1\\rangle$ , the controlled Z gate applies a $Z$ gate in the same case.\n",
        "\n",
        "Let's see how to do this in Qiskit:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "4ydAMP8ETyFx",
        "outputId": "6d46cd4f-d0a0-4a70-b754-24861bbadd94"
      },
      "outputs": [
        {
          "data": {
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            "text/plain": [
              "<Figure size 203.683x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 2,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# a controlled-Z\n",
        "qc.cz(c,t)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zfK00LvlT6Ex"
      },
      "source": [
        "where c and t are the control and target qubits. In IBM Q devices, however, the only kind of two-qubit gate that can be directly applied is the CNOT. We therefore need a way to transform one to the other."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "LUqQnz0_T8SY"
      },
      "source": [
        "The process for this is quite simple. We know that the Hadamard transforms the states $|0\\rangle$ and $|1\\rangle$ to the states $|+\\rangle$ and $|-\\rangle$ respectively. We also know that the effect of the $Z$ gate on the states $|+\\rangle$ and  $|-\\rangle$ is the same as that for $X$ on the states $|0\\rangle$ and $|1\\rangle$ respectively. From this reasoning, or from simply multiplying matrices, we find that:\n",
        "\n",
        "![Screenshot 2023-07-18 at 17-24-49 Basic Circuit Identities.png](data:image/png;base64,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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "MQ7_cOqEUT8Q"
      },
      "source": [
        "The same trick can be used to transform a CNOT into a controlled-$Z$. All we need to do is precede and follow the CNOT with a Hadamard on the target qubit. This will transform any $X$ applied to that qubit into a $Z$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "sY7KHnvZT1JB",
        "outputId": "78a1bb14-ffa9-4165-d1f0-a31e176b1892"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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h9AN4/1f5aLH0orTiIg7/sRkNV77AyIj9y2bCZwbgkYf02PrUYizPmedS/Mi7GDFSpEh9MF4tzMCrhRno67+FMxc7cLmlG4M3h+Hnq0ZoiD/SF4cjam4wf0dBcIwYKd6MID98NX0Ovpou75cCknfwnBgRCY0RIyKhMWJEJDRGjIiEprLxKxumHJvNBoj2Q6f8tXS34j7gOEaMiITGl5NEJDRGjIiExogRkdAYMSISGiNGREJjxIhIaIwYEQmNESMioTFiRCQ0RoyIhMaIEZHQGDEiEhojRkRCY8SISGiMGBEJjREjIqExYkQkNEaMiITGiBGR0BgxIhIaI0ZEQmPEiEhojBgRCY0RIyKhMWJEJLT/B6jsXIl+cAj2AAAAAElFTkSuQmCC",
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 3,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "# also a controlled-Z\n",
        "qc.h(t)\n",
        "qc.cx(c,t)\n",
        "qc.h(t)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0EWKDVCcUdbI"
      },
      "source": [
        "More generally, we can transform a single CNOT into a controlled version of any rotation around the Bloch sphere by an angle $\\pi$, by simply preceding and following it with the correct rotations. For example, a controlled-$Y$:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "dkhwwCkTUaBq",
        "outputId": "d2f4fb1d-283d-427c-ff2d-9ffea8979ba3"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 4,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "# a controlled-Y\n",
        "qc.sdg(t)\n",
        "qc.cx(c,t)\n",
        "qc.s(t)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "gusSICKMUkzZ"
      },
      "source": [
        "and a controlled-$H$:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "0UvJBieRUjRS",
        "outputId": "095984c4-61f0-4383-e900-120cbf887390"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 5,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "# a controlled-H\n",
        "qc.ry(pi/4,t)\n",
        "qc.cx(c,t)\n",
        "qc.ry(-pi/4,t)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4Ur7caW5UpgH"
      },
      "source": [
        "## Swapping Qubits"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {
        "id": "n0Hb0gL7Unqx"
      },
      "outputs": [],
      "source": [
        "a = 0\n",
        "b = 1"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "eUmwqv8-Ut4p"
      },
      "source": [
        "Sometimes we need to move information around in a quantum computer. For some qubit implementations, this could be done by physically moving them. Another option is simply to move the state between two qubits. This is done by the SWAP gate."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "QCEe59X4UsQR",
        "outputId": "b7877fc4-1856-4de3-9be5-b356b70b447d"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 203.683x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 7,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "# swaps states of qubits a and b\n",
        "qc.swap(a,b)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "PZLohq3nUzqA"
      },
      "source": [
        "The command above directly invokes this gate, but let's see how we might make it using our standard gate set. For this, we'll need to consider a few examples."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-no4T8BjU1YI"
      },
      "source": [
        "First, we'll look at the case that qubit a is in state $|1\\rangle$ and qubit b is in state $|0\\rangle$. For this we'll apply the following gates:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "bWS8UsslUwub",
        "outputId": "30fb2227-a461-44fa-8138-34294bf493d2"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 287.294x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 8,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "# swap a 1 from a to b\n",
        "qc.cx(a,b) # copies 1 from a to b\n",
        "qc.cx(b,a) # uses the 1 on b to rotate the state of a to 0\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tqtO7F6GVBoA"
      },
      "source": [
        "This has the effect of putting qubit b in state $|1\\rangle$ and qubit a in state $|0\\rangle$. In this case at least, we have done a SWAP."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "zASNj6WIVHEQ"
      },
      "source": [
        "Now let's take this state and SWAP back to the original one. As you may have guessed, we can do this with the reverse of the above process:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 9,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "ZIh0_7kuU8tx",
        "outputId": "24e09221-e40f-4f6b-d565-6fce3f491829"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 454.517x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 9,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "# swap a q from b to a\n",
        "qc.cx(b,a) # copies 1 from b to a\n",
        "qc.cx(a,b) # uses the 1 on a to rotate the state of b to 0\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "6Seu1zFHVMc5"
      },
      "source": [
        "Note that in these two processes, the first gate of one would have no effect on the initial state of the other. For example, when we swap the $|1\\rangle$ b to a, the first gate is cx(b,a). If this were instead applied to a state where no $|1\\rangle$ was initially on b, it would have no effect.\n",
        "\n",
        "\n",
        "Note also that for these two processes, the final gate of one would have no effect on the final state of the other. For example, the final cx(b,a) that is required when we swap the $|1\\rangle$ from a to b has no effect on the state where the $|1\\rangle$ is not on b.\n",
        "\n",
        "With these observations, we can combine the two processes by adding an ineffective gate from one onto the other. For example,"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 10,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "dg6cBHDfVJ0y",
        "outputId": "68f534c8-d233-4a9f-c35a-3df2c068dbdb"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 10,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "qc.cx(b,a)\n",
        "qc.cx(a,b)\n",
        "qc.cx(b,a)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "YbN5t8hwW_qD"
      },
      "source": [
        "We can think of this as a process that swaps a $|1\\rangle$ from a to b, but with a useless `qc.cx(b,a)` at the beginning. We can also think of it as a process that swaps a $|1\\rangle$ from `b` to `a`, but with a useless `qc.cx(b,a)` at the end. Either way, the result is a process that can do the swap both ways around.\n",
        "\n",
        "It also has the correct effect on the $|00\\rangle$ state. This is symmetric, and so swapping the states should have no effect. Since the CNOT gates have no effect when their control qubits are $|0\\rangle$, the process correctly does nothing.\n",
        "\n",
        "The $|11\\rangle$ state is also symmetric, and so needs a trivial effect from the swap. In this case, the first CNOT gate in the process above will cause the second to have no effect, and the third undoes the first. Therefore, the whole effect is indeed trivial.\n",
        "\n",
        "We have thus found a way to decompose SWAP gates into our standard gate set of single-qubit rotations and CNOT gates."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "ES8_IcLJVeYZ",
        "outputId": "51fbce3c-883f-4779-a5fe-b755037382c0"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 11,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "# swaps states of qubits a and b\n",
        "qc.cx(b,a)\n",
        "qc.cx(a,b)\n",
        "qc.cx(b,a)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "M_YPYRqJXXOh"
      },
      "source": [
        "It works for the states $|00\\rangle$,$|01\\rangle$ ,$|10\\rangle$ and $|11\\rangle$, and if it works for all the states in the computational basis, it must work for all states generally. This circuit therefore swaps all possible two-qubit states.\n",
        "\n",
        "The same effect would also result if we changed the order of the CNOT gates:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 12,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "kU623nrfXU9e",
        "outputId": "f3827601-b2a6-4b56-fd03-aa95d002f3f7"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 12,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "# swaps states of qubits a and b\n",
        "qc.cx(a,b)\n",
        "qc.cx(b,a)\n",
        "qc.cx(a,b)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FE3ObjpHXjnR"
      },
      "source": [
        "This is an equally valid way to get the SWAP gate.\n",
        "\n",
        "The derivation used here was very much based on the z basis states, but it could also be done by thinking about what is required to swap qubits in states $|+\\rangle$ and $|-⟩$. The resulting ways of implementing the SWAP gate will be completely equivalent to the ones here."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "m1hWdM9OXpwB"
      },
      "source": [
        "## Controlled Rotations\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xCBR6mapyY-w"
      },
      "source": [
        "We have already seen how to build controlled $\\pi$ rotations from a single CNOT gate. Now we'll look at how to build any controlled rotation.\n",
        "\n",
        "First, let's consider arbitrary rotations around the y axis. Specifically, consider the following sequence of gates."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 13,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "tIHgD6fZXh5T",
        "outputId": "5db2eb6b-ad26-4115-fb2a-d07f36a99c78"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 454.517x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 13,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "theta = pi # theta can be anything (pi chosen arbitrarily)\n",
        "qc.ry(theta/2,t)\n",
        "qc.cx(c,t)\n",
        "qc.ry(-theta/2,t)\n",
        "qc.cx(c,t)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xaxVAt19yg4e"
      },
      "source": [
        "If the control qubit is in state $|0⟩$ all we have here is $R_y(\\theta/2)$\n",
        "immediately followed by its inverse,$R_y(-\\theta/2)$\n",
        ". The end effect is trivial. If the control qubit is in state $|1⟩$, however, the `ry(-theta/2)` is effectively preceded and followed by an X gate. This has the effect of flipping the direction of the y rotation and making a second $R_y(\\theta/2)$\n",
        ". The net effect in this case is therefore to make a controlled version of the rotation $R_y(\\theta)$\n",
        ".\n",
        "\n",
        "This method works because the x and y axis are orthogonal, which causes the x gates to flip the direction of the rotation. It therefore similarly works to make a controlled $R_z(\\theta)$\n",
        ". A controlled $R_z(\\theta)$\n",
        "could similarly be made using CNOT gates.\n",
        "\n",
        "We can also make a controlled version of any single-qubit rotation, V . For this we simply need to find three rotations A, B and C, and a phase $\\alpha$ such that\n",
        "\n",
        "\n",
        "![Screenshot 2023-07-21 at 17-33-25 Basic Circuit Identities.png](data:image/png;base64,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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SXJv3CO6zDQX"
      },
      "source": [
        "We then use controlled-Z gates to cause the first of these relations to happen whenever the control is in state $|0⟩$, and the second to happen when the control in state $|1⟩$ . An $R_z(2\\alpha)$\n",
        "rotation is also used on the control to get the right phase, which will be important whenever there are superposition states."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 14,
      "metadata": {
        "id": "GH9Bi5GzzB8x"
      },
      "outputs": [],
      "source": [
        "A = Gate('A', 1, [])\n",
        "B = Gate('B', 1, [])\n",
        "C = Gate('C', 1, [])\n",
        "alpha = 1 # arbitrarily define alpha to allow drawing of circuit"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 15,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "hkACgPC5zV5G",
        "outputId": "370c5be6-aa2e-4896-d3a2-2237fbb2d660"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 538.128x200.667 with 1 Axes>"
            ]
          },
          "execution_count": 15,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(2)\n",
        "qc.append(C, [t])\n",
        "qc.cz(c,t)\n",
        "qc.append(B, [t])\n",
        "qc.cz(c,t)\n",
        "qc.append(A, [t])\n",
        "qc.p(alpha,c)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "klhtLCBezZYV"
      },
      "source": [
        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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ivhynKI6zb3-"
      },
      "source": [
        "Here A, B and C are gates that implement A,B and C, respectively."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "V0aQsHWQzhEV"
      },
      "source": [
        "## The Toffoli\n",
        "\n",
        "\n",
        "The Toffoli gate is a three-qubit gate with two controls and one target. It performs an X on the target only if both controls are in the state $|1⟩$ . The final state of the target is then equal to either the AND or the NAND of the two controls, depending on whether the initial state of the target was $|0⟩$ or $|1⟩$. A Toffoli can also be thought of as a controlled-controlled-NOT, and is also called the CCX gate.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 16,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 255
        },
        "id": "8E823Op7zXr_",
        "outputId": "f5c331ff-6333-42d4-a1b1-7449ac64ae9a"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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",
            "text/plain": [
              "<Figure size 203.885x284.278 with 1 Axes>"
            ]
          },
          "execution_count": 16,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(3)\n",
        "a = 0\n",
        "b = 1\n",
        "t = 2\n",
        "# Toffoli with control qubits a and b and target t\n",
        "qc.ccx(a,b,t)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FgIEjgCxztFW"
      },
      "source": [
        "To see how to build it from single- and two-qubit gates, it is helpful to first show how to build something even more general: an arbitrary controlled-controlled-U for any single-qubit rotation U. For this we need to define controlled versions of $V = \\sqrt{U}$ and $V^{\\dagger}$\n",
        " . In the code below, we use cp(theta,c,t) and cp(-theta,c,t)in place of the undefined subroutines cv and cvdg respectively. The controls are qubits a and b, and the target is qubit t."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 17,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 255
        },
        "id": "djp2tW7PzrsG",
        "outputId": "9b39c14b-af2d-4655-b934-6a8a30024f15"
      },
      "outputs": [
        {
          "data": {
            "image/png": 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/n8GDB9PQ0EBGRgYJCQYell7AuiPw863QVu7UO5rarTvin7q84c8r/L8Dg6ar2FdvKgvItk30QSn86jNo66DxlAPmboEtdv/UFQzeWV1CTa3/z02crnfwxvsH/b5dubCgDfLi4mIKCgpITk5m/vz5LbbJyckBYPDgwWe/dyb4hw8fTocOHbBYLH6p15sanTB/R9NDD1zhAH6zo6mfCTYF8FaYzbt1jtAVpxthwU7X2zc4m8agHk/lmkCeq9ataO1P0Ab5kiVLcDgcTJ06lbi4uBbbdOzYEWge5F988QVvv/02qampDBs2zC+1etu6I3DEzdkvWy2sN+Co3Ol0smV34A7dArltk3xU3nRxmztKqmHrMd/UE2y2FAVuHG4N4LalZUEb5KtXrwZg1KhRrbaxWq1A8yC/9tprKS8vZ9myZYwZM8a3RfrI8i896/fuYe/W4Qs2ey3Hv/bdPbNt2b2/MmDbNslyD8fSux6O3VCz+4vKgG27+OAJPVehnQnaq9YPHWq6NalHjx4tLm9oaGD9+vVA8yAPC/P+e5vc3FxsNpvX19uazg8tJ7LHELf7rdpQyJKJ471fkBc1hCVB4qxWl29eMuGCt9WkJnc8+//hD+5stZ3NfpJh31923vcPlJSSlpbmRsWhKfm/PiYitY/b/ZZ+tJ6XJtzhg4qChxM4mfRYq8t9/RpwOJyk9+hFGGbew9pl0SEsYeE0OhpJS2s5HwIhNTWVLVu2eNQ3aIO8pqbpnt/a2pbnmAsKCrDb7cTHx5OZmenTWmw2G6Wl/rs0PP7UaTz5zKL606f8WqdHouohsfXFZ+6RbUtEeJhL7b7N0djY/n9H7cAl9Q0e7VxO1dXp9+uKpNYX+fo1AFBeVgbOwM2MXYwuZy7EcDqDZqwFbZCnpqZy/Phxtm3bxsiRI5stKy8vZ86cOQAMGjTI5xe0paam+nT93xZe49k5rPBqO927d/dyNd7VaInjQnMbNvvJC/ZPTe5IRHgYDY0ObPbWLyRobT3hYY2ktvPfUXsQVnUE6O92v8iTx9r9GGwPypyncFo6tLjM168BnA66dbsUC4ZemXhmf2+xtKuxdjE5EbRBPmbMGIqLi1mwYAFjx46lb9++AGzevJlp06ZhtzeFnT8eBOPpdImn1trgZ5vc7/fc9Dyu+YXV+wV5kdPpJPnaP1NxouVHs7Y0FXiuwx/cSVqXWGz2WtLH/sXt7d809gre+V37/h21B3+3wn9uc7/fa7MnkzN/svcLCjJXTlnW6t0bvn4NDOjTmZ1LDbigphVnPsY0PCz87HVSpgvai93y8/Pp3Lkzhw8fJjs7m4EDB9KnTx+GDx9Oz549GT16NND8/HiwuKoLdO3oXp9uMTDyUt/U400Wi4WcrM4B234gt22S0V0hKcq9Pj3j4Qr9el0S2NdAcsC2LS0L2iBPS0tj7dq15OXlER0dTUlJCUlJSSxevJgVK1awb98+IDiDPNwCjwyGMBfPGIRZ4OFBTf1McOXAwL3jGD4gJWDbNklUOMwdBK4Oqciwf7U3ZAwGWmBfAwry9iZogxygf//+LF++nKqqKqqqqti4cSM//vGPqampoaSkhLCwMAYMGBDoMn1i5KXweA5EtfEXjgpramfC0fgZP8jrFZDtdr80htHDuwVk2yYa3Q3mDW37DWJ0OPx2mI7G3THxez2I7ej/M6MdosKZ/G89/b5dubCgPUd+Ibt378bpdNK3b19iYs6/TeOtt94CoKioqNnXGRkZ5Obm+q/QizSmG/SKhzcOworDTc+2PtekDJicCZmGfWBFv8xExozoxoef+vdxqT+ZdDkREUH93tfrxqdDvwR4owRWWps/8z8uAm66DCZnQHrLz2ySViTERTFtfG+ee3OPX7c7+fpMkjtF+3Wb0raQDPKdO5ueHdnatPqkSZNa/Pruu+/m1Vdf9Wlt3pYZDw8NghlZTZ9C9e//bLoP1ULT90314A+y/RrkMdER/OjWfn7bXjDpcwk8OhhmZsHold+Mv5XXQwAOKoPGT6dk8cLSvTT68dnKM6dm+W1b4rqQPLxoK8idTmeL/0wL8XPFRkBO8jfnLE0/FZl37WVMut639/+f6zcP5tI1xb3Pb5bm4iObjz+F+MXJ6tWJ/Hv892585pQscrN1jUh7pCAXYz3zyEi3p/ls9pNYj9S0ea/tub6bm8oDd+pIRNqfedOHkt0r0a0+nrwGeqXH8/hMc04rhhqL06nPGwolZ+6hDAM2TQh0NRfvo0/LuPGBVZz20eeEp6fGsv618aSn6iSuNwTb+GsPdn9xnGt/uKLVZytcrPjYSNa8dCNXBMltZ8E4BkPyiFyCx/dGdOPNhaOJivT+UE7rEsuHz49TiEu7lt27E3//w7/RKcHNG/ddkBAXyXvPXB80IR6sFORivAmjerDquRvodqn3zmEPH5DCuj/m0TfjEq+tU8RXhg1I4ZNX8ujfM9Fr6+zTI4E1L93INVf49xHT4j4FuQSF64Z1ZdfSW7nnZvc/cetcHaLCWfDgMNa/Np4e3Qy7L09C2oA+SWwruJm59w0izNWnQbXAYoGfTcum8I2JDO2vI3ETKMglaHRK6MArv7yWdX8czx03ZBIR4frOLDE+ip9Ny2b3O7eSf+8g3S8uRoruEMH8WcP47M1b+Mmky916aEx0h3DundiXbQW38N9zRhCj2wqMoYvdQkwwXujRGpv9JG9/WMKW3Xa2FtnZc/AE9Q1NF8XFdoxgUN8kcrKSGTEohYmjM7Tj8oNQGn/twYmq07z9YQmbdn3F1iI7u744Tt2ppqfydIgKJ7tXIjlZyQzLTua2sZkkXdLyJ6oFk2Acg9pzSdBKTY4577axhgYHYWGWi5p6FDHFJfFR3DuxL/dO7Hv2e42NTW9mw8M16xQsFOQSUjRlLqFOAR589BcVERExmIJcRETEYApyERERgynIRUREDKYgFxERMZiCXERExGAKchEREYMpyKVFGRkZ9OvXjyFDhpCVlcUzzzzTattJkyaxYcOGNtf5ne98h4MHD3qzTAlivhiDrtA4lTNM2Q8qyKVVBQUFFBYWsnLlSh555BF27NhxXptNmzZRUVHByJEj21zf7NmzmTdvni9KlSDl7THoCo1TOZcJ+0EFubSpR48e9OvXj3379p23bPHixUyZMgWAyspK0tLS6Ny5M0OGDKF///506NCBH/3oRwDk5eWxcuVKTpw44df6xXyujkFXaJyKJ9rzflBBLm3auXMne/bsYfDgwectW7NmDVdeeSUAiYmJTJkyhQcffJDCwkJ+97vfMWLECF588UUAIiMjGThwIGvXrvVr/WI+V8egKzROxRPteT+oZ61Lq+644w46duxITEwML7/8Mn36nP9Z31arlS5dupz9urCwkJkzZwKwdetWhg4d2qx9amoqVqvVt4VL0PBkDI4cOZLPP/+8xfVt376d9PR0jVNxmQn7QQW5tKqgoIAhQ4ZcsE1MTAx1dXVnvy4sLDw7aLdu3cqECc0/J7Curo6OHTt6vVYJTp6MQVcuONI4FVeZsB/U1LpclEGDBrF3714ASktLsVgsdO/eHYAdO3YwcODAZu2Li4tbnJoS8dS5Y9AVGqfibYHeDyrI5aLcfvvtrFq1Cmiatjx3CikxMZFnn3327NclJSU0NjZqBylede4YdIXGqXhboPeDFqfT6fTa2qTdG74MHDS9g9s0oa3Wbauuruaqq65iw4YNxMbGXrDt3Llz6d2799mrNyX0eHv8gXtj0BUap8Et0GPQF+NLR+RyUeLi4njyySddesBBt27duPfee/1QlYQSd8agKzROxV2B3g/qiDzE+OLdqIirNP4k0IJxDOqIXERExGAKchEREYMpyEVERAymIBcRETGYglxERMRgCnIRERGDKchFREQMpiAXERExmIJcRETEYApyERERgynIRUREDKYgFxERMZiCXERExGAKchEREYMpyEVERAymIBcRETGYglxERMRgCnIRERGDKchFREQMpiAXERExmIJcRETEYCER5Ha7nfz8fHr37k10dDTp6enMmjWLmpoa7rvvPiwWC4sWLQp0mT7hdMKOCvj5Nsh7Hxz/+r6Dpu87nYGsTkKB0wnbjsHDW5qPv8e2Q1FlAAuTkOF0wsavYM6m5mPw15/BvhOBrMw7LE5ncO/KCwsLGTduHDabjdjYWPr27UtZWRlHjhwhLy+PiooKNmzYwNq1a7nmmmsCXa5X2etg7hYorGi9zeAkWJALydH+q0tCh60W8jdfOLCHJ8PjuZAY5beyJIRYa+A/NsEXVa23uaYL/OoKiIv0X13eFNRBbrfbGTp0KFarldmzZzNv3jzi4+MBeOKJJ3jooYeIiIigsbGRyspKEhISAlyx91ScgvvWweGattumx8JL10BSB9/XJaHjSC3cu67p/7b0jIcXr4YEhbl4kbWmaQxWnGq7bf9LYPHVEBPh+7q8Lain1mfOnInVamXGjBksXLjwbIgD5OfnM3jwYBoaGsjIyAiqEAf4xXbXQhya2v1iu2/rkdDz6FbXQhzgQBU8vsO39UhocTqbZoNcCXGA4hOwcJdva/KVoA3y4uJiCgoKSE5OZv78+S22ycnJAWDw4MFnv/fWW29x22230aNHD2JiYrj88st59NFHqa6u9kvd3lBSBf886l6ffx6FEnN+RGnniisvfEqnJavLmqbiRbxh2zHY97V7fVZaodLF4G9PgjbIlyxZgsPhYOrUqcTFxbXYpmPHjkDzIF+4cCHh4eE8/vjjrFy5kunTp/OHP/yBG264AYfD0eJ62pu3D3nYr8SrZUgIe6vE/T4O4B0Px67It3kyBusd8LcvvV6Kzxl4NsA1q1evBmDUqFGttrFarUDzIH/33XdJSUk5+/V3v/tdUlJSmDp1KuvWrePaa691u5bc3FxsNpvb/TyV9H+XEtV7uNv9/vj+Rp684TYfVCShpvPDq4hMz3a737PvfMyvn5nmg4ok1CQ/to6IlAy3+z3xP+/y6EvTvV9QG1JTU9myZYtHfYM2yA8danpr36NHjxaXNzQ0sH79eqB5kJ8b4mfk5uYCUFpa6lEtNpvN476eSLBE4Mk1Qw2WSL/WKcErMTwKTy4APk24xqB4Redwz67ePe0MM24MBm2Q19Q0XelVW9vySbeCggLsdjvx8fFkZmZecF0ff/wxAP379/eoltTUVI/6eSq8sc6zfg21dO/e3cvVSCgKr3fxSstviWys0xgUr7Cc9mwMRjlOB2QMXkxOBO3tZ1lZWRQXF7No0SIeeOCBZsvKy8vJycmhvLycq6++mnXr1rW6ntLSUoYOHUpOTg4rV670ddle8doX8FSR+/1mZcG03t6vR0LP4j3wwj73+80dBLdneL0cCUH/vQteP+B+v19eAePSvF+PLwXtxW5jxowBYMGCBezb980eZfPmzYwaNQq73Q7AkCFDWl1HdXU1N998M1FRUbz88ss+rdebJqRDlJt/2agwuOky39QjoeeWHhBuca9PbIR5O1Bpv27LcL9PYhR8r6vXS/G5oA3y/Px8OnfuzOHDh8nOzmbgwIH06dOH4cOH07NnT0aPHg00Pz9+rtraWm666SYOHjzI+++/T9eu5vx1Ezu4f1QzKUNP1hLv6dIRbkp3r8+Unk1hLuINPeLgejdnyO/qDVHhvqnHl4I2yNPS0li7di15eXlER0dTUlJCUlISixcvZsWKFWeP0lsK8vr6em6//Xa2bNnCypUrycrK8nf5F21mFoxy8ZTLqK7wU/N+RGnn8gfCyPOvHW3RuDS4v59v65HQ81+D4YrOrrW9tQdM6+XbenwlaM+RX0h1dTUJCQlYLBaqqqqIiYk5u8zhcHDnnXeybNky3nvvvbNH7iZqdMILe+GNg/B1/fnLEyJhcmbTDtTdaVARVzQ44JliWHoIahrOX54Y1XQkfk8fCNMYFB841QhPFzXdH17beP7yzh2aAnxqL7AYOgZDMsg3btzIiBEj6NevH3v27Gm2bPr06Tz33HPMnTuXm2++udmyXr16tXh7WntX1wDvlzU96aimoWn68orOcH03iNZUpvjByQb4uxU+q4CTjRAXAcNTms5HmjiVKeaprof3rLDreFOgx0XAVZfCdV0h0vC56ZAM8hdffJH777+fyZMnU1BQ0GxZRkbG2XvQv+2VV17hnnvu8UOFIiIirgnJ47GdO3cCLZ8fLykp8XM1IiIinjN8QsEzFwpyERERk4Tk1LqIiEiwCMkjchERkWChIBcRETGYglxERMRgCnIRERGDKchFREQMpiAXERExmIJcRETEYApyERERgynIRUREDKYgFxERMZiCXERExGAKchEREYMpyEVERAymIBcRETGYglxERMRgCnIRERGDKchFREQMpiAXERExmIJcRETEYApyERERgynIRUREDKYgFxERMZiCXERExGAKchEREYMpyEVERAymIBcRETGYglxERMRgCnIRERGDKchFREQMpiAXERExmIJcRETEYApyERERgynIRUREDKYgFxERMZiCXERExGAKchEREYMpyEVERAz2vzp68RE4+KrOAAAAAElFTkSuQmCC",
            "text/plain": [
              "<Figure size 621.941x284.278 with 1 Axes>"
            ]
          },
          "execution_count": 17,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(3)\n",
        "qc.cp(theta,b,t)\n",
        "qc.cx(a,b)\n",
        "qc.cp(-theta,b,t)\n",
        "qc.cx(a,b)\n",
        "qc.cp(theta,a,t)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IgNLQnQ5z8gF"
      },
      "source": [
        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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "nEieLVFLz_fW"
      },
      "source": [
        "By tracing through each value of the two control qubits, you can convince yourself that a U gate is applied to the target qubit if and only if both controls are 1. Using ideas we have already described, you could now implement each controlled-V gate to arrive at some circuit for the doubly-controlled-U gate. It turns out that the minimum number of CNOT gates required to implement the Toffoli gate is six"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hP0CJazO0BV9"
      },
      "source": [
        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U9wlMe4CALKANWvWYBAAAMDszM7O6u11p+fF7OysJm/Y0dGxbdu2uro6PD6oNOmkrKwMDwwAAAAAIO3cuHFD749YWloiF1rpTlar9ciRI3h2WYmZcmk0rI4HAAAAAAAyGVO0JSRtQLU1u1OvFLJPs1OlAQCYFzPWxgEAAJAIA2JnaO5lxlb/j0QiwWCQ09rsboAHDCoNAACowYDaOAAAAPTGyLLCme+j0OloI2WxHA4H5htUGgBAhoIIUgAAMC+0L40BsM0rM9PVTyPEioqKNHxbUhbr3nvvxXyDSgMAyFA07K8MAAAgiykuLt64cSO5zs/Pz8A7HB4eJheFhYUavu327ds57UpX3yWYqeLZ0NBQTU0NnhkAZmRychKDAAAAQD4Wi6WgoIBcRyKRtGg1kUgkbnRZNBq9du1afX09+a8eMXKjo6OYA9mp0gAAzMuqVaswCAAAABTx1ltvcRzncDhefvllEo5VUlLC9jcbHx9fXFwsKyu7devWmjVrlpaW5IQ3l5aWkp6hk5OTS0tLQ0NDCwsLo6OjJNffbrfbbLaBgQHyXzmEQiENtZqZmRmO4+R/OoBKAwAwCBp+jZKUwOyQBuQAAMN4/fXXjey63tvbq+j1Xq9X26NtamqKQ3kAhZgjl4ZW8QMAAADSC9RyAAyjp6fnm9/8ppH6jCLsdvuFCxecTqceci/KAyjCHF4a1sMIAAAAGE84HMYgAGAwNTU1NTU1fr+ftMEhcWL0t0NDQySTXszMzMzU1NTCwoLP50v6KS6Xi+M4Er1GPCSURO9fVFR03333Wa1WPCOoNAog9bkBAACAdGFkOw4AMp/JyUnDPJa0OpTgE+VUjers7KTXkUjEYrFYLBb638wspAayVqXJWIcjAAAAAMBdCOstMQsCBSZj9RlEJ6kAfWkAAIaC6B0AAAAAQKUBAJgYRO8As5Moth4AADQBCRdQaQAAGYqcRgEAAAAAQMIFVBoAAAAAAKA7RUVFGAQAlQYAcHdhxkRSAFhmZ2cxCABQCgsLMQgAKg0AAABgJm7cuEEuSktLMRoAAACVBgBw9zI2NoZBAKYmJycHgwAAAFBpAAAAAAAAAHEIhUIYBKg0AAAAgPYMDQ2RC3hpAAC60tvbi0FQyjJoqwAAAIB8rFarqe8/HA4nbQ8VjUZZza2goCBj+6wDAIBpVBpoqwBkDWVlZRgEYEY8Hg/HcXa7Xe8PikQiV69evXHjBvULxb0Zl8s1MDAQDAbTMho8z7e1tZWXl2NiAACg0gAAAABmwmaz6fr+fr//2LFjchQVomKli2AwWFFRceHChb/4i7/ArMBaABoSiUTIxfbt2zEaUGkAABnK2NgYDkIAErG0tBQMBnme19v9wvN8ZWXlwsKCz+ejP3S5XPJ1p6eeemrnzp2BQAC+mrsKCech0IS5uTkMQtaqNHa7HbFnAGQHubm5GAQAEkESdZxO565du2jGSzQavX37Nvf73ji5ublr164V/OHs7GxhYeHk5KT4V5Ti4mKLxSL4YWdnp/QtnThxQvCTUCj08ccf79y5k+O43t5eqDQA6EFRUREGIdtUGqLPIAQfgCxgaWkJgwBAXEKh0MMPP8xx3IEDB+7cuaPob4nz0xgXqM1mm5ycxPMCQFcKCwsxCPIxUxFnGHcBAABkMUkLkWUOly5dIhewNgIAoNIAAO46YJsAZsSYXgLXrl0zy4AsLi5iRQMAoNKoYWZmBg8MALMjEegPADALAwMDGAQAAFQaZZA+AFNTU3hgAAAA0gjirAi0IBsymAEAUGnkQvwzOEgAAACkF8RZCUAG890GHHSGgYYHWajSEGsQDhIAAABZDDp+ALOIZEA/TFQmJKNAq00AgKGyGsxOAGhIOByORqMFBQX5+fni30YiEdK2Lycnx2q10v8SsYlUI8jNzd2zZw9GEoAMgTShAlBpAACZSF5eHgYBAJmQDFI5+kxNTU3qVnO3233kyBF1f1tcXIzndVfhcDh8Ph/GQT9IR12gFBRxBgAYwYYNGziOc7lcGAoAkiLTmTk3N6dJFND169cVvT4SidBri8WC53VXce+992IQdAV5FuqAlwYAYBwLCwsYBAC0YtOmTYFA4MaNG0NDQwMDA4nUmx07dmzfvj3Rm2zfvr2yslLR59LoNZ7n8RQA0Ba0Osh+lQZqKwBmx+fzdXZ2YhwA0Iry8nKO42pqasS/CoVC69ev5zjO4XDU1dXp8elKdSEAQFImJycxCCowQeBZOByG2goAACDr0cmNqblBEBWZANCPpaUljuMcDgeGIttUGmydAAAA7gbMknVNSqVxHCcRzwayG8Qc6g1ylpSCXBrFjIyM3L59O249ipmZmampKXI9MDBQV1fndDoxYgAAAADIJtCdBkClUUZ/f/+ZM2fItc/n2759e9yIYSP1mc2bN8tf8Hv27EEXDgAoMkvTAhVEIpFLly6RiIXKysq4XUqAWSgpKcnk2yPTjOO40tJSPCxs5kBbysrKMAjZptJ0dHTU19fT/3o8Ho7jAoEAyYZMCwUFBTzPyzFO2O32mpoa6DMAsKIPVoROhEKhJ598km5NPM/7/X6r1YqRMRE0cZTjuDVr1qT+hrOzs+SiqKhI21sdGxsjFzk5OXhwiojFYtlR9hqbuX7Q9QWyRKUJhUKsPkNpaGh48803FRkgY7HYxMQEx3HXrl2bmZk5ePCg6g3FarX6/f65uTmJfZw0acbcMh2skbuoqCiNmrPpxs1iscRdU2yr8ps3b3Ic9+mnn2aZDBqNRgVLPhaLzc/Py0kCnJ2dvXHjRlFRUWFhYSo7SSwWY/UZjuOCwaDT6Tx9+rSKvS4SiUSjUdqQnnzHyclJcYEWwbkriLyVqIVFyg3b7faZmRlyIS0e5eXlkb5GicYqO6QrzRNHaYB03EHTBKg0ikSRw4cP33///Vu3bt26dSv8qABkv0oTCoVycnL6+/vj/jYYDB47dkyckjg0NETcOHJIJcXFarXe5RpLLBbjsq69mt/vr62tZX/icrlaW1vN+DW7u7sfeughImooVbCJ/Er/Ozk5SSNMyCoTr9be3l757+/1er1eb6LfxhVtxd02du/evWHDhr/8y7+kP8nNzRUL3MXFxRaLJRQKsSI4kYkF34uI48QuTuTysrIyUiRqfHx8cXFRfKvydxvNEY/Sr3/9azpE69at++ijjziO6+3tbWlpSZS9Tb6X0seniKTebPrR2t4Dz/OVlZXkCZaWlrIydyr2puLiYnKBUCuOqcxm0tOQ2h2Ki4unp6eJPnn79m1W9yM7IS2EkIjc3NylpSVWn+d+XzWB3TAHBgaWLVv27rvvCuZqXV1dKmbWjGVkZOSNN94QbJ4knmpsbOyBBx6IazSkVhLiVySPA2ZikE6VJhaLtbS0CNY2lRumpqbEq12RfKBOmCByQF5e3sGDB/HgU3m4hw8f5jiuqqqK/jCR6ZQ9G65du7a0tDQ2NkY3tbivLCkpiRtxQcSIFC2CxOrMiVz/ghBHOs3y8vKOHDki882JR6KgoGBubm52dvb27dtr165lRWfBmUfEgtHR0biSn8PhkKh2QsQ1+oZUMHU4HG+99ZbSRuCZg0zR9vz58+fPn+/o6Lg716D0KBF9Ju16VxoJBoPqEpddLlfcnwvUwkAgQMXcuFsf3WdSIcMVJ1KZzYCaV+FweOXKlUTroHppXHcWOVPENgjBKpAZOm7kXK2vr+/p6UlvnrD0uUb0Per0IwrtBx984Pf76TFUVlYWDofPnj2r9/CK1ym1QNH/IigOKo1mvPrqq8YfpVQEjGuVRBwRgbVYkzNgaWlJomUBzVGLRqM3b95cWlr6oz/6I3KYmaXYaFyOHz9eXV1NrPgjIyNxQxw5jrt+/TrrtWD1E6Ke6Sc1qhte8lf333+/hloNWVlk5ojPCcFZQsnNzV21ahVVQXNycmZmZr7xjW9wHHf69OktW7YI1F3peyB2UOqK+fnPf/7hhx+Su9LKwE+DoFihx263k/eP6z6i0VDSty1wFAigb8sG6cWFRKnF/dXIyMiPfvQjrcYhrjTAuq0Sfeu4vjKlCOaD2DHIKfcNykTmWk60XSSCeI0EOlJc9xHbX88UAV2VlZUCL2hpaenNmzeXL19+48YNMmEEU4LsomL7Dmvo0fVwSZc+QzbS/fv3b9q0KRwOX7x4cWlpqaOjg9xPbW1tX1/fnj170iIV0Idy69Ytq9VKjrakq+zs2bNnz55Nup/EfZPq6uqkf5viOk20AKnZlNgj4ALKJu65c+eO5m8ai8WeeOIJTl58M51e4uOwo6NDHKPidrsbGhqyz0ubFL/fTxQMIohzohghsfQzNDSk3/GwevXqp556ShwUlFTnjCsOUg0hxc1LDlarlU3DNQaxWEOFm6R/Oz4+vmLFilu3bsWNgKKLiKygnJycnJwci8XC2jUF0rBAnCLZIIL3LCgoSJQho45IJLJy5UqO47xer67FzROtC7pwsg86tgIaGhqeeeaZRH+VHWd5XFWQdQvrpwLpysTEROqWZhpPq8m7sULw+vXrTTSYLpeLPUoEejtV14kNglwQrYwNf5KwNRCfPPd7D5LNZqNOe2l/XSwWO3PmzIEDB8gZ0dbWZoztNRwOv//+++fPn9f8hHU4HDSCQ6a3JBaL0dOKvJ79CRVmdDVnkClBZwLRdtLo6iEr1+VynThxAopKmlUaTqN0C/G+qbcwlLF0d3eTXU9zIVsQ6DU+Pk6NNBJaqM1m27FjRxoFRJqWvXLlSqW3ETfMTOaIbdy4kSpmZPsjpyBbchFWn4TbzT33EPEC27TmhEIhj8fD2i8cDkd7e/tdaP1JOlCJVP1EUNVoeHj4/PnzUGkUNTOQkH2lOwnKbOJJduCioqLly5fTBMKcnBwTJd+/+OKLzc3N5Frvmq4jIyNHjx5VpBWw4j6R9f/zP//zH/7hH8i8Sq+piLVoUAfv0NCQfGOrIsVY8BNx2RKtTn+oNOrQqzyAJlPcZrN1dXX96Ec/+vDDDzmOq6+vzxB9ZmRkJCcnh9THZO03rCYwPj7OcZxWDqV169YpOipmZ2fpHkSl7WwStVP5Ik6nc//+/dSPIRAEWX7+858/+OCDUFFA5mOz2drb28lMXr169UsvvVRVVQV9Ju5Axb2Wr5NzHCc2BSbyDQq0JkHAlR6ClwHQyECv12uz2RRFG2I7FdPQ0LC4uEgcJh6PRyeVJhKJHDt2LK5bxuVyCeImBCEhAqiok3bXd35+Pqu7btq0ieM4QVYSVXvYxUjdPvIjWeR7tMThu4IIbYloZHZIaREOkGaVRivq6upWr169d+9ejuMkSoIaiVIbv/z8cgnKy8tZMxspmxuLxVAFUvU+SAUal8sVd0dzuVzV1dUYK83J8B6C5oWGajz11FOZmWqcxSQS0xVpTaYL6EIvaU2wWCwnTpwgIVW9vb39/f3aJtXEYrFXX31VILeQCDF1mfTEYkv2nMyfAFTtYW+V3SE7OztpqBvV1lKJ29cqNM7n89EboD6ivLy8nTt33nfffTANmE+l4Thu1apVVK9N+83EYrHR0VGZLyb1GbX6aHY1kvUJE6xWAzs8PHzo0CHWXMrzPCnsBjRHkx6CAIB0MTMzo0JhA9J873vfI6Lw888/r2Fc9+DgYENDA3u6ORwOl8uVyrNLlNVpaq2SDAgdFqLzdHZ2ioXAuHmq4gKk2iL2Ednt9qNHj65fvx6ioJlUGurjzoQKMBaLpb29fd++fYKmFiQze3Z2lobzalLHExjDpk2b3nzzze985ztkJ/r+97//T//0T9gmNDcHYBAAkEDzluGCc0orEhUrA6mwZ88eUsYgGAy2tbVpEt8h6LdmZAWCbIUqP3K0+rgxbxRxwXFFkajEp8dxnNvt3rlzp+keKxusq5XAvMxE37+goCBDJnQipzBMVuYlPz+fPr5NmzZBn4FKY17y8vIwCGZE80gEzXUkoCuHDx8mxvjm5uZHH32UZIaolhdbW1tp2BLP801NTcivM16uEMe8yWFwcLCiooLjuCtXrnzpS19ifzU5OXnp0iW2/w+pLeFwOJ5++unMUWxoIRaZ3S+6urpSD2taZq7JgRUCgEmhdWl0shwDinS3HJCxCEqra/jOpGAMRjjDsVqtXV1dpLrpoUOHLoRKMkUAACAASURBVF68qO59BMFmdru9o6MDqRcmgrp0vvSlLwlWrs1m27NnT0NDw+XLl1977TWqtZLEG57nX3nllVSUYUVqMykFXlJSQj236uLu7Hb7Qw89lPotLcPUAQAYCSzHAMSFjUS4efNm6m9I/XVyalWDTGDfvn1+v7+3tzcYDPr9fqV1PsLhcHt7O2sC93q9Bw8ehHMmy7BYLOXl5eXl5e3t7efOnaPhhcFgcPPmzS6XizR11fZDSVfWS5cuXb9+XXXDQ1LngJRop830tNK3odIAAEA2QG38NHsbmAs2EmH58uWpv6Guadx2ux2PTA9R1eVyETv3sWPHFKk0gv51cM7cJROmpqYmGo2eO3fu2LFjxDXn8Xg8Ho9WEyAWi12+fNnj8SR1v5BmU0Rdob16jKz0bQKVhpYPB0BXaGe3oaEh1MAFpoMeXbdu3cJomB3pjvUy0SmramBggEPuqG6Ul5fb7XbiqBkcHJSZHcH26+TgnLkrFZuqqipWsSH1A/r6+lKpCR4Oh51Op0CZoZ6W0tJSY4LcskelAQAAIJ8VK1ZgEIB+EIGJNnEGmlNXV0eESDmdN2OxWEtLCw02c7lcDocDCuddq9hUVlb+4he/oP66vXv3ulyupqYmFeno4XC4pqaGJmV5vd49e/akvb+qBF/AJAAAgCwgEomQC9TYNSm0RpBWUMcd27k8RcLhMLlYvXo1HplOVFVV8TzPcVxvb+/IyIj049i5cyfVZ3p6ek6cOAF9JmtQ8Sjz8/Pr6uqmp6fJFCKK8cqVK+nKlY/T6ST6jMPhmJ+fdzqdNpstk11/UGkA+B20IzLQg+LiYgyCrtCacgAQHnvsMZfL5Xa7NexGT0uxrV27FiOsExaLxel0kusrV64kelksFmON6H19fQiZBgSr1XrhwgW3280ewYq0Glq7zOFwtLe3m6LmMFQaAH4HTaVdWFjAaOhxSGMQjIFmhQFzoblxPT8//8SJE5o0bQQGQ2va+v3+RK959dVXiT7D8/zExISGiivIjjP3yJEjExMT1F1TU1MjX6uhtUmffvppsxzfUGkAEAqCqqsTAgnoTlpSUoLRAEAM2tECCs267u3tjRuRGAqF6uvryfUrr7xiZLCZuE8iyFhsNpvf7ydaTTAYdDqdMveZc+fOkYstW7aY5ctCpQEAGAGNV1mzZg1GAwAx09PTmX+Tk5OT5AKhpHrj9XrJxeuvvy7+LTW9ud3ujKo6BTINq9V68uRJmp3V1taW9E8ikQiZYC6Xy0QRFlBpAIi/njEIwKRKI8DT1I+PP/6YXCCUVG+qq6vJRXNzs8Cy3t/fT1wlPM83NDRgrLIMzZuX2Gw2qsmIp5MYWmPGXGHMUGkAiAMyrYHpuHbtmk7HIeCYyEn9WkyysUP0aWYaNMIe6I3VanU4HOSa9eCFw+Hnn3+eXL/yyivG65Y0NwOYiPLycjqdaFBZImhRCnPVaodKAwAAWQUahugBdZvomrRAZY6MhVRPgVBrDDt27BBrkq2traQqgMvlSkvIWWVlJR6NGdm3bx+5qK2tla4T0NHRQZa5ueJLodIAAAAActG8ewzL/fffn8nfPRaLkQh7CLXGIG4oNDg4SLNoDh8+nN7bg8vOXOzZs4c6mVtbWxOFn4XDYaIzV1dXmyu+FCoNAAAAkISCggJyQXo16MRf/dVf5eXlcRy3fPnyDByEd955h1zAE2gMRUVF5KK7u5tcsF01aStVAGRy4sQJcuHz+ehyFvD++++Tiw0bNpjr2y3DAwYAgCyAGnQ1bBUPKMbk1y0sLJAGWfn5+YrcQQUFBeI/KS4uFhtZY7HY9PT07OzsjRs3xO8zPj5OO3QJYEv33rp1C1PCAGj9XFrKmWjUPM9XVVVhfIBSbDab1+sl5b/PnDkTt5fR0tKSQKOGSgMAAMA40M09C6DCxLZt2zAawGKxdHV1HThwgOO4119/nWqbTU1NqDgH1LF//36i0vh8PpfLJZEcWFhYaK6vhsAzAADIKuCl0RVdM+Mz3yy6bdu2vr4+p9OJmWAMNKW7ubmZFm6GiwaoJj8/3+12k+vGxsZs6vALLw0AwAhmZ2cxCMaMMNw1uqJrZnx5efnExMTJkyevXLlCracLCws0I9wwtS0YDJJ/yU/sdrvNZtu9e/eOHTvgHzASi8XicrnYqD+4aECKPProo83NzRzH9fb2Xr58uby8nP3t+Pg4ucjJyYFKA4ApYbt5oDG25sQN3AcYYSDAZrP98Ic/FPyws7NTzt+SdItoNCrR1iY3N3ft2rU5OTlscnk4HI5Go7rWpwaqYYsxZIKLhtWvgBnZtGmT2+2mWo1ApSE/5zjOdPUnoNIAEAfYwAAAZlSHqMii6A9ROyuTYUNJ6+rqMud4QuE78/L4448T1cXj8dAyaBxTpN7lcpnuS5kgl4ZUtAQAAAAAuNtgNYeKigoMSNZjgB+MDUVh227SAObt27dDpdGeFStWYH4DAABII0gGA+mClT6V+t+AGXE4HHp/hMVioW03L168SH9OA5jNWGbGBCrN1NQU5jcwErrOAQBAcNKXlJRgNICRTExM0GtF3YqASSHlQPQO/aKV4mnteBYzlplBEWcAfsfAwAC52L17N0YDmA56LNE+90AP1qxZg0EARnLlyhV6DW8h0IoNGzaQi3feeSc7vpEJVBoSRQrDOdAbWq4UIgswI2NjY+QiPz8fowFA1sBKnB999BEGBGgC9dL4fL5IJEKuaQVnqDQ6ntMoLgn0BmozAACAjCIWi7Fdifx+P8YEaILVaqViD41SoenrZvT2I/AMgN8BtRkAAEBGQRNpSER0b2/vyMhIelUsem26VoxAQE1NDbmgzhmavm5Gbz9UGgCAERQVFQm2TgDMCHpxACOhiTS1tbXk4o033kjj/UxPT9NrtDPSCcNiRnbt2kUuaHtNUn/CpEErUGkA+B0LCwsC4RtoSGFhIblYXFzEaAAAgBxoIs1f//Vf8zzPSp8gW+nt7TXmg1ilNBKJhMNh8tEmLZIElQaA30HjlanwDQAAhKGhIQwCMBiaSGO32/Pz8ysrKzEmQFuoQ+bUqVO0Qc3mzZuh0gBgYogBDAAAAMgEaCINLU6VOejdNQUYQ11dHbmor6//7LPPyLVJDbumUWny8vIw84Cu0CLOAAAggLY4REo0MIxAIEAutm7dyv48vRUCgDEY09W3qqqKXs/NzZl6lzOBSkMyHGhLIAD0BiILAEAAjW5HSjQwjO7ublaloaUpqKoDshhaT1lXLBYLDVGhXkGT7nImUGnYiuwAGABEFmBGSGMBtFfSA7ZwLQDGMDIyQmIHHA4Hqaj72GOPkV+dP38e45P10O7JekNztK5du2bqEUMuDQDACCYnJzEIukKkH7RX0gO2cC0AxkDLN+/bt49c5OfnE4O6YRWxwN3A9u3bycV7770HlcYIcnNzMe2ArsC8rStLS0sYBAOgtcgBAKaGGukffPBB+kNqUIfnEGgFLT7xq1/9ijNz4YdlGX5/kUiEXKxduxbTLjuIxWKsybO4uHh+fj4ajYpfWVxcbLFYDLsxmLeNgRqEgB7ce++9GAQAsuCg9Hg8HMfxPB83Fnp+fj69MdLoOZs1WK1WnudphSTzPtlMV2lo+YWCggJMO2N0yKmpKRJPOT4+LrMr4sDAQDAYtNvtYm943B+miMPhOH78OIktBmYBjlZgXlAyBBgMNfwJetHQ6q9x7YAGQEOIsaVnE5WVlVSlMe+TXWaWG4X8agAdHR319fWq/zyu6qJHyK/P5/v888//+Z//2aSzQuCnigs5rm7fvn3jxo3c3Fxx1FZRUdGNGzcEP2S7Aebl5a1YsWJqakr85nl5eXFLCObm5u7YsUMnzxgCz4B5oeIjulcBY5idnSUXArc23brTpdJgJ89K7HY78QqammV4kIASV/zNTObn5x955BG/32+W6mSDg4O9vb2Zv2U4HI729nY9tJrS0lIsMb3PJCQN6w3atwNjGB4ejrtzFhUVkYtr165t2rTJ+BujNwD0xsh+jGxvTdpwEyqNxqTLDnF3snLlSo7jhoaGSDg+8S+rM8mUlpbevn1b3IA2JydHtRISDofJfLhw4cKhQ4c4jjt79qzT6cz8ge3v79+7d68p5oDP5ysoKPjBD36guVZDq0OmS7cJh8M5OTk0llUpk5OTEmthZmaGWATy8vIWFxfLysoEvvuioiK6HAoKCvRwMJJkMAS4A5AF0I40gn6LaW/rvnz5cjwdYzCyHyObS2zeUs6ZrtKYvUi2iejv729qauI4rqen58SJE1zmpctTXai/v59crFmzJvW3JWFgY2Njv/71rzmO27dvXygU0rYywYMPPuhwODZu3Hjr1q3FxUXShlydQZ3n+bh2YhJmpm5AlpaWaGmdoaGhY8eOzc3NdXZ2avv4qIivX1pCJBI5derU1NRUhnvDeJ7Xz8GYZQHuNEozJyeHWDRQxgNkPeFwWNCRJnNAXlm24nA4SB/Ijz/+GCqNlnLJ3Nzc0NDQ2NgYEf44jvP7/fTAXrt2LU41zTFRgGzqAXKxWGxiYiIQCJw/f16gWpw5c+bMmTP0v263e+vWrSlOue7ubp/Px/P8m2++mcn5P5FIhHjqfD6f5iqN3kULw+FwcXGxKSZwMBj84Q9/+IMf/MBETV2JahHXVbW0tBQOh6VLiSwsLBDfbygUstls9L/c74uLpHJvbMnRw4cP6zqqcIIBA3j//ffJxY4dOxK9ZmhoqKamBmMFtILuyZcuXYJKo4FEcvHixXPnzhE1UUBtbS37X57nq6urt27dunXrVlQO0ITx8XGz3Cq1wasQL0KhkM/nk2/Fb25uplOuqalJ3RFC3iQYDM7NzaV3uo6MjEiEX1NdUfMc6HA4/Nvf/pZc62Tks1qtTqdz1apVn3zyycLCQtxtRAVit1hJSQn1hhEHlzjMTLCyWHF/YWHhnnvu6ezsHBkZyXAVlzI4ONjQ0JCi4mHAhkAUpNOnT69bt06nzzLSCSanjgjHcbOzs8uXL7927VrcOiLkhgU/F8zJsrKyRH3KE5Vcr6ys1HDq0qDipExOTq5du3Z2dlZQHIUkeAiKqVD/My2IEreMJ7uiBWzbti0tdofz588nGn9a/ZUafAHQhC984XedKt99993BwcHy8nKoNGpQKmUS6ZCer11dXfv27TOygUlWIrNes3lRMc0EU662ttbhcLhcLkUem8w5eAYHBysqKgKBQKKtisZ5apIDPTIyEggERkdHBdpFcXGx3W6niR/bt2/XxOk6ODjY0dHBcdzExITNZhN7mSKRSIboD42NjRmi4spkeHg4Y/UZAWvXrn3iiSc0j+szLNgmFAq9/vrrZ8+ezfABTzF4ksSIip3kGcj09LTxWs3AwAC5EO+N+fn5pBAIaoEAbWGrAng8Hqg0igmHw+3t7SkGvh84cMDv9584cQLRaHcD4XBYxV/5/X6Bo48ezMTdt3btWo/Hk9S07/P5fD6fSbVostAktqqZmRlNVIuktd3Eh7HD4Xj22WdTKeBD33NsbCzuVpA5yoPpamWuW7du48aN/+///b9169aJY8+WlpbU+S4EXlZB7RBSzkH6qQnsBVevXq2uruY4rr29nSQEaoUxQm0oFFq/fr0ppkQqwZM0wNUU/PCHPzxw4ICR4l0sFiMKbaIm7nR/S6+ZBqXPsoxAICBxRkOlSUJ/f//zzz8vtkU5HI4dO3YQ263MLZ5YLLxe78GDB+GuyW7Y+AQ5DVjD4XBra6tAV+F53ul0CvwDo6Oj9Lqnp4eI+N3d3eIpeuDAgY6ODhOVkGYPwm3btiV6AQ08U5cwMDg46PF41G2FRFd0uVxNTU3qDmmzhGGwOnkqVphIJHLp0qVQKERN3atXr+Y47qc//SltT7R9+/aysrIUbT2hUOjRRx/lOK6lpeXixYuGWY7kLC7BzSSKntIQXQPPbDbboUOH1q9f/+mnny4uLi4sLMzOzqpbUNQRKiAvL89qtar+FnRqRaPRl156SV3wZH5+fkNDw5/8yZ8UFhZOTU2FQqFU5CeS0yyW/hN130oKG5y2uLjY2dnZ2dk5PDxsWMXkiYmJVPZhw0h77TWguZ1CsPObzk+QHpUmFou1tLQITJU8z7/wwguCTn+0Ty2RMskKn5ycvHTpEk1yoNTX13d3d5tO0LwLIaH5qT+ppEdpKBR68skn2YVKUmKqqqoEqm8kEmFfRnNmnE5n3Ii1YDBYXFzc19e3Z88e6XugHdPSCFlx7733Hsdx7733Xn9/v3TBA6UST1y9keJyubZv3x4IBNra2qTfx+PxDAwMvPDCC0lHNa5dw3Q6uTqN6OzZs+JwPu73ZWrOnz9PA/HZR2C329VZmukmnPmxZ9TNqJUsGIvFLl++LEjbIMpzKvXoE9Hf3//KK69waQp2kgPdGO+55x5ObfDkyMgI2Qq00hM0r2VCITGiHMfdvn3bsEF+4403yAUxUkhgcOTqyMjItWvX5ufnxeJZdsTIkNpUxcXF09PT2lY9VbrJZ8KHDg0NQaWRdUgcPnxYcB739PTETbwWRDiQ8bXZbHv27HnmmWeOHTumWtAEAhYWFgxbORUVFURbOHv2rK6KE/kgisPhaGlpiSsrSLQrsdlsJ06cKCsrO3DggOBXe/fu9Xq90r1xBPJQWmhra6Mr5dSpU6dOndJQbIrrbrXb7TU1NQ899BAVWdj6E8SkGjeJPxgM7t27V+JJmR1WDlB63J46daq+vl7F33o8Ho/Ho25U014LMRaLtbW1Pf7440nPV+pmTMWXEolEpqamAoFAXA8t22DKbrfv3r178+bNW7Zs0UT6+clPfkIu3n///Uye/JFIhF7LcZULIPsPl75mkYoWDrkw0iNBzbWJyp3RTjWTk5O6Cp2kQMXQ0JDf7xfbjNi1oG3FplAoJK4AQbejW7duqa56mkrQr8PhoDXB8vLy6urqdFqk1OxlpJuOfui3vvWt//iP/+A4zu/319XVmex8vWMs09PTdrudvQGe5ycmJhK9nsT/ULVH+gUsXV1dd4ASWHFT/tN0u93RaFTRB1GvurrpR//cbrdL35tgSni9XulbTboupqen4wY3x52ZcaeoxFTXe905HA4ScrZ7926Xy9XV1SUeDfrtpL8RJRqNut1u8b4/PDwsfjFbRY3+cH5+PhAICPYEujPEfZ+kj0/mzacLOh8cDoeKv4o7UF6v1+v1ytzze3p6FK1Z9qPTMmIOh0PmvkTvU9HMYfeWRNkLSXG73fPz81ptwhk+h1Pcw8kDzfyvSd0RHMel/nBlMjw8TGdUotf09fXpLefMz8/L31LER4Dqky7RcZCBOByOcDis6/oyUmCg+zy7B5pPjjXyw6LRKN3L6NhJH65JVRqy9gRvSwgEAlBU9FNppqeniZAqXwUSbMfEMKB6tUt8Lr03RZNBpqofd6OXeH92Dis9F4eHhycS8Oabb/b09PT19Q0PD/f09PT09LjdbtfvkdBYJMQIupfJGS7xcpZ+c/ZlgnGIRqOJjk+v15utKo38hRZXzna73X19fXQk2czOffv2uVwuiUrcdrtd/mGZxhOuq6uroaGB3nNPT8/09LScOaBUFCDWmRRFHJ7nU5x7ZpnDdLJJ25Xucs1N9ZxPugnTG5O/PSqS07xeb+p1/JPaEOUfr5nJvn37eJ6X3pFMpNLQ05yNndHj2+mKcYFn4nizpBE7AhI1TsnPz+/s7Ny4caMgJKOiosLIlL6sIS8vT87L3n77bRKVQQJa5Lu/V61axS4YpX7zpKkI4XC4pqaGjRiRKFusAqfTuXnzZkGbjoqKClI7WPpvFXnkOzo61EUZUVR7jZNGWYiXM8/zr7zyiszlJojSsVgsTqezoqLi6NGjgvCG+vr6NWvWJG0HlK0tGsQNYRIlg3300Uf0+tvf/jYZsUgkcvXqVXHNBlJPJcMDdLu7u9lQT1q1VvPbTtR1hy2H2N/fn3Qxqq7zbro5TCOCUGVUczGJzvkHHngg6etT7zotPj2dTmfcvESSEllWVjY5OcmGnCWC5Da3tbUpOnxJR6n//u///r//+z9B4BnpMpSoUZI0RUVFgkONrRpHws4lJrNgbc7Ozn7wwQd/+7d/y+lQXJFjSsYb1jyaJAyTazbc8eLFiyYL/zZeBVRkXWAt3EmNmqydkp5JKuwEd7mXRqbljLXdKrIlsKYvFUYIOiUSTSGB21q+IVCgOMkMD2At33H/SnXcTlzfo3yDsdi+ktRLI/O5iOPNHA5H0hFjX5/o/RO5a5J6jdhJlTVeGvGG5nA4Ejn62PUoHoFEcWtJ/eTsO/M8b+RAJeoyKTEZlG4s0WhU7AHjeb6rq0vwDuzL+vr6iF9UIkpNxSTMyjkML426EAaJqDN2DLVdktRBJBDVhoeH2V2C3Uy8Xi9ZDm63O5FjR350Ol3yBm81xi8BaWjEo/G+QSJWUSEqw1do2rw0fr+fNeg6HA5F/hmZlJeXBwIBNiM8GAy2tLRorkODTCBuv2dBFqPX601q4I9LUvfRpk2bBJOtt7f3zJkzGqbTtbe3V1VViX2Vi4uLFy5cuHjxIsdxDQ0Ny5Yt437fAJtmExYUFKSSoyn93QX+GYfD0d7enjQ9mrSHS+q9cTqda9asETQRkukEI6hLDRd3as/JyYlGo9FolHYgJZSWlubk5NB6warbqEubG8XFLaS7IdGM4bjU1NRMTEyIOy95PJ68vLwjR47IuWFNGrDKx2q1Tk9P9/f3Hzx4kBqJZXZzl1neXeDOJUe42AMmMNPSaoE1NTWtra3nzp0T97yqra11u90NDQ3qygag44eGfo+JiYkUgzVSsS4p4syZM+Ti8ccfl7OdBoNBTSrtxi3a5Ha7n3nmGfHmxrYvW7NmDXWZHjlyJK7Ds7m5+fr163IKk7z//vtUckujc3jXrl1Jb5UGDWmVwR8Oh+fm5gKBAOt58/v9dEO477779HOY0ONMcOikvTZMJnppotEoq77LMeiq89IkMm0iqSabvDTUnNDX1yftolFqPmHnjMykF7HxW+wbYU1fWj0sOviKBlC+l0b+V5a/nBVNGPEqTuQE02RSxc0Lko+iT6RuKIkHoWITk5N2mChUXdpnTh+cOnskyQcLBALifLBAINDD4Ha7vV6vYDDpk5WzL8lfaOJ0O+lCNeyLxS+bmJiIm9Os6LBLcQ6nxUStLpfDMC8NXdfqZABdLfEStnk5GUp0Icc9B1Pf/STelnXRi5/g/Px8XO+lnLSTtFciIYc1z/NJBQClpXQktseuri6ZFRHsdrvX6w0EApoHH9GvI5hd6pLl0inHGulLTSTzaavSCIQnMz4V42ETVGQuUekzXr+TW0KaF1jZlWa2qRBS41a8kJiNmazSyCklx5YAkrn1q9aBxWK9RDBGipNKLODKR2nRPzoOiYQG8ReXU79LpkpD3l/8ZSUkmFRUGvF3kWMRFxz5MpU6dvkrfdxJA/CSzt5oNBo3uk++VqN6Dstcg+Q2otEoVSlTV2lUyHMqzhryVypqjsmP48oElUaRlkJXVooVAsRHmN1ulz465Ww1cTeZpFpN2lUaMqpyalEq2pwTfZbq2m48z6euysa1BYv3bXNVCPiCAY4gtu+b1+s1INmotbWVXU69vb3UfwcSxVkp/ZO0992jKXQUQZcbA2aaxWJpaWlhJ5vH42GbNpgRiTAG2lCCusX1a/RWXl4ukBGbm5t1SqG2Wq1tbW20Xhz7QB966CGXy+V2u9mCci6Xq6enZ2JiYnp6+siRI+qCi9auXRs3/EDg+g8EAtrWOCkvL/f7/QKBY+/evXr0d2OLFsg8qgUGKRr1l7RkBZWVpQsxkwRodvvyer0nTpxIsbGMxWKpqakRS3I+n+/w4cP6rdbBwcGVK1feI4OcnBzy7/rfs23btsHBQRUfmqhUj05nDQmLeuSRR5ROUdJ8luO4xcXFVAbZgPYg4XCY1J/geT5ROxqW++67j1yMjo6mEm/W0tIiiCI+ffp06kcn2WQEInswGKypqZE4H4eGhtJ48HV3d5PQ6MuXL/v9fpnnuFgOScrIyEh1dXVFRYXq9tCke1tjY6NWwga5E7pzlpeX02dHQtzNgu65NJFIhO1tFDf/QQ9B8+TJk+vXr2eXirqcirsQvbfv2dnZVP6ctgQVb7vs5q66v4QKadjpdLKlmRJ1dE69LKYm+2lSJEredXd3s/Y51Sdf0rJ1hJqaGofDwZ64Pp8vbmqczDeUPoMFlXlIf/SWlhZttw6qlcV9du3t7QJ9RkWxvqQSp9Vq9fv9gkyS1tbWuF3YU2lOR1LLaHbT0NBQX1/flStXtm3btnPnTsGLS0pKqqurdbVExGIxwbdWWnhTjiTX2toqmLRVVVU6HUCvvfaa6r8NBoPDw8MqJhjVEIzJ+WlrayPjmWiKSugJ5GJgYCCVG0ilc6tMqODodDrlaNdWq5Xn+WAw6PP5nn32WXVWj5aWFnZ1y8yKlH8ynj59uq2tjXYOJVPu2LFjGZje3N/fTw/xy5cv19bWulwuze8zFoudOXNG3LObSCy7d+9+8MEHyeYsTs+LuzMPDAw0NTWluLfQI4mV/UxazFB3lUbQlJ00+zMAm80mEIaAfmqGIuK2BJaPxDNlf6Wu1KMeBx7dL/TIrlYk/5ENq7S0NO5vaYLghg0bEtmWWFkwFdVXftfw48ePj46O0s9NVDFckMGvIZrnR1LLnPjZjYyMsBJGT0+PuuLjcmaF1WoV2H10krzZghk1NTXkC371q1/VXFygE1hCJ29ra0tdn5Fu2W61Woleym5HtbW1Setb0PtXRHt7+8aNG2/dukXVDGkVlHZA3759e2lpaYoOwKSuM01UBSru+3w+RSoN3crUxRSk4oxSCrUW7dq1S+afNDU1EcH31KlTKp5jf39/6vqM9BBZLJYjR46sWLGCLYDu8Xi2b98uvc+oiMgi5yytixOJRC5dunTp0iWyLkj5nKGhoVAoRHdgh8OxdN3zdQAAIABJREFUb98+Ut5gx44drMRot9s1d67GLZDtcDjEGinrrRodHf3yl79M9odz584J5B9aOD4VXZQaBEkFbWpgIhdsNQioNMJt2sgS12TjBgaoGcaTdMtTaleLxWKqb0ZauFftXNacuro6ObVcEg2dIC/CGCtOfn7+Cy+8wLZBGBsbM+ajiRFUnaCpDnaE7Xa7au1C5j3bbLauri7WZHjs2LG49b40R9dFkUgnD4VCrMHY4XCQWmqaa7kWi0Ws1TQ2Np4+fVrzsSVFAtmfEF2RdrXq6+sjIY4pVkFkoU6PFL3EcWMvM2cLpVpiIjOQVvT395PvyPO8fBmJ1sP0eDxNTU2KHm4sFnv++efZva6lpUXF5JQT0UfmJ6vV1NbWTk9PS3xTpTu8uh5uPp/P5/OR7oVkzVZVVSn1z8gprkjMVYcOHRJ0GJPTye3LX/4yGQ2bzVZTU+NyucSFK30+3+jo6MmTJ9WdjFeuXCEXNJqR47jNmzeTC817H5lbpdGQTz75JG4kfaLz+4MPPqDXw8PDra2tv/jFLx5++GGbzfbEE08YcGyDpJqDzO2AQiNHNRdqk0Z4kztnpw2djVevXmVfKd5xMgc552UiOYPd2lTE9dGIQZ3kHopW7ckqKytJjVQ95r94AGOxGBvXp2E1cAn27dv3zjvv0OkaDAYlCpFL14nOfF5//XVWntAwxiaupiHIUujt7T137pxh8c90tdKS0xpCJTOTteFTix7xveyhRk02r7zyiqI55vV6iSj/3HPPKfJfvfPOO6x4ffLkSV0f5cGDB8+fP8+qphp2qIxEIqp7UjscDrqtqdsN5GiS4or8Dofj+PHjKkwMNputvb39/vvvZ60zZEmuX79eXXN5kmouUKdv375N5ZxYLGYWgVl7lYZ++XA4/Ktf/UrgtJIjH4yNjRG/28DAAGsPe/nll19++WV1d3X+/HlSpeDSpUscx3388ccyWzGARBAbdqLfhkKhsbGxmZkZerJSh+/+/fsVbQcsNI4xadSTzHgh2o3k8uXLArMHuRgYGFAatJBIn9EwFi7F0PC4JE3NZHUSFd9FPCziVjCTk5PLly8X+AwFe0hPTw+51YWFBZ/Px/P8mjVr2CxJrTZf8h21tRBLaM7T09PsTDMmS0EseQssROzAGpMJmfoETrQ5sOVD2tra9D6krVar2Akms4uRYV3DgZwdT9fHQWuuuN1upfLo/v37iTTv8/l27Ngh3w5CBCGC1+vV2+9tsViOHj3K7qUej2f//v2aFD7Jz8+fnp7u7u4WuIzKysq+9KUvffvb3+Y47t/+7d+++MUvlpaWFhQUUME9Eomo81sqsnOFw+GGhgb2J11dXalYrEg43+OPP/7kk08KhJNDhw75/X5F2unIyAh5LoJb2rp1K7XFtLW1mUVg1lil6ejoGB0draqqunPnTktLy3vvvSd4ARu6nRaBe9++fV/5yleMsYBmPuFwmIRRsgqAzN2N2LDp9jQ6OkpyRZImE7/77ruqb5jm/PzXf/2XdGjZuXPniIhTVlaWm5vLKldybjKVlGj5ip9qyHuqLgEpTSLXmUzXE6uozM7O3r59W6xe/vu//7uc9EdpCYAditWrV5tr6Yk9HgJdYsuWLcbcifT5J0iGNJ5t27bxPF9ZWZmi1DU4OMiuxBSTQGRSV1cncIJdunSJtiaUsMJksk2U5tyra+WkNG9Tj0J8Sq0w+j2OkZER6mEQCL4ypXna8fnAgQO3bt2SU2AjEomwNn5jTBWbNm0SaPhvvPGGVrUcrVZrXJmblrrdvHmzeANJPQ4zabSCuJ+v0ooviTL3bDbbhQsXBA1SSU25N998U+ZXi8ViL730ErkW7Ev5+fkkyI3juObm5rq6OlO4ZLVUaUKhEDUYpP5udrudPDOaGlhdXf3UU0/l5uYKAlGKi4vFO04sFmOdxS6Xq7W1lexNJnKihcPht99+W9oiTk8IYtsWyO4qxHQV40OeuEzZfevWrdevX1cnkVP7/WuvvSZd6keTSUhkKXJNnwI1Bufl5ZGo/ePHj7OFDicmJtg5SQpn6YG2pjVqeVq5cmXSF//0pz8dGhoqKyu7desWmW/EZyLzs9j2o6k/I6fTOTIy0tbWlvkrmq7WpGKE0jWoOokzFos9/PDDb731FvnvwsICa4NkDZ9paWlvtVrln9AS64ItJ83zvGH1fFwuF7suJLzHRmZtpQJNJlaXrcr6YOUEHqdezDB1UUS/8/3QoUN0S1QnlpSXl1NVob6+vr6+PqkTQCAhGFa0qa6ujlVpxHk4WlkSM4RIJJKiPiO9Y1gsls7Ozo0bN7Jxd4pqyp05c4bsTi6XS7wltra20ify9ttvm8IToKVKY7PZAoHAhQsXiJz32WefkZ8/8cQT9DU0VVGCnJwcqg42NjZSYfHhhx+WH4gsiCMqKyuj+4VZ9BnxejAGrcbH5XKxyhjRRQsKCp577jnVEjmV2zZv3rx79262eklSHA7H7OxsYWHh7Ows/eiSkpLNmzcXFhZeunTpb/7mb+iLBZ03WQSTMBKJsA6HuPW4TIGiCCsayamOr3/964888gj3+0I04hfk5ubSrXxoaIg96k6fPk08GGy6MxUH1VmODYPKc2wlABIoNTIyQn+yb9++UCjE7oRJYcUUUvKLaibRaJRUhFtaWqIDlUgFJSmzcT/CGM+GGJn6jHQ0CKsw6FF7MBFK0wUzfw4L5q0BT1Y1qRRrot4hnfZz1n7vcrlSkRfJ33Z0dJB3I+4aiVJ+gvqQRlrf7XZ75tTLUY2c4oocxx07doyV37q6utRVsJRGXH1BTk05juMGBweJhsnzfFNTU1xR0O12E4fegQMH9u3bl/nCs8aBZ+LeDoJjZtWqVao3CEXuUUGLt4ceesh0y8ZisWzcuFFDlYZkHbB7NPUzzM/PU3ORCvr6+h588EFqTovrN2NlJtUfROW28+fPi4/DxsZGKvva7XZB502dEETmyOmSloHQ8/vZZ58lxSsFNuOZmZmf/exnEl23SHFYqqIIxJ3JyUm2atnrr78ufx8QmKm2bNki8bcZXueQfpe2tjYJt9KZM2fOnDkjthHE1UZ4nt+4cSP7k+bmZkHyqGpYB44pdHKd9AHVxS3E21Rc1YuNKdB2DkuXnFYB3efVdWsxsiZsKsWadPUOsfqMw+EgISSpUFdXR0p9kG22vr5+amqKmBRLS0vXr18vXxil4ehiyMmeSl17k9r7EpGouCJZ5qwlzuv16ufliFtTbn5+XsJkwFYsaGtrS/TKurq6s2fPkon6zjvvJAqazVqVJqmN6sqVK9p2wk7EO++8w/7XsBwebVWa9vb2p59+Oq5xlEicpaWlgmIs0upEImjIqTrWrl1rcJxl3BXIitG9vb2qk/8U0d/fz/7XyH44eogpu3btoiUjxfIBVWkcDoeiAjupIJicBmdOaxt2QhXFP//zP/+f//kfRX+bKCojGAzKN3ywsZRE//zss89YR7rD4aC6U0FBwdzcnLk2z9/+9rdKHVxyiGuIiUQirEVjcnLy448/Zm0B4uIiEilkcRvwpY7SgoFJoTZ+dYGIStUMXauNyUHb3tOxWOzy5csNDQ1Un9Gq8p7FYtmzZw+tgRZ3uyDx/AKlurGxkRXEFXlRfvazn3k8HvK2dM7X1dUJDNBEUCkoKBCYBtjES1ZW1LzftyIFVbX9goWtr2i329XVi5fPwYMHu7u72d3m5ZdfTpTTHw6HqT4jHQtntVrr6urI2+7du1dd6+esUmkEMqUx4cLhcJg9gVwul0lLNlsslkQTKO3WjtSL26quCZsoJ09gNbx69aq65Sff0CsoH2lkjL62TE5OJhVT2GNGac+7VGCPWIfDYfBa1jZMgqwanueJcsie6P39/exc6unpifsOJFlOrDn39PTQwgler3fPnj1EFiRnucS0FCzkHTt2sC8WFCjPWOi3iOvg4v6wgMSFCxdYMY46q8XjTH4rCPpXXdlCmi9+8Yt6vK1+fZyMCUSUqdKwvoXZ2dkbN27k5uZ++umnSrd08Zaozhk1MjIisN6GQqHZ2VmqzJB94Pjx49puaE6nc/PmzR6PJ+7GFfeHqWSwfPzxx+K3lWlh6evrS/TR6sZc4n1kNncmDqjR0VFpe2jS4oqxWIwNEjFABLVYLILWyc3NzY8//rh47YdCoSeffJKeGkllpOrqanowVVRUSDcUyn6VhjxOOnFV9IRSgSDiaPfu3RzQmtSPSRWFVqSLFwv2F4/Ho06lkR/48Ytf/IL9rzggVduuJvq9J40lkBBTBFXFQqGQ6jkg3/LKZphwaYrrC4fDWm3i5OynfhKLxULHUGDu2bZtm6IPZXvarFmzRv6jERTvZhtIm4ikVlgiexEuXrwoEUKpyZHHSsasikUyDIuKigoLC6PR6O3bt8mKy8nJ0alljeZtIn/5y1/KWcWRSGRqakqQtsFx3M2bN6lAL+fjBNJCKBSKRqPko8mSEeTaxd2yVMTy0S2RHiuJNl525Q4NDckslEJKFukh6ZL4f3K3JI9OsMYFw0WmK/HP8DxfXV29c+dOkvVXWlp68+ZNMhQzMzOjo6MatlyjHR71gz5EmToSiQ3btWtXijLq5cuXja+vaLPZenp6WIOL2JwxMjJCe2h6vV45e47VaqVV9TiOq6mpaWpqMqy/ViaqNAJB89SpUxKJa9J/Kwex4dyk6Q3pIpMrwkmbf2w2G5t92Nvb29/fryL60+PxyCkYEolEOjo62JlG2zlnJYIIFkXWX0EAqnxhXVCyOS1xfZobpeImlQpEz7m5OUWfqzpFQaBKPfDAA+x/01USQClUdB4dHc3PzxdrOM899xw1dW3btu25555jiyXI2RPotdfrjZswlpOTs3LlSsHm2dHRwao0EuVDdKoEo23gViwWo4GgJASUJNFxCmsechy3ceNGFTegKAzyj//4j1V/0/HxcXLR0NDwxhtvaDiGbrf70Ucf1TsCn06zTZs2CQTQ7du3s4KvIitzZ2cnm7NKHMLsC2ZnZy9cuCBwbEo4gmgo7IYNG2h0luZ6uPzwS0UZL4nWMlvWz8jYDWkJZHBwkBYK7+npka+WlJeX9/X1kTStYDBIJg99cPRQpuWI2co9WajSPPbYY+x/6+vr9+zZI/MZqwhUO3bsGPvfF154waRRZ0bC7iDT09NKV6D8U5Pt3KcHLpeLdYI///zzO3bs0GMCxGKx5557jpVCmpqaxB+UTdmQArWku7tb/p6oboMLh8PsWchGbJsROvnjJpUK4jBlRkpQ2D5RimA9n3a7XfCklHYRSRdUo8vPz4+rCj788MNUpbl48aIiK2MkEmHnofzzi+M4QWFAiUwwNpxBQ9gCj6kzPz8vMK4rUmN4nv/yl79MMl1HR0d1mgxko2AFdxWmECqUK9JniJz3ySefTE1NPfXUUxs2bGD9VLm5uVu3bk2XtEcRRBfPzc0puiXWWyV2CNtsNnFwxIkTJ0Kh0MmTJ3/0ox/RH/74xz9++OGH6S7Hvm3aE6jkj0BcWLeYkfUVE4k6g4ODbCyiipQYNk2LmmCkTxyHw7Fjx47HHnvMyAlvhEqTn59PK8FRlV2nQPzu7m72VHA4HJlfoiETYHcQmbuJdPnCROjduW/Lli1sd8tgMCiz8S1r1pUTeN3W1ibI10oqJKVSKCYuqpORVC9kVurq7e31+/26OqDb29sFMl+iV5qioYH05Bfs++fOnVM9tvLztvv7+9ljSfyJrLlRE+jyTCVwUQxNOk/k2hKEuSoKJlR9JMdiMdbCIh1SLx1VqxS6P2v7BOmgkTIStD44hXiuiCZcWFgofsSNjY1EpVEn6lGnEP241atXr127VvxBKZrPyOP41re+9eMf/1hg2cnPzyeVIeRUoTCmHpIiBK5XpTXx1C1bm832m9/8hv3JP/7jP5pUXtK1uKLYWMOWIZmcnFxaWhK3Z0ykWc3MzLBlA+x2+9GjR9XNSafTuWfPHp/PJ/O0pf0AyMEdN2WRE9W4SrGyyzJjZkBdXR2r0vh8vh07digtaZc0GpKW2aa0tLRAXVGKzNogO3fudDgcPp/P4XAYky4Wi8WSivIWi6WtrY3GfXIc19zcvGHDhqTSIXswJw287u7uFhTJldjdqAynVcpjIilNE6TbaNjtdnY7O3bsWFVVlRwnmIoW4P39/QLzhIRhSQ8Lt+YqqBxxjerJPp/v+PHj6oRp+dFiP/nJT9j/ShuzNTGdUm+SwS0UBUFiv/rVrwzYtQTt0aSHV9vAM52+HbVPV1VVEdE2rniknzdVvjE0RfMZeRwbNmyI+13y8/PT7mxRjeAbhUIhAyy/kUhEYAQ0u6R07733ii0jRP2gtSU4jvvggw/6+/vZNmuhUKiwsPDee+8lZh3pKnOkfaq6O2TPRNJ8RuZ5LTFzTpw40dTURBYX0a/Ir8bHx4ljUxyAquhodrlcMvuEplOlsVqtAqfVgQMH1q1bp8j5Ja2VsmXpCF1dXZlcmSFjkdkbrry8/L777tu4caOi6oTFxcVUxFdqpaO1oaTPy/LycoGAW1tb29fXp9WuHQqFBJpzIBCQuCXVEUFGQjcm6XO6vLyczVZS5wSTw8jICNvHhud5meYJDWsnGN/Nvaqqij0M5DeBFiAz1mhwcJA9St1ut/RfaauEGBxbIvi4CxcuqK5GKr+MuOAsl9mmXZMKtpobUEA2wcYydHd3y09vFiA/6UWQFSku2mSWGFd6xHz++edytoKzZ88a0yIv0YOuq6urqKjQ0FtI9flE50VnZ2ckErl69eprr72mKDDV5XLFbfqZcSoNF69sdkVFhSJBU9o9KliTujY2ym7kG5+sVqvSrdBisTQ1NdXW1rrdbtVWrqQu19bWVkFHiL1797rd7oaGhhTzasLhMFv+lZMRlqpJkXu9pRYivsupQSTIVpLpBFM6yILGr01NTdLmCc37GKQFgZLv8XgkssklkBNrFA6HaaooIe6GmTWSsdVqZcW45ubmuro6XW1eIyMjgqgzmR9369YtHEMZslWatM9YUqqrq9nwbHV1dOQTi8XSUrRJUOpNE6hl5/r16yr+XBA5yR5hY2NjZL6xxRsEBRho5r2gKRbh8uXLbJMxRTUANBcjSeW9zs5OsZ1RXfvEzFJpLBYLibxXLWgmEmRjsVhLS4vA3Kja6gD0pqamJhqNpjKhk9p3xTXaiRBz/fp1l8sVV0aUc+bFYjHBBPZ6vUltvRpWvSTYbDaHwzE6OqrhqUD0Ljl+s7hOMA07cMViMafTyQ6yw+FIWkqOPj4je5OrJlGuS35+vsCb3djYePr0aaWLJamCF4vFWltb2UFO5NPWvFGjthkjYnFB4re0Zxyhvb1dnRNMTgGVWCx29OhR+ffGkkrPe6BIeJBQ+LN7QHbu3Mn+V3UdHZlVTNra2tj/xi2lY7riiq2trQUFBTt27GBPnzVr1pSVlQkKIdy5c0f1x23evFl+6KOg8bfmheNUSyyGfdYXjPxiVqtVMLOJoPnEE08MDg6qftu2tjZBzL3A+ggUoUfbE7HKoeKvaCCQHGOnzWYLBAJi7WL9+vUvvviioNuJzGE5fPgwKxKlUXNub2+/cOGChnYOonfJ9HU0NTUJ/DkVFRVsX5RUePXVVwW2bUXdtfUoL6NVmT46gSUO7/3797P/7e3tFe+ZcVEUr9zS0sJq2na73TCftq5BmNJZcIKx9Xg8KvYB+aeSYBonPdeNyTbWCtXSEq1bkMklPQxO9DIeEkLMrsq4DWo1ob+/n009TdTtwCyBZ2yQdmdnZ11dXc3vcTqdNTU1NptNoObRNGAVKNL0BFUEFVU8zw6+YPxC8nq9gh/29vZWVFR897vf9fv9ip59KBRqbGxkV4vdbte8HS8w767d1dUl/nlzc/PmzZurq6sHBwfpfEuUCx6LxQYHBxsbG9evX89KgfI1Z5kd5ZTqhHpMcpnhGfn5+bQ3BeXAgQPf/e53U5T+Ozo6BPEJOnWjU4TmZfokIrDz8/MFk7a5uZltf5Q6grKQZNgTvVhmZl3akeP8IU4w9ieHDh1SMWOTBtD7/X7BqdTa2pqiPpZpqE6FolWP9NgYzSgipwtBgv6BAwdUaPhJNdtQKMRmRZLVEXdL17y4oibTVYwc06rADz8xMWHAA43FYgIR5S6UhL9g/Ec6nU6xVsNxnM/nq62t3blzZ3d3dygUEus2ZIeKRCIjIyN+v7+6unr9+vWCkg4dHR3mrUOSRlijVMaWhCdah6KDsK6uLhAIxP0Tokjn5ORUV1f7/X42YCkUCvn9/hdffLG6ujonJ6eiokIgAjocDvmuAyPL0huG1Wrt6+sTL+GVK1d2d3erU2y6u7sF+kyiw08M7YunIdSczJavSQUS2G2326W/VF1dncBgX19f/+KLL6Zi6qOEw2G2uAXP89PT0xJns95V17WCOH+SuhkFjppgMPjcc88lHViB41r68Q0ODrKh8Fyy2s3i+ZbJaKgnyPfX0eXAOhaSoquInAUIHDVEw5cTcceehtLSQjgcfvLJJ9mfeL1eOUEWmltSDK4UJXCtCAqd6wSpjU7J7sbfCbmTJhIJmoKTgI34FJfIYCFn8x2gClY8nZ+fz8ybJFYll8ul9A/n5+c1LBnp9Xqj0aj8T6d/2NPTk5kDSxXaQCCg6A/j2ibo4p2YmKCvFJipBJ/e19cnFlYULWe3200/V6thGR4eJlKUosetyQSORqPi7dFut0uMicAomOgbCd426RNnH1yit1UEfdAabtf0JuVMYLEqTvqryBwE6UNT/OC6uroUTQ+t5nBPT48e2w59W9XHBH0H+eIH2cCVHvGpHGo0aFmTOZ+x0CKiMjcZ+VuNYErLWWjsxNBq2Oluo7kckvQ9BaOq6CPkjzCLYOe5O+VhLo2fPT8/LyESKcLhcGgldtydaC646AFZsarP++np6RQVG7vdrkI+yHyVhj59FY+ePYfijpjX6x0eHhbE+E5MTAQCAa/Xmyh/QKlyRW9DQ5WG7FEabiyKdPJEsQperzeufJb0FJyenhaceV6v1/idgQiLDocjjROYKsAyJTmZKk00GhXMZ/n6jOYqDVHINd92yE3yPJ+6UqRIzotGo0pXIlUvVYxn6pqbGa2Z1Dosvf3KFLgF0l3SJc/eiVZCiB62e5kqjWArUHSisQY+mUMhyBzW9hyESqNYsVEdVmu325VKP0B68ff19WXyHaa4Vufn5wOBgKJMXLvd7na7VU8zuj3JkSDTAj2/1R0kExMTigJCpOF5XsVt0PyTTN7KlboZxfUt2BNLMCEl5Izp6Wmx8UjmbGTtuFrJGRMTE9paoJSqNGLdg0r/cW9sfn4+qUojfk+lapu2Kg0dEz1UmlTuUCfrQyK9rqurS4VaQm/ybjj641qW3W53onGTI3AL3pPn+aRPQQ+7qrqVKGcJJH1Pwe7N87z8TU+pl0bsHM56VTwRxhVxTkR+fr7T6XQ6naFQqL+/f2pqStBRJK7QU11dvWHDhm3btqGZpoaiJBl2zcu2agKN8U2xAwlbLj0cDl+8eHFoaEjQ79bhcGzcuHHNmjWlpaUlJSUpZmfRSkfZWpvVZrOdPXt2cHDQ4/FIdEGWg8vlOnz4sIpFbXxbTBWQFHb5E7i8vDwQCDQ0NIj3Q4/H4/F47Hb77t27KyoqBJVtSPLP5OTkpUuXrl+/Li4jLr9Ynx6ln/Sr6SkzBN9isbS0tIyOjgoGtra21m63f+9739u6dSu76pMOQiQSOXbsmGAPaW9vT++q9Hq93d3dMvt7Go8BpTU3bdqUSodBDS01mczBgwdHR0cFu0Rzc/PZs2edTueuXbsEG7J0ofxYLHbmzBlxVmQak5z1KLyR9D3Ly8vZRljBYPDcuXN6dImJxWKCWqxdXV13bUr5ssy5FZvNRo+6kZERklA1Pj4+Nzf3la985etf/3pubu7atWv169Fzl/PEE08Eg8G8vLzPPvssA2/v7bffJherV6/W6j2tVqvVaiW7TGdnp053nvnN2mhufSqyZnl5+ZYtWyYmJq5cudLR0aG0Vi9pKKb6BkpKSgyTk9QRiUTImCjq+ldeXv7mm28eO3YsbsXb3t7euDqkoMSQeKizrG0XVWgVtQkW90ljh9Tr9VZXVxNhLpFKEw6H5+bmTp06JXg6drtdUeVxgublAZxO58GDB/U4LjW51RTNHwZsiUb200gjFouFqN8CrSYYDJJSIi6Xa//+/VQ5jLu3k86P/f39rDJD8Pv9aSkJoNNZIL+z1gsvvMBuxbW1tcPDw0p17KTlml599dV0VeSHSmOQcQWoYOXKlRzHLS4usnsBbVVLzu9oNJqTk5OTk2OxWOT0m4srBESjUdamXlpaevv27Rs3biwtLSWytXs8HlpCZPny5eYaWFrm8vPPP8/A24vFYmfPnuW0qKlqsVjIyq2rqwuFQrOzsxcuXGCr2bLbrs1mKykp2bx583333Ze6r5VOxYyVk1SXDsvPzz9x4oTdbk/dCcbzfFNTU7r6SesHMRsrncBEq2ltbY3bDLe+vp4IZ263e8WKFeyvvv/97//v//5vomgCu93e0dGhQpEge6/m0qoe8hwtxJyt6PEsMl+ruf/+++Nu18QnzMXrevzyyy8vW7YsFArF3Zp4nm9ra5O5vWdZcUWO4/bs2WO329mROXTokEwFj92mJH4r1iEFHX6h0oC7FKpOfP3rX9fwbXmeJ1WME+16MqHFQ2/fvm2ugaXGJw39S0pJZK+anZ0dHh4me7S2xaaJ07W8vPzIkSOCG2D1ZK1goyVjsVgGOnLpBBa0LJAJiZYcGRm5cuWK3+9XupSI6a6qqkrpyLCm08nJycy0W5OQThUT2Gq1dnZ2VlVVHTt2LJFfUSzkvfDCC4neUFF5d9MRi8V07ZSaOSiNEc0OrebIkSOPPvro0aNHE20vPp9PoP9LNAIm8WZZlhpAD7J169bJef2JEyfYwQwGgzU1NSdPnpS/kUYikUTHZTjO5Bh5AAAH70lEQVQcFjjkA4HAXe4MgEoDfodOXRGCwaBWp+C2bdv+5V/+pby83FwDS6NWCgsLGxsb2V+VlZXl5ubOzMwQmaykpOTWrVvEJLxmzZqk71xUVFRYWBiNRq9du0Z8XALLscPhiGuBljjFdUJvUZiVvDNToKQVqBR1gxbAOsEmJydDoZA4zIPqMDabrays7KGHHkolH4z9w0TtaDNHzlBHTU1NVVXVuXPnBP1klNpumpqaVCiNYhYWFjJznGmtiKQNFs2OihjR7GDTpk0kMTJuCp98vF7v/v3705vRUVBQQLJZ9FBNZW7jNpstEAhUVFSwU2v9+vU9PT0yveUS+owgfrivr8900pH2oNIXoLz11lv79u1zOBwul0tRTTBd4Xne5XL19fWZtE63MZ2DU+TP/uzPMrbSnUzIjLXZbGm8h+np6enp6Wg0OjExQWvOTExMqOjFoehDJ/4QbVcKrcSVmTOEmgy0KhSmqAKnw+FIpRxi3Kp9GXsup1LtPW4lqMzc0nUqFmc6pqene3p65NdIsNvtqZzUelQ8m56eVlRIXaclELemnKB1W6KKZ4k2PcFzydh6qnddxTOQOezateub3/wmtTKyGfMS3k9qJRVkwszMzIhdDaWlpTk5OaTGg8C2mq25mMXFxV6v9yc/+cnVq1cz8w4bGhqef/55s9dIIf6op556yu/3a/WeAvPe2NhYooyvpHUaWeuatsEY/7+9u4dlrw3jOH4YRL2lxEIQMTSGSgyayD9qEmLCImIQwyEGIpGwsBgsVgYJFoMYpJGwECQSEpF4CYZqIt4aiaCxUOMzXHnupzna/kt7np7W9zO1FUdb7TnnOvd1/26zWzv+/PmTm5v7/v6enZ1twf+7WjM7ITkc0jAplz/Pzs729/dlBFXNJcjPz6+trc3Ly3M6nYnNqomxlSWJEpIrGHp525o9oqpI/oWjNIYdi4rP8fv9u7u78gGQfV1HR0dZWZnD4UhUNKhJL8GkufLfCjMImyknM5R0Xe/t7f3W6MrXcNHYEyzTXoahIgRghkAgsLGxIWddz8/PT09P0Ws/9aP7+/tYpiGpCfelpaWqk03Ig2F/y7KHop/syzIyNE2z2+1Wntqr67p52Xom8Xg80pH18PBgavkUtn8stK/S6XT6/f63tzfp2PR6vUdHR8fHx5LwHnsvh2V3EZLRommaNY/L7e3tsiP6+Pj4cSkSDAbdbvfx8XGc2zHP7Oys9HNeXV39ktAzK/D5fCqM3rLvvNoZfvcbKlHLkfrAJTu+srJSXrUcy8Tr62tRUZHf77+7u7u+vv6aJpre8/e+i1Ea4P9QVFSkLhdxmDTDzMzM0NCQZesZl8s1MjLS1dWVuu9weXl5ojYlFbi6G/sYV3rvImpqai4uLpqbm5P1HCSRMmxheX5+ri6sxHP+ZLPZGhsbpaRJ4ihNpPlXl5eXc3Nzcvuv+bn4baKvyRP9Yz8/P+92uyUX20Blx7tcrtCJN5qmdXZ27uzsRNos9QwlDYA0NDg42N3dvby83NDQIKtaqYOQGrMKHcNxOp2hjTRhF34JW5lESdaS0YOwj1PHGg7hCdxaXV3d1NSU2+1O9bfl4uJC07SWlhZDjogMtKqmx4qKivv7e0kTiSVPWZZ0k8Fh2Yhh4PdbOSLxk6/nwMDA3NycfB8NXxz5zkrbZ+zdX9LVrP07DO71em9ubp6fn+WVypQww6rK0ZmxziwiCS0gLR6uqKYXfldPT099ff3CwkKkw83XOKUo9Uz6rTAWPxrPAOA/wWBQkp1ycnLKyspkCTl193e+J+vr6/Pz85mZmZrFVv5pa2trampSa2Km/PE4pOHEsuI/kUqJHtHR0dHJyUmufyfl82/ZJtK+vr6FhYXR0dHp6el4tuPz+VZXV8OuAhTjh1PXdS6TUdIAAH5S6WlfOo5C1+SVwq+4uNhmswWDQbVwXujMMa/XK2exMqnMcDdK1qoZaxlZ0NLS0vj4uEwNsiBd14eHh+OPxPV4PBMTE5aNS0mDHtEUNTY2JsMXm5ubSWy/jLIPlKGk+Esa5eDgYG9v7+3t7a9tAnL5xuFw1NXV/YadISUNAACpXTouLi7K+ZxEI6gfRckRkXayWLqqpHPSbrdXV1cb4vvkwbC/VVVVVVxcnNihMI/HI9XR7e2tWvJInlKk8LrDw0Mt5uYxXdcLCwvtdntBQUFoXIoWuUFUOvS4+J0sa2trHR0dmqbt7e01NjZa7eltbW3J0pZmDCIFg8GTkxO5+nN4ePj5+SlpcrL6HJ9JShoAAJDOAoGAzWZT44fR1xuAlak8sdPT09raWvP+UNgYDMMVBK/Xqyr8x8fH7e1tn88nQ4upHq6YxogHAAAAKclQwFDPpIH+/v5ERSDKYJ26G8uKCKCkAQAAAOKSwEj3xEb5lZSUzM7Otra28j+yJhrPAAAAkGQrKytLS0tZWVlWG0txuVw9PT1pE65ISQMAAACY5Wu4YqQVYA1U1kXoWmQGssSTzLkPffz29rayslJuOxyOQCAgt19eXn5J3CIlDQAAAAAkWSZvAQAAAABKGgAAAACgpAEAAAAAShoAAAAAlDQAAAAAQEkDAAAAAJQ0AAAAAEBJAwAAAICSBgAAAAAoaQAAAACAkgYAAAAAJQ0AAAAAUNIAAAAAACUNAAAAAFDSAAAAAKCkAQAAAABKGgAAAACgpAEAAAAAShoAAAAAlDQAAAAAkGT/AJLXnAUSvcevAAAAAElFTkSuQmCC)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "dU4kr0Xi0DxV"
      },
      "source": [
        "This is a Toffoli with 3 qubits(q0,q1,q2) respectively. In this circuit example, q0 is connected with q2 but q0 is not connected with q1.\n",
        "\n",
        "The Toffoli is not the unique way to implement an AND gate in quantum computing. We could also define other gates that have the same effect, but which also introduce relative phases. In these cases, we can implement the gate with fewer CNOTs.\n",
        "\n",
        "For example, suppose we use both the controlled-Hadamard and controlled-Z| gates, which can both be implemented with a single CNOT. With these we can make the following circuit:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 19,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 255
        },
        "id": "SGZdEH8pz6sP",
        "outputId": "ecb05555-290a-403b-838c-0af75cc95da6"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 371.107x284.278 with 1 Axes>"
            ]
          },
          "execution_count": 19,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(3)\n",
        "qc.ch(a,t)\n",
        "qc.cz(b,t)\n",
        "qc.ch(a,t)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IePM1Wsb0KIV"
      },
      "source": [
        "For the state $|00⟩$ on the two controls, this does nothing to the target. For $|11⟩$, the target experiences a Z gate that is both preceded and followed by an H. The net effect is an X on the target. For the states $|01⟩$ and $|10⟩$, the target experiences either just the two Hadamards (which cancel each other out) or just the Z (which only induces a relative phase). This therefore also reproduces the effect of an AND, because the value of the target is only changed for the $|11⟩$ state on the controls -- but it does it with the equivalent of just three CNOT gates."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RnQhC9DX0d9t"
      },
      "source": [
        "## Arbitrary rotations from H and T\n",
        "\n",
        "\n",
        "The qubits in current devices are subject to noise, which basically consists of gates that are done by mistake. Simple things like temperature, stray magnetic fields or activity on neighboring qubits can make things happen that we didn't intend.\n",
        "\n",
        "For large applications of quantum computers, it will be necessary to encode our qubits in a way that protects them from this noise. This is done by making gates much harder to do by mistake, or to implement in a manner that is slightly wrong.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qDYWr6cs0iAt"
      },
      "source": [
        "This is unfortunate for the single-qubit rotations $R_x(Θ)$,$R_y(Θ)$ and $R_z(Θ)$ .\n",
        "\n",
        "It is impossible to implement an angle $\\theta$ with perfect accuracy, such that you are sure that you are not accidentally implementing something like $\\theta + 0.0000000000001$. There will always be a limit to the accuracy we can achieve, and it will always be larger than is tolerable when we account for the build-up of imperfections over large circuits. We will therefore not be able to implement these rotations directly in fault-tolerant quantum computers, but will instead need to build them in a much more deliberate manner.\n",
        "\n",
        "Fault-tolerant schemes typically perform these rotations using multiple applications of just two gates: H and T.\n",
        "\n",
        "The T gate is expressed in Qiskit as .t():"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 20,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 127
        },
        "id": "fjQjVrOS0H9-",
        "outputId": "53fb51f1-f9c6-4dc4-efe5-704155019ab4"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 185.453x117.056 with 1 Axes>"
            ]
          },
          "execution_count": 20,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(1)\n",
        "qc.t(0) # T gate on qubit 0\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cqyRDTKN02c-"
      },
      "source": [
        "It is a rotation around the z axis by $\\theta = \\pi/4$, and so is expressed mathematically as $R_x(\\pi/4) = e^{\\iota \\pi /8 Z}$\n",
        ".\n",
        "\n",
        "In the following we assume that the H and T gates are effectively perfect. This can be engineered by suitable methods for error correction and fault-tolerance.\n",
        "\n",
        "Using the Hadamard and the methods discussed in the last chapter, we can use the T gate to create a similar rotation around the x axis."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 21,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 127
        },
        "id": "BLLhRvXe00lP",
        "outputId": "b3aa80cc-9deb-4142-a9d2-4bcac8d65e91"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 352.675x117.056 with 1 Axes>"
            ]
          },
          "execution_count": 21,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(1)\n",
        "qc.h(0)\n",
        "qc.t(0)\n",
        "qc.h(0)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "iQIQeeaC1F19"
      },
      "source": [
        "Now let's put the two together. Let's make the gate $R_z(\\pi/4) R_x(\\pi/4)$\n",
        "."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 22,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 127
        },
        "id": "6lGof2lL1EBe",
        "outputId": "2dccec51-b647-4d42-c6db-489f46e6b1d7"
      },
      "outputs": [
        {
          "data": {
            "image/png": "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",
            "text/plain": [
              "<Figure size 436.286x117.056 with 1 Axes>"
            ]
          },
          "execution_count": 22,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "qc = QuantumCircuit(1)\n",
        "qc.h(0)\n",
        "qc.t(0)\n",
        "qc.h(0)\n",
        "qc.t(0)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "U07-YUxo1QfV"
      },
      "source": [
        "Since this is a single-qubit gate, we can think of it as a rotation around the Bloch sphere. That means that it is a rotation around some axis by some angle. We don't need to think about the axis too much here, but it clearly won't be simply x, y or z. More important is the angle."
      ]
    }
  ],
  "metadata": {
    "colab": {
      "authorship_tag": "ABX9TyOb/23UEVWFs9DMH+0z5Fgx",
      "include_colab_link": true,
      "provenance": []
    },
    "kernelspec": {
      "display_name": "Python 3",
      "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": "3.12.2"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}