Basic_Circuit_Identities.ipynb

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      "authorship_tag": "ABX9TyOb/23UEVWFs9DMH+0z5Fgx",
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      "name": "python3",
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      "name": "python"
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      "cell_type": "markdown",
      "metadata": {
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      "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": "code",
      "source": [
        "%pip install qiskit\n",
        "%pip install pylatexenc"
      ],
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        },
        "id": "eY3ssiL7TXJS",
        "outputId": "363bd14e-2852-47c3-a3ed-d1c5698cb585"
      },
      "execution_count": 1,
      "outputs": [
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          "text": [
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    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "3bsZWDvWTKTq"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "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",
      "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:"
      ],
      "metadata": {
        "id": "sz7C7KUpTeXJ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# a controlled-Z\n",
        "qc.cz(c,t)\n",
        "qc.draw('mpl')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "4ydAMP8ETyFx",
        "outputId": "6d46cd4f-d0a0-4a70-b754-24861bbadd94"
      },
      "execution_count": 3,
      "outputs": [
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          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 203.683x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 3
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "zfK00LvlT6Ex"
      }
    },
    {
      "cell_type": "markdown",
      "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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)"
      ],
      "metadata": {
        "id": "LUqQnz0_T8SY"
      }
    },
    {
      "cell_type": "markdown",
      "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 $Z$."
      ],
      "metadata": {
        "id": "MQ7_cOqEUT8Q"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "sY7KHnvZT1JB",
        "outputId": "78a1bb14-ffa9-4165-d1f0-a31e176b1892"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 4
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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-$Y$:"
      ],
      "metadata": {
        "id": "0EWKDVCcUdbI"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "dkhwwCkTUaBq",
        "outputId": "d2f4fb1d-283d-427c-ff2d-9ffea8979ba3"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "and a controlled-$H$:"
      ],
      "metadata": {
        "id": "gusSICKMUkzZ"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "0UvJBieRUjRS",
        "outputId": "095984c4-61f0-4383-e900-120cbf887390"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Swapping Qubits"
      ],
      "metadata": {
        "id": "4Ur7caW5UpgH"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "a = 0\n",
        "b = 1"
      ],
      "metadata": {
        "id": "n0Hb0gL7Unqx"
      },
      "execution_count": 7,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "eUmwqv8-Ut4p"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "qc = QuantumCircuit(2)\n",
        "# swaps states of qubits a and b\n",
        "qc.swap(a,b)\n",
        "qc.draw('mpl')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "QCEe59X4UsQR",
        "outputId": "b7877fc4-1856-4de3-9be5-b356b70b447d"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 203.683x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "PZLohq3nUzqA"
      }
    },
    {
      "cell_type": "markdown",
      "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:"
      ],
      "metadata": {
        "id": "-no4T8BjU1YI"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "bWS8UsslUwub",
        "outputId": "30fb2227-a461-44fa-8138-34294bf493d2"
      },
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 287.294x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 9
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "tqtO7F6GVBoA"
      }
    },
    {
      "cell_type": "markdown",
      "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:"
      ],
      "metadata": {
        "id": "zASNj6WIVHEQ"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "ZIh0_7kuU8tx",
        "outputId": "24e09221-e40f-4f6b-d565-6fce3f491829"
      },
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 454.517x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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,"
      ],
      "metadata": {
        "id": "6Seu1zFHVMc5"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "qc = QuantumCircuit(2)\n",
        "qc.cx(b,a)\n",
        "qc.cx(a,b)\n",
        "qc.cx(b,a)\n",
        "qc.draw('mpl')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "dg6cBHDfVJ0y",
        "outputId": "68f534c8-d233-4a9f-c35a-3df2c068dbdb"
      },
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 11
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "YbN5t8hwW_qD"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "ES8_IcLJVeYZ",
        "outputId": "51fbce3c-883f-4779-a5fe-b755037382c0"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 12
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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:"
      ],
      "metadata": {
        "id": "M_YPYRqJXXOh"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "kU623nrfXU9e",
        "outputId": "f3827601-b2a6-4b56-fd03-aa95d002f3f7"
      },
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 370.906x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 13
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "FE3ObjpHXjnR"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Controlled Rotations\n",
        "\n"
      ],
      "metadata": {
        "id": "m1hWdM9OXpwB"
      }
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "xCBR6mapyY-w"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "tIHgD6fZXh5T",
        "outputId": "5db2eb6b-ad26-4115-fb2a-d07f36a99c78"
      },
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 454.517x200.667 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {},
          "execution_count": 14
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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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)"
      ],
      "metadata": {
        "id": "xaxVAt19yg4e"
      }
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "SXJv3CO6zDQX"
      }
    },
    {
      "cell_type": "code",
      "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"
      ],
      "metadata": {
        "id": "GH9Bi5GzzB8x"
      },
      "execution_count": 15,
      "outputs": []
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 191
        },
        "id": "hkACgPC5zV5G",
        "outputId": "370c5be6-aa2e-4896-d3a2-2237fbb2d660"
      },
      "execution_count": 16,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 538.128x200.667 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 16
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        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)"
      ],
      "metadata": {
        "id": "klhtLCBezZYV"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Here A, B and C are gates that implement A,B and C, respectively."
      ],
      "metadata": {
        "id": "ivhynKI6zb3-"
      }
    },
    {
      "cell_type": "markdown",
      "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"
      ],
      "metadata": {
        "id": "V0aQsHWQzhEV"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 255
        },
        "id": "8E823Op7zXr_",
        "outputId": "f5c331ff-6333-42d4-a1b1-7449ac64ae9a"
      },
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 203.885x284.278 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {},
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "FgIEjgCxztFW"
      }
    },
    {
      "cell_type": "code",
      "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')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 255
        },
        "id": "djp2tW7PzrsG",
        "outputId": "9b39c14b-af2d-4655-b934-6a8a30024f15"
      },
      "execution_count": 18,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 621.941x284.278 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {},
          "execution_count": 18
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        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)"
      ],
      "metadata": {
        "id": "IgNLQnQ5z8gF"
      }
    },
    {
      "cell_type": "markdown",
      "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"
      ],
      "metadata": {
        "id": "nEieLVFLz_fW"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        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)"
      ],
      "metadata": {
        "id": "hP0CJazO0BV9"
      }
    },
    {
      "cell_type": "markdown",
      "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:"
      ],
      "metadata": {
        "id": "dU4kr0Xi0DxV"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "qc = QuantumCircuit(3)\n",
        "qc.ch(a,t)\n",
        "qc.cz(b,t)\n",
        "qc.ch(a,t)\n",
        "qc.draw('mpl')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 255
        },
        "id": "SGZdEH8pz6sP",
        "outputId": "ecb05555-290a-403b-838c-0af75cc95da6"
      },
      "execution_count": 19,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 371.107x284.278 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 19
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "IePM1Wsb0KIV"
      }
    },
    {
      "cell_type": "markdown",
      "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"
      ],
      "metadata": {
        "id": "RnQhC9DX0d9t"
      }
    },
    {
      "cell_type": "markdown",
      "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():"
      ],
      "metadata": {
        "id": "qDYWr6cs0iAt"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "qc = QuantumCircuit(1)\n",
        "qc.t(0) # T gate on qubit 0\n",
        "qc.draw('mpl')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 127
        },
        "id": "fjQjVrOS0H9-",
        "outputId": "53fb51f1-f9c6-4dc4-efe5-704155019ab4"
      },
      "execution_count": 20,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 185.453x117.056 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 20
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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."
      ],
      "metadata": {
        "id": "cqyRDTKN02c-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "qc = QuantumCircuit(1)\n",
        "qc.h(0)\n",
        "qc.t(0)\n",
        "qc.h(0)\n",
        "qc.draw('mpl')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 127
        },
        "id": "BLLhRvXe00lP",
        "outputId": "b3aa80cc-9deb-4142-a9d2-4bcac8d65e91"
      },
      "execution_count": 21,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 352.675x117.056 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 21
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now let's put the two together. Let's make the gate $R_z(\\pi/4) R_x(\\pi/4)$\n",
        "."
      ],
      "metadata": {
        "id": "iQIQeeaC1F19"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "qc = QuantumCircuit(1)\n",
        "qc.h(0)\n",
        "qc.t(0)\n",
        "qc.h(0)\n",
        "qc.t(0)\n",
        "qc.draw('mpl')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 127
        },
        "id": "6lGof2lL1EBe",
        "outputId": "2dccec51-b647-4d42-c6db-489f46e6b1d7"
      },
      "execution_count": 22,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<Figure size 436.286x117.056 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {},
          "execution_count": 22
        }
      ]
    },
    {
      "cell_type": "markdown",
      "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": {
        "id": "U07-YUxo1QfV"
      }
    }
  ]
}