Introduction_to_Qiskit.ipynb

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        "<a href=\"https://colab.research.google.com/github/NSPFD/IBM_-QUANTUM/blob/main/Introduction_to_Qiskit.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "source": [
        "# Introduction to Qiskit"
      ]
    },
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      "source": [
        "**A brief review**\n",
        "\n",
        "1. Qubits: A *normalized* column vector $\\quad$\n",
        "    $|0> = \\begin{pmatrix} 1\\\\ 0 \\end{pmatrix}$\n",
        "    $|1> = \\begin{pmatrix} 0\\\\ 1 \\end{pmatrix}$ $\\quad$ $|\\psi> = \\begin{pmatrix} \\alpha \\\\ \\beta \\end{pmatrix} = \\alpha|0> + \\beta|1>$.\n",
        "\n",
        "2. Measurement: Qubit collapses to $|0>$ or $|1>$; $\\quad$ measure $|\\psi>$ $\\Rightarrow$ $|0>$ with probability $|\\alpha|^2$, $|1>$ with probability $|\\beta|^2$ $\\Rightarrow$ $|\\alpha|^2 + |\\beta|^2 = 1$.\n",
        "\n",
        "3. Gates: Quantum gates are unitary operators $U|\\psi> = |\\phi>$. Action of any quantum gate is a rotation on the Bloch sphere."
      ]
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      "source": [
        "from qiskit import QuantumRegister, ClassicalRegister, QuantumCircuit\n",
        "from qiskit.visualization import plot_histogram, plot_bloch_multivector\n",
        "import numpy as np"
      ]
    },
    {
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      "source": [
        "### The first quantum circuit"
      ]
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    {
      "cell_type": "markdown",
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      "source": [
        "We can define the number of quantum and classical registers that we want. A quantum register is a qubit, and a classical register is a bit that will hold the measurement outcome of the qubit. A QuantumCircuit is an object that takes the quantum and classical registers."
      ]
    },
    {
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      "source": [
        "# initialize a quantum circuit with 2 qubits and 2 cbits\n",
        "\n",
        "qr = QuantumRegister(2)\n",
        "cr = ClassicalRegister(2)\n",
        "qc = QuantumCircuit(qr,cr)"
      ]
    },
    {
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      "outputs": [],
      "source": [
        "qc.draw('mpl')"
      ]
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    {
      "cell_type": "markdown",
      "id": "4948f9f1-dd65-4878-b3a4-49f6582f2353",
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      "source": [
        "All qubits are initialized in |0>. In the Bloch sphere representation, this is the north pole. This is a 2-qubit system, so there should be two Bloch spheres, one for each qubit -- both of them pointing to the north pole."
      ]
    },
    {
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      "source": [
        "plot_bloch_multivector(qc)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "e04e345e-017b-4ab6-8730-cad996ecfea7",
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      "source": [
        "Let us now measure the qubits. Since the qubits are in state |00>, the outcome should be 00 itself with probability 1."
      ]
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    {
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      "source": [
        "qc.measure(qr,cr)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
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      "source": [
        "In order to simulate the circuit, we need to define a simulator to run the circuit. We shall use Qiskit Aer for simulation."
      ]
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      "outputs": [],
      "source": [
        "from qiskit_aer import Aer"
      ]
    },
    {
      "cell_type": "markdown",
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      "source": [
        "We will use Statevector simulator to simulate statevectors"
      ]
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      "source": [
        "simulator = Aer.get_backend(\"statevector_simulator\")\n",
        "result = simulator.run(qc).result()"
      ]
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    {
      "cell_type": "markdown",
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      "source": [
        "To retrieve the statevector, we will use the `get_statevector()` method."
      ]
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      "source": [
        "sv = result.get_statevector()\n",
        "print(sv)"
      ]
    },
    {
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      "source": [
        "Let's create a bell state $\\ket{\\Phi^+}$"
      ]
    },
    {
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      "source": [
        "qc = QuantumCircuit(2)\n",
        "qc.h(0)\n",
        "qc.cx(0,1)\n",
        "qc.draw('mpl')"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "36fbf146-e64e-428d-b117-79ebbaf8797a",
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      "source": [
        "Let us verify our notion using Bloch sphere."
      ]
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    {
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      "outputs": [],
      "source": [
        "plot_bloch_multivector(qc)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "c3f4fa32-4199-40cf-b1ed-a7cc62f5f300",
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      "source": [
        "When we measure these two qubits, the first qubit should collapse to 1, and the second to 0, with probability 1."
      ]
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    {
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      "outputs": [],
      "source": [
        "qc.draw('mpl')"
      ]
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    {
      "cell_type": "code",
      "execution_count": null,
      "id": "ce912e80-c498-4e89-9456-a5dbed956176",
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      "source": [
        "result = simulator.run(qc).result()\n",
        "sv = result.get_statevector()\n",
        "print(sv)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "52a8ad11-4f64-450e-a308-f2be4a2e7171",
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        "id": "52a8ad11-4f64-450e-a308-f2be4a2e7171"
      },
      "source": [
        "To simulate the same circuit using shot based simulater, we will use QASM simulator"
      ]
    },
    {
      "cell_type": "code",
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      "source": [
        "simulator = Aer.get_backend(\"qasm_simulator\")\n",
        "qc.measure_all()\n",
        "result = simulator.run(qc).result()\n",
        "counts = result.get_counts()\n",
        "print(counts)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "60211a88-afad-4d6b-9ec6-5bf4a69c775b",
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      "source": [
        "### Try out questions\n",
        "Let's build some circuits and get our hands dirty.\n",
        "\n",
        "Q1. Let's create all the bell states $\\ket{\\Phi^+}, \\ket{\\Phi^-}, \\ket{\\Psi^+}, \\ket{\\Psi^-}$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "97851822",
      "metadata": {
        "id": "97851822"
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      "outputs": [],
      "source": [
        "simulator = Aer.get_backend(\"statevector_simulator\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "3deebf21-4e0c-4b30-a4f0-684d4d5908ae",
      "metadata": {
        "id": "3deebf21-4e0c-4b30-a4f0-684d4d5908ae"
      },
      "outputs": [],
      "source": [
        "phi_plus = None # Implement your circuit here\n",
        "\n",
        "result = simulator.run(phi_plus).result()\n",
        "sv = result.get_statevector()\n",
        "assert sv.data == [1/np.sqrt(2), 0, 0, 1/np.sqrt(2)]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "72e03fe9",
      "metadata": {
        "id": "72e03fe9"
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      "outputs": [],
      "source": [
        "phi_minus = None # Implement your circuit here\n",
        "\n",
        "result = simulator.run(phi_minus).result()\n",
        "sv = result.get_statevector()\n",
        "assert sv.data == [1/np.sqrt(2), 0, 0, -1/np.sqrt(2)]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "a4b84c4e",
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        "id": "a4b84c4e"
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      "outputs": [],
      "source": [
        "psi_plus = None # Implement your circuit here\n",
        "\n",
        "result = simulator.run(phi_minus).result()\n",
        "sv = result.get_statevector()\n",
        "assert sv.data == [0, 1/np.sqrt(2), 1/np.sqrt(2), 0]"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "66d6b093",
      "metadata": {
        "id": "66d6b093"
      },
      "outputs": [],
      "source": [
        "psi_minus = None # Implement your circuit here\n",
        "\n",
        "result = simulator.run(phi_minus).result()\n",
        "sv = result.get_statevector()\n",
        "assert sv.data == [0, 1/np.sqrt(2), -1/np.sqrt(2), 0]"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "300a86d0-6913-40c5-b722-71f0c5c205b5",
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      },
      "source": [
        "Q2. Let's implement logic gates using quantum circuits"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "f17f1504",
      "metadata": {
        "id": "f17f1504"
      },
      "outputs": [],
      "source": [
        "simulator = Aer.get_backend(\"qasm_simulator\")"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "cd76b15c-4a70-4a30-b228-453efe53cb84",
      "metadata": {
        "id": "cd76b15c-4a70-4a30-b228-453efe53cb84"
      },
      "outputs": [],
      "source": [
        "not_gate = None # Implement your circuit here"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "c4274a40-f6ca-4ab8-bd14-b4a2ea71d939",
      "metadata": {
        "id": "c4274a40-f6ca-4ab8-bd14-b4a2ea71d939"
      },
      "outputs": [],
      "source": [
        "xor_gate = None # Implement your circuit here"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "fef75c00-2f87-4c21-b551-289b93e8fa9e",
      "metadata": {
        "id": "fef75c00-2f87-4c21-b551-289b93e8fa9e"
      },
      "outputs": [],
      "source": [
        "and_gate = None # Implement your circuit here"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "e319ce83",
      "metadata": {
        "id": "e319ce83"
      },
      "outputs": [],
      "source": [
        "or_gate = None # Implement your circuit here"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "6954324f",
      "metadata": {
        "id": "6954324f"
      },
      "source": [
        "Q3. Implement GHZ state $\\frac{1}{\\sqrt{2}}(\\ket{000} + \\ket{111})$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "fafb97ec",
      "metadata": {
        "id": "fafb97ec"
      },
      "outputs": [],
      "source": [
        "ghz =  None # Implement your circuit here"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "691e37fc",
      "metadata": {
        "id": "691e37fc"
      },
      "source": [
        "Q4. Implement W state $\\frac{1}{\\sqrt{3}}(\\ket{001} + \\ket{010} + \\ket{100})$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "09301706",
      "metadata": {
        "id": "09301706"
      },
      "outputs": [],
      "source": [
        "w_state =  None # Implement your circuit here"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "b079e724",
      "metadata": {
        "id": "b079e724"
      },
      "source": [
        "Q5. Design a Quantum Circuit which corresponds to this statevector"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "4b42eebc",
      "metadata": {
        "id": "4b42eebc"
      },
      "outputs": [],
      "source": [
        "required_sv = [ 0+0j, 0+0j, 0.707+0j, 0+0j, 0.707+0j, 0+0j, 0+0j, 0+0j ]\n",
        "qc = None # Implement your circuit here\n",
        "\n",
        "\n",
        "simulator = Aer.get_backend(\"statevector_simulator\")\n",
        "result = simulator.run(qc).result()\n",
        "sv = result.get_statevector().data\n",
        "assert sv == required_sv"
      ]
    }
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