assignment2_shor_algorithm/shor_algorithm.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "nQE-3gffBxVm"
      },
      "source": [
        "# **Assigment $|02\\rangle$ 🔐 Shor's Algorithm 🔐** &nbsp;[![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/18520339/uts-quantum-computing/blob/main/assignment2_shor_algorithm/shor_algorithm.ipynb)\n",
        "\n",
        "[![image](https://github.com/18520339/uts-quantum-computing/assets/50880271/e0588281-1148-4acb-a4cd-54fecd46c2ce)](https://rand.cs.uchicago.edu/publication/peng-2022-shor/)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "image/jpeg": 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",
            "text/html": [
              "\n",
              "        <iframe\n",
              "            width=\"858\"\n",
              "            height=\"540\"\n",
              "            src=\"https://www.youtube.com/embed/BYKc2RnQMqo\"\n",
              "            frameborder=\"0\"\n",
              "            allowfullscreen\n",
              "            \n",
              "        ></iframe>\n",
              "        "
            ],
            "text/plain": [
              "<IPython.lib.display.YouTubeVideo at 0x245b8f5f210>"
            ]
          },
          "execution_count": 1,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "from IPython.display import YouTubeVideo\n",
        "YouTubeVideo('BYKc2RnQMqo', width=858, height=540)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "OOs1NvImopfu",
        "outputId": "3b9f489d-30c4-413b-fb56-b7ddee27a15d"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m4.3/4.3 MB\u001b[0m \u001b[31m26.0 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m2.1/2.1 MB\u001b[0m \u001b[31m51.7 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m116.3/116.3 kB\u001b[0m \u001b[31m10.7 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m49.7/49.7 kB\u001b[0m \u001b[31m4.6 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m39.4/39.4 MB\u001b[0m \u001b[31m14.1 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m107.5/107.5 kB\u001b[0m \u001b[31m9.9 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m12.3/12.3 MB\u001b[0m \u001b[31m73.9 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m162.6/162.6 kB\u001b[0m \u001b[31m3.1 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25h  Preparing metadata (setup.py) ... \u001b[?25l\u001b[?25hdone\n",
            "  Building wheel for pylatexenc (setup.py) ... \u001b[?25l\u001b[?25hdone\n"
          ]
        }
      ],
      "source": [
        "!pip install qiskit --quiet\n",
        "!pip install qiskit-aer --quiet\n",
        "!pip install pylatexenc --quiet"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aPieNzTlfiTt"
      },
      "source": [
        "**Shor's algorithm** can be divided into 2 main parts:\n",
        "1. **Classical Part**: This involves reducing the problem of factorizing an integer $N$ to the problem of finding the period $r$ of a specific function.\n",
        "2. **Quantum Part**:\n",
        "- This involves using a quantum computer to **find the period $r$** efficiently. The algorithm creates a superposition of states to encode information about the period into the quantum state.\n",
        "- The **Quantum Fourier Transform (QFT)** is applied to the quantum state to extract the period $r$:\n",
        "\n",
        "    $$\\left| \\tilde{x} \\right\\rangle = QFT_N \\vert x \\rangle = \\frac{1}{\\sqrt{N}} \\sum_{k=0}^{N-1}  e^{2\\pi i \\frac{xk}{N}} \\vert k \\rangle$$\n",
        "    $$\\text{where N = number of qubits; } \\vert k \\rangle = \\vert k_1 k_2 \\cdots k_n \\rangle (\\text{binary qubit state })$$\n",
        "\n",
        " - It is a series of **Hadamard** and **Controlled Phase Rotation** gates to the output state of the **Modular Exponentiation** circuit. This transforms the state into a form where measuring the qubits can yield the period $r$ (just a change of basis).\n",
        " - This **Modular Exponentiation** is crucial for creating the periodic function. It involves applying a series of **controlled-U** gates, where $U$ represents modular multiplication.\n",
        "\n",
        "**The steps are as follows**:\n",
        "1. **Choose a Random Number $a$**: Select a random integer $a$ such that $1 < a < N$.\n",
        "2. **Check if $N$ is valid**: If $N$ is even or a prime number or a power of a prime number, then $N$ is not valid for factorization.\n",
        "3. **Compute the Greatest Common Divisor (GCD)**: Compute $\\text{gcd}(a, N)$. If $\\text{gcd}(a, N) \\neq 1$, then $\\text{gcd}(a, N)$ is a non-trivial factor of $N$.\n",
        "4. **Quantum Period Finding**: Use a quantum computer to **find the period $r$** of the function $f(x) = a^x \\mod N$.\n",
        "5. **Check the Period**: If $r$ is even and $a^{r/2} \\pm 1$ is not a multiple of $N$, then compute $\\text{gcd}(a^{r/2} \\pm 1, N)$ to obtain the factors of $N$.\n",
        "\n",
        "👉 If the above steps do not yield factors, repeat the process with a different $a$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "cellView": "form",
        "id": "s_bn4OMbCswD"
      },
      "outputs": [],
      "source": [
        "# @markdown ### **1. Import `Qiskit` and essential packages** { display-mode: \"form\" }\n",
        "from qiskit import QuantumCircuit, transpile, assemble\n",
        "from qiskit.circuit.library import QFT\n",
        "from qiskit_aer import AerSimulator\n",
        "\n",
        "from fractions import Fraction\n",
        "import random\n",
        "import sympy\n",
        "import math"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "cellView": "form",
        "id": "TfS2XoGuBATI"
      },
      "outputs": [],
      "source": [
        "# @markdown ### **2. Controlled Modular Multiplication for $U(x) = a^x mod N$** { display-mode: \"form\" }\n",
        "class CtrlMultCircuit(QuantumCircuit):\n",
        "    def __init__(self, a, binary_power, N):\n",
        "        super().__init__(N.bit_length())\n",
        "        self.a = a\n",
        "        self.power = 2 ** binary_power # Convert binary to decimal\n",
        "        self.N = N\n",
        "        self.name = f'{self.a}^{self.power} mod {self.N}'\n",
        "        self._create_circuit()\n",
        "\n",
        "    def _create_circuit(self):\n",
        "        for dec_power in range(self.power):\n",
        "            a_exp = self.a ** dec_power % self.N\n",
        "            for i in range(self.num_qubits):\n",
        "                if a_exp >> i & 1: self.x(i)\n",
        "                for j in range(i + 1, self.num_qubits):\n",
        "                    if a_exp >> j & 1: self.swap(i, j)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "cellView": "form",
        "id": "lTDB8evCV0ni"
      },
      "outputs": [],
      "source": [
        "# @markdown ### **3. Quantum Phase Estimation with `Modular Exponentiation` and `Quantum Fourier Transform` to find period $r$** { display-mode: \"form\" }\n",
        "# @markdown ![](https://reneroliveira.github.io/quantum-shors-algorithm/images/shor_circuit.png)\n",
        "\n",
        "class QPECircuit(QuantumCircuit):\n",
        "    def __init__(self, a, N):\n",
        "        super().__init__(2 * N.bit_length(), N.bit_length())\n",
        "        self.a = a\n",
        "        self.N = N\n",
        "        self._create_circuit()\n",
        "\n",
        "    def _modular_exponentiation(self):\n",
        "        for qbit_idx in range(self.num_qubits // 2):\n",
        "            self.append(\n",
        "                CtrlMultCircuit(self.a, qbit_idx, self.N).to_gate().control(),\n",
        "                [qbit_idx] + list(range(self.num_qubits // 2, 2 * self.num_qubits // 2))\n",
        "            )\n",
        "\n",
        "    def _create_circuit(self):\n",
        "        self.h(range(self.num_qubits // 2)) # Apply Hadamard gates to the first n qubits\n",
        "        self.x(self.num_qubits - 1)\n",
        "        self.barrier()\n",
        "\n",
        "        self._modular_exponentiation() # Apply controlled modular exponentiation\n",
        "        self.barrier()\n",
        "        self.append(\n",
        "            QFT(self.num_qubits // 2, inverse=True),\n",
        "            range(self.num_qubits // 2) # Apply inverse QFT to the first n qubits\n",
        "        )\n",
        "\n",
        "    def collapse(self, simulator):\n",
        "        self.measure(range(self.num_qubits // 2), range(self.num_qubits // 2))\n",
        "        transpiled_circuit = transpile(self, simulator)\n",
        "        self.collapse_result = simulator.run(transpiled_circuit, memory=True).result()\n",
        "        return self.collapse_result"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "9VNFVz61mFrT"
      },
      "source": [
        "This is at the heart of Shor's algorithm. It estimates the **eigenvalue** corresponding to an **eigenstate** of a **unitary operator**. In this case, I use it to estimate the phase $\\phi$ related to the order $r$.\n",
        "\n",
        "1. **Initialize qubits**: I start with an equal superposition of all possible states.\n",
        "2. **Apply controlled-unitary operations**: These operations encode the **Modular Exponentiation** into the qubits.\n",
        "3. **Inverse QFT**: Apply the inverse $QFT^{-1} = \\frac{1}{\\sqrt{2^n}} \\sum_{j,k=0}^{2^n-1} e^{-2\\pi i jk / 2^n} |j\\rangle \\langle k|$ to the qubits to extract the phase information."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "cellView": "form",
        "id": "BRzKIInVBATI"
      },
      "outputs": [],
      "source": [
        "# @markdown ### **4. Completed Shor's Algorithm for Integer Factorization** { display-mode: \"form\" }\n",
        "class ShorAlgorithm:\n",
        "    def __init__(self, N, max_attempts=-1, random_coprime_only=False, simulator=None):\n",
        "        self.N = N\n",
        "        self.simulator = simulator\n",
        "        self.max_attempts = max_attempts # -1 for all possible values of a\n",
        "        self.random_coprime_only = random_coprime_only # True to select only coprime values of a and N\n",
        "\n",
        "\n",
        "    def execute(self):\n",
        "        is_N_invalid = self._is_N_invalid()\n",
        "        if is_N_invalid: return is_N_invalid\n",
        "        \n",
        "        # Only coprime values remain if random_coprime_only is enabled, \n",
        "        # Otherwise select a random integer in [2, N) as initial guess\n",
        "        a_values = [a for a in range(2, self.N) if not self.random_coprime_only or (math.gcd(a, self.N) == 1)]\n",
        "        print(f'[INFO] {len(a_values)} possible values of a: {a_values}')\n",
        "        self.max_attempts = len(a_values) if self.max_attempts <= -1 else min(self.max_attempts, len(a_values))\n",
        "        attempts_count = 0\n",
        "\n",
        "        while attempts_count < self.max_attempts:\n",
        "            print(f'\\n===== Attempt {attempts_count + 1}/{self.max_attempts} =====')\n",
        "            attempts_count += 1\n",
        "            self.chosen_a = random.choice(a_values)\n",
        "            self.r = 1\n",
        "\n",
        "            print(f'[START] Chosen base a: {self.chosen_a}')\n",
        "            if not self.random_coprime_only:\n",
        "                gcd = math.gcd(self.chosen_a, self.N)\n",
        "                if gcd != 1:\n",
        "                    print(f'=> {self.chosen_a} and {self.N} share common factor: {self.N} = {gcd} * {self.N // gcd}')\n",
        "                    return gcd, self.N // gcd\n",
        "\n",
        "            print(f'>>> {self.chosen_a} and {self.N} are coprime => Perform Quantum Phase Estimation to find {self.chosen_a}^r - 1 = 0 (MOD {self.N})')\n",
        "            if not self._quantum_period_finding():\n",
        "                a_values.remove(self.chosen_a)\n",
        "                self.r = self.chosen_a = self.qpe_circuit = None\n",
        "                continue\n",
        "\n",
        "            factors = self._classical_postprocess()\n",
        "            if factors: return factors\n",
        "            a_values.remove(self.chosen_a)\n",
        "            self.r = self.chosen_a = self.qpe_circuit = None\n",
        "        print(f'[FAIL] No non-trivial factors found after {self.max_attempts} attempts.')\n",
        "\n",
        "\n",
        "    def _is_N_invalid(self):\n",
        "        if self.N <= 3:\n",
        "            print('[ERR] N must be > 3')\n",
        "            return 1, self.N\n",
        "\n",
        "        if self.N % 2 == 0:\n",
        "            print(f'=> {self.N} is an even number: {self.N} = 2 * {self.N // 2}')\n",
        "            return 2, self.N // 2\n",
        "        \n",
        "        if sympy.isprime(self.N):\n",
        "            print(f'=> {self.N} is a prime number: {self.N} = 1 * {self.N}')\n",
        "            return 1, self.N\n",
        "        \n",
        "        max_exponent = int(math.log2(self.N)) # Start with a large exponent and reduce\n",
        "        for k in range(max_exponent, 1, -1):\n",
        "            p = round(self.N ** (1 / k))\n",
        "            if p ** k == self.N: \n",
        "                print(f'=> {self.N} is a power of prime: {self.N} = {p}^{k}')\n",
        "                return p, k\n",
        "        return False\n",
        "    \n",
        "    \n",
        "    def _quantum_period_finding(self):\n",
        "        while self.chosen_a ** self.r % self.N != 1: # QPE + continued fractions may find wrong r\n",
        "            self.qpe_circuit = QPECircuit(self.chosen_a, self.N) # Find phase s/r\n",
        "            result = self.qpe_circuit.collapse(self.simulator)\n",
        "            state_bin = result.get_memory()[0]\n",
        "            state_dec = int(state_bin, 2) # Convert to decimal\n",
        "            bits_count = 2 ** (self.N.bit_length() - 1)\n",
        "            phase = state_dec / bits_count\n",
        "\n",
        "            # Continued fraction to find r\n",
        "            self.r = Fraction(phase).limit_denominator(self.N).denominator # Get fraction that most closely approximates phase\n",
        "            if self.r > self.N or self.r == 1: # Safety check to avoid infinite loops\n",
        "                print(f'[ERR] Invalid period found: r = {self.r} => Retry with different a.')\n",
        "                return False\n",
        "\n",
        "        print(f'>>> Output State: |{state_bin}⟩ = {state_dec} (dec) => Phase = {state_dec} / {bits_count} = {phase:.3f}')\n",
        "        return True\n",
        "\n",
        "\n",
        "    def _classical_postprocess(self):\n",
        "        # Classical postprocessing to find factors from the period\n",
        "        print(f'>>> Found r = {self.r} => a^{{r/2}} ± 1 = {self.chosen_a:.0f}^{self.r/2:.0f} ± 1')\n",
        "\n",
        "        if self.r % 2 != 0:\n",
        "            print(f'[ERR] r = {self.r} is odd => Retry with different a.')\n",
        "            return None\n",
        "\n",
        "        int1, int2 = self.chosen_a ** (self.r // 2) - 1, self.chosen_a ** (self.r // 2) + 1\n",
        "        if int1 % self.N == 0 or int2 % self.N == 0:\n",
        "            print(f'[ERR] {self.chosen_a}^{self.r/2:.0f} ± 1 is a multiple of {self.N} => Retry with different a.')\n",
        "            return None\n",
        "\n",
        "        factor1, factor2 = math.gcd(int1, self.N), math.gcd(int2, self.N)\n",
        "        if factor1 not in [1, self.N] and factor2 not in [1, self.N]: # Check to see if factor is non-trivial\n",
        "            print(f'[DONE] Successfully found non-trivial factors: {self.N} = {factor1} * {factor2}')\n",
        "            return factor1, factor2\n",
        "\n",
        "        print(f'[FAIL] Trivial factors found: [1, {self.N}] => Retry with different a.')\n",
        "        return None"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "xDVtd62PmxMu"
      },
      "source": [
        "By measuring the state after applying the **inverse QFT**, I obtain the phase $\\phi$, from which I can deduce the order $r$: $\\phi = \\frac{s}{r} \\implies r \\approx \\frac{1}{\\phi}$\n",
        "\n",
        "The continued fraction representation of a rational number $\\frac{p}{q}$ helps in extracting the period $r$ from the phase $\\frac{s}{r}$.\n",
        "\n",
        "$$\\frac{p}{q} = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{\\ddots + \\cfrac{1}{a_n}}}}$$\n",
        "\n",
        "Using the order $r$, I apply **classical postprocessing** to find the factors of $N$. If $r$ is even and $x^{r/2} \\neq \\pm 1 \\text{ (mod N)}$, then the factors are given by: $\\gcd(x^{r/2} \\pm 1, N)$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "cellView": "form",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "8mprSlwHBATJ",
        "outputId": "47c3b3e5-b439-469b-e59e-7107b29ce25b"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "\n",
            "===== Attempt 1/11 =====\n",
            "[START] Chosen base a: 20\n",
            ">>> 20 and 21 are coprime => Perform Quantum Phase Estimation to find 20^r - 1 = 0 (MOD 21)\n",
            ">>> Output State: |00111⟩ = 7 (dec) => Phase = 7 / 16 = 0.438\n",
            ">>> Found r = 16 => a^{r/2} ± 1 = 20^8 ± 1\n",
            "[ERR] 20^8 ± 1 is a multiple of 21 => Retry with different a.\n",
            "\n",
            "===== Attempt 2/11 =====\n",
            "[START] Chosen base a: 5\n",
            ">>> 5 and 21 are coprime => Perform Quantum Phase Estimation to find 5^r - 1 = 0 (MOD 21)\n",
            "[ERR] Invalid period found: r = 1 => Retry with different a.\n",
            "\n",
            "===== Attempt 3/11 =====\n",
            "[START] Chosen base a: 13\n",
            ">>> 13 and 21 are coprime => Perform Quantum Phase Estimation to find 13^r - 1 = 0 (MOD 21)\n",
            ">>> Output State: |01100⟩ = 12 (dec) => Phase = 12 / 16 = 0.750\n",
            ">>> Found r = 4 => a^{r/2} ± 1 = 13^2 ± 1\n",
            "[ERR] 13^2 ± 1 is a multiple of 21 => Retry with different a.\n",
            "\n",
            "===== Attempt 4/11 =====\n",
            "[START] Chosen base a: 8\n",
            ">>> 8 and 21 are coprime => Perform Quantum Phase Estimation to find 8^r - 1 = 0 (MOD 21)\n",
            ">>> Output State: |01000⟩ = 8 (dec) => Phase = 8 / 16 = 0.500\n",
            ">>> Found r = 2 => a^{r/2} ± 1 = 8^1 ± 1\n",
            "[DONE] Successfully found non-trivial factors: 21 = 7 * 3\n"
          ]
        },
        {
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            "text/plain": [
              "<Figure size 2126.94x953.167 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "# @markdown ### **5. Run the Factorization** { display-mode: \"form\" }\n",
        "number_to_factor = 21 # @param {type:\"slider\", min: 15, max: 55, step: 1}\n",
        "max_attempts = -1 # @param {type:\"slider\", min:-1, max:100, step:10}\n",
        "random_coprime_only = False # @param {type:\"boolean\"}\n",
        "# @markdown ***Note**: `max_attempts` will be limited to number of available values.\n",
        "\n",
        "shor = ShorAlgorithm(number_to_factor, max_attempts, random_coprime_only, AerSimulator())\n",
        "factors = shor.execute()\n",
        "try: display(shor.qpe_circuit.draw(output='mpl', fold=-1))\n",
        "except Exception: pass"
      ]
    }
  ],
  "metadata": {
    "colab": {
      "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": "3.11.7"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}