tutorial-ja/317_boolean_ja.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "041_boolean_reduction",
      "provenance": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_-zOj3w97u4Q",
        "colab_type": "text"
      },
      "source": [
        "#３体問題の２体問題への分解\n",
        "\n",
        "QUBOでは通常２体問題と呼ばれる２つの量子ビットの掛け算までしか計算できません。しかし、時々問題を作っている際に、３量子ビット以上の掛け算が出ることがあります。その際には数学的なテクニックを利用して３体問題を２体問題に分解をします。"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pDPWveBOD_Ke",
        "colab_type": "text"
      },
      "source": [
        "##２体問題の解き方\n",
        "例えば下記のようなコスト関数がある時には、そのままその係数をQUBOで計算できます。これは、$q_0$のバイアスと$q_0,q1$間の結合係数を上三角行列で表すことで実現できます。詳しくは他のチュートリアルを参考にしてください。\n",
        "\n",
        "$$E = q_0-q_0q_1$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "gxoMW9sN2dLt",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 136
        },
        "outputId": "3c866918-f483-4b28-83ec-93909f7e10b7"
      },
      "source": [
        "!pip install blueqat"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Collecting blueqat\n",
            "\u001b[?25l  Downloading https://files.pythonhosted.org/packages/bb/86/1b72a7cbe500b861d63e84cc6383fbf3730f08ae69fcd85146ae8e3b8873/blueqat-0.3.10-py3-none-any.whl (46kB)\n",
            "\r\u001b[K     |███████                         | 10kB 15.5MB/s eta 0:00:01\r\u001b[K     |██████████████                  | 20kB 6.5MB/s eta 0:00:01\r\u001b[K     |█████████████████████▏          | 30kB 9.0MB/s eta 0:00:01\r\u001b[K     |████████████████████████████▏   | 40kB 5.6MB/s eta 0:00:01\r\u001b[K     |████████████████████████████████| 51kB 3.6MB/s \n",
            "\u001b[?25hRequirement already satisfied: numpy~=1.12 in /usr/local/lib/python3.6/dist-packages (from blueqat) (1.17.4)\n",
            "Requirement already satisfied: scipy>=1.1.0 in /usr/local/lib/python3.6/dist-packages (from blueqat) (1.3.3)\n",
            "Installing collected packages: blueqat\n",
            "Successfully installed blueqat-0.3.10\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "I0g-6vN97eU1",
        "colab_type": "code",
        "outputId": "0d20126f-cbe0-4a98-ad87-9e5c998b8286",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "import blueqat.opt as wq\n",
        "a = wq.opt()\n",
        "a.qubo = [[1,-1],[0,0]]\n",
        "a.sa()"
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[1, 1]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 2
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "QrfaOgkRDw9k",
        "colab_type": "text"
      },
      "source": [
        "##３体問題の解き方\n",
        "では、３体問題という量子ビットが複数掛け算されているものはどうでしょうか？\n",
        "結合係数は２量子ビット同士の係数しかないので、そのままでは設定できません。\n",
        "\n",
        "$$E = q_0 -q_0q_1q_2$$\n",
        "\n",
        "そこで、今回はこの３体問題を数学的に２体問題に分解します。"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kCrwoRiw3DnF",
        "colab_type": "text"
      },
      "source": [
        "##分解方法\n",
        "分解に際して新しい量子ビットを追加します。下記のように新しい量子ビットを加えて、表現します。\n",
        "\n",
        "$$q_1q_2 = q_3$$\n",
        "\n",
        "ただ、置き換えただけではなく、上記条件を満たすための新しい拘束条件を式の最後に付け加えます。\n",
        "\n",
        "$$E = q_0-q_0q_3+\\delta(3q_3+q_1q_2-2q_1q_3-2q_2q_3)$$\n",
        "\n",
        "係数の$\\delta$はその時々によって調整するハイパーパラメータです。ここでは$\\delta$を0.5にしてみると下記のようになり、\n",
        "\n",
        "$$E = q_0-q_0q_3+1.5q_3+0.5q_1q_2-q_1q_3-q_2q_3$$\n",
        "\n",
        "これを計算にかけると、"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "S7f4vccTDaZZ",
        "colab_type": "code",
        "outputId": "7dcc9532-31af-4fe6-e5a7-6cd106c44f4d",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 34
        }
      },
      "source": [
        "b = wq.opt()\n",
        "b.qubo = [[1,0,0,-1],[0,0,0.5,-1],[0,0,0,-1],[0,0,0,1.5]]\n",
        "b.sa()"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[1, 1, 1, 1]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "SJ5BJf3E6ArS",
        "colab_type": "text"
      },
      "source": [
        "実際に、これは条件を満たしています。\n",
        "\n",
        "$$q_0*q_2=1*1=1=q_3$$\n",
        "\n",
        "全体のエネルギーの遷移は図で確認することができて下記のようになります。"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "IIb4PqrFEzLs",
        "colab_type": "code",
        "outputId": "37e76e9d-0a4a-482e-c8e9-998cbf50f8f4",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        }
      },
      "source": [
        "a.plot()"
      ],
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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u1yIDzVFlibtLqnV6oL0v1TeFkPbpspuSCE1T7I0ediN4p6zR3L9tYtqSNleU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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    }
  ]
}