introduction/PaddleQuantum_Custom_CN.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 利用 Paddle Quantum 的 `gate` 和 `channel` 模块构建量子操作\n",
    "\n",
    "_Copyright (c) 2023 Institute for Quantum Computing, Baidu Inc. All Rights Reserved._\n",
    "\n",
    "量桨的 `gate` 和 `channel` 模块可以用于构建量子操作。它们可以直接作用在一个 `State` 上或者插入到一个量子电路当中。量桨提供了方便的接口可以快速地创建常见的量子操作，用户也可以创建自定义操作。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "import paddle_quantum\n",
    "from paddle_quantum.ansatz import Circuit\n",
    "from paddle_quantum.gate.custom import Oracle\n",
    "from paddle_quantum.channel import *\n",
    "from paddle_quantum.state.common import zero_state\n",
    "from paddle_quantum.qinfo import qft_generation, random_channel\n",
    "from paddle_quantum.linalg import dagger"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 自定义量子门\n",
    "之前，我们已经看到如何创建一个量子门并将其插入到电路中。用户也可以使用 Oracle 类来实现自定义的量子门。\n",
    "\n",
    "例如，考虑 n 量子比特下的傅里叶逆变换 (Inverse Quantum Fourier Transform, inverse QFT)，它是一个酉算符，其矩阵形式为\n",
    "\n",
    "$$\n",
    "\\frac{1}{\\sqrt{N}}\\begin{pmatrix}\n",
    "    1 & 1 & 1 & \\ldots & 1  \\\\\n",
    "    1 & \\omega^{N-1} & \\omega^{N-2} & \\ldots & \\omega \\\\\n",
    "    1 & \\omega^{2(N-1)} & \\omega^{2(N-2)} & \\ldots & \\omega^2 \\\\\n",
    "    \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\\n",
    "    1 & \\omega^{(N-1)(N-1)} & \\omega^{(N-1)(N-2)} & \\ldots & \\omega^{N-1}\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "其中， $N=2^n$ 是系统的维度，$\\omega = e^{2\\pi i/N}$。"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "下面的代码创建了一个实现 2 比特傅里叶逆变换的 oracle 。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "unitary = qft_generation(2) # 通过量桨内置函数得到2比特QFT的矩阵\n",
    "unitary = dagger(unitary) # 通过量桨内置函数对矩阵进行共轭转置，得到 inverse QFT 的矩阵 （对于么正矩阵来说，共轭转置就是求逆的过程）\n",
    "QFT_inverse = Oracle(unitary)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "它是一个量子门，我们可以把它插入到一个量子电路当中并且打印电路。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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hKk9HE0U7BodIAoNDJIHBIZLA4BBJYHCIJDA4RBIYHCIJDA6RBAaHSAKDQySBwSGSwOAQSYjINdl6DmvmkGkygqHBMZlMsFqt9x0FGiqr1QqTyaRrHxTfDB2PA9y5sYLet/IxmUx8VDvpyvDgEI0FPDlAJIHBIZLA4BBJYHCIJDA4RBIYHCIJDA6RBAaHSAKDQySBwSGSwOAQSWBwiCQwOEQSGBwiCf8DKvOyd3u1eAwAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 194x165 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cir = Circuit(3)\n",
    "cir.h()\n",
    "cir.cnot([0,1])\n",
    "cir.append(QFT_inverse) # 插入 oracle \n",
    "cir.cnot([2,1])\n",
    "cir.plot()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意**：假如需要打印电路，那么它不能是不连续的（比如作用在第 1 和第 3 量子比特上），否则将报错。"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "用户可以通过改变 gate_info 的内容去修改打印电路时 oracle 的样式。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 254x165 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gate_info = {\n",
    "    'texname': r'$QFT^{\\dagger}$', # 更改打印的时候 oracle 的名字\n",
    "    'plot_width': 1.5, # 调整打印的时候 oracle 的宽度\n",
    "}\n",
    "# 更新 oracle 的 gate_info\n",
    "QFT_inverse.gate_info.update(gate_info)\n",
    "# 创建和之前一样的电路\n",
    "cir = Circuit(3)\n",
    "cir.h()\n",
    "cir.cnot([0,1])\n",
    "cir.append(QFT_inverse) # 插入 oracle\n",
    "cir.cnot([2,1])\n",
    "cir.plot()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "另外一种插入 oracle 的方法是在搭建电路的时候直接使用 `Circuit.oracle`。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 254x165 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cir=Circuit(3)\n",
    "cir.h()\n",
    "cir.cnot([0,1])\n",
    "# 直接使用 cir.oracle 去构建一个 oracle 门。构建的同时可以给出 gate_info 的自定义内容\n",
    "cir.oracle(unitary, qubits_idx=[0,1], latex_name=r'$QFT^{\\dagger}$', plot_width=1.5) \n",
    "cir.cnot([2,1])\n",
    "cir.plot()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 量子信道\n",
    "\n",
    "量桨同时支持构建通用的量子操作，即构建量子信道。\n",
    "\n",
    "构建信道的方法有两种。一种是通过量桨构建固定模式的信道，或者也可以构建一个自定义信道。"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意**: `gate` 类可以在 `state_vecotr` 或 `density_matrix` 后端下运行，而 `channel` 只能在 `density_matrix` 后端运行。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "paddle_quantum.set_backend('density_matrix') # 更改后端为 'density_matrix'"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "例如，我们可以构建一个比特反转信道，然后作用在 0 态上。常用信道的详细列表可参见 [量桨API](https://qml.baidu.com/api/paddle_quantum.channel.common.html)。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.49999997+0.j 0.        +0.j]\n",
      " [0.        +0.j 0.49999997+0.j]]\n"
     ]
    }
   ],
   "source": [
    "bit_flip_channel = BitFlip(prob=0.5, num_qubits=1) # 定义一个单比特比特反转信道，反转概率为 0.5\n",
    "input = zero_state(1)\n",
    "output = bit_flip_channel(input) # 作用信道在 0 态上\n",
    "print(output)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "自定义信道可以通过其 Kraus, Stinespring 或者 Choi 表达式构建。下面我们展示如何使用这三种不同的表达方式构建一个自定义信道。我们使用量桨的内置函数去随机生成这些表达方式。"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意**： 量桨里对 Choi 矩阵的定义是信道作用在第二个系统上，即\n",
    "$$\n",
    "    J(\\Lambda) = \\sum_{i,j} |i\\rangle\\langle j| \\otimes \\Lambda(|i\\rangle\\langle j|).\n",
    "$$\n",
    "其中，$\\Lambda$ 表示信道。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 随机生成一组 Kraus 算子\n",
    "kraus = random_channel(num_qubits=1, target='Kraus') \n",
    "# 通过 Kraus 算子构建信道\n",
    "custom_kraus = KrausRepr(kraus)\n",
    "\n",
    "# 随机生成一个Stinespring 保距映射\n",
    "stinespring = random_channel(num_qubits=1, target='Stinespring')\n",
    "# 通过 Stinespring 保距映射构建信道\n",
    "custom_stinespring = StinespringRepr(stinespring)\n",
    "\n",
    "# 随机生成一个 Choi 矩阵\n",
    "choi = random_channel(num_qubits=1, target='Choi')\n",
    "# 通过 Choi 矩阵构建信道\n",
    "custom_choi = ChoiRepr(choi)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "之前看到我们可以把一个量子门插入到电路当中，我们同样也可以把一个信道以相同的方法插入到电路当中。不过，目前量桨并不支持打印含有信道的电路。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    " # 默认情况下，信道会作用在第一个比特上。我们可以通过在构建信道的时候给出 qubits_idx 的值来指定它作用在哪个比特上。\n",
    "custom_choi = ChoiRepr(choi, qubits_idx=[2])\n",
    "cir = Circuit(3)\n",
    "cir.append(bit_flip_channel) # 在电路的末尾插入信道\n",
    "cir.h()\n",
    "cir.insert(index=2, operator = custom_choi) # 在电路指定位置插入一个信道\n",
    "cir.bit_flip(0.5, qubits_idx=[2]) # 在构建电路的时候直接构建信道"
   ]
  }
 ],
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