{
 "cells": [
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   "source": [
    "# 量桨线性代数与量子信息模块简介\n",
    "*Copyright (c) 2023 Institute for Quantum Computing, Baidu Inc. All Rights Reserved.*\n",
    "\n",
    "`linalg` 和 `qinfo` 模块是量桨封装各类矩阵操作和量子信息相关功能的模块。`linalg` 包含了矩阵的转置、共轭、迹等常用的函数，而 `qinfo` 模块则主要包含了偏迹，各类量子态的熵，保真度，迹距离等常见的量子信息相关的函数。"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 矩阵的常见运算\n",
    "\n",
    "我们知道量子力学和线性代数有着密不可分的关联，在这里，我们通过简单的例子介绍量桨 `linalg` 模块和飞桨内常见的矩阵相关的运算函数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "c:\\Users\\liugeng02\\Anaconda3\\envs\\2023q2\\lib\\site-packages\\openfermion\\hamiltonians\\hartree_fock.py:11: DeprecationWarning: Please use `OptimizeResult` from the `scipy.optimize` namespace, the `scipy.optimize.optimize` namespace is deprecated.\n",
      "  from scipy.optimize.optimize import OptimizeResult\n",
      "c:\\Users\\liugeng02\\Anaconda3\\envs\\2023q2\\lib\\site-packages\\paddle\\tensor\\creation.py:125: DeprecationWarning: `np.object` is a deprecated alias for the builtin `object`. To silence this warning, use `object` by itself. Doing this will not modify any behavior and is safe. \n",
      "Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations\n",
      "  if data.dtype == np.object:\n",
      "c:\\Users\\liugeng02\\Anaconda3\\envs\\2023q2\\lib\\site-packages\\paddle\\tensor\\creation.py:125: DeprecationWarning: `np.object` is a deprecated alias for the builtin `object`. To silence this warning, use `object` by itself. Doing this will not modify any behavior and is safe. \n",
      "Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations\n",
      "  if data.dtype == np.object:\n"
     ]
    }
   ],
   "source": [
    "import paddle\n",
    "import paddle_quantum\n",
    "from paddle_quantum.state import State\n",
    "from paddle.linalg import eigvals, eig\n",
    "from paddle_quantum.linalg import dagger, NKron, density_matrix_random, unitary_random\n",
    "from paddle_quantum.qinfo import (\n",
    "    partial_trace,\n",
    "    state_fidelity,\n",
    "    trace_distance,\n",
    "    von_neumann_entropy,\n",
    "    relative_entropy,\n",
    ")\n",
    "\n",
    "\n",
    "paddle_quantum.set_backend(\"density_matrix\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们先通过量桨内置的函数创建两个随机酉矩阵，在这里两个矩阵A和B均是 `paddle.Tensor` 形式。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "矩阵A为：\n",
      "Tensor(shape=[2, 2], dtype=complex64, place=Place(cpu), stop_gradient=True,\n",
      "       [[ (0.2921386957168579-0.1418590545654297j),\n",
      "         (-0.7422900795936584+0.5861198306083679j)],\n",
      "        [(0.9016497135162354-0.28558504581451416j),\n",
      "         (0.28133758902549744-0.1622287780046463j)]])\n",
      "\n",
      "矩阵B为：\n",
      "Tensor(shape=[2, 2], dtype=complex64, place=Place(cpu), stop_gradient=True,\n",
      "       [[(-0.30973920226097107-0.29339805245399475j),\n",
      "          (0.8378809690475464+0.34049180150032043j) ],\n",
      "        [ (0.8848108649253845-0.18732032179832458j) ,\n",
      "          (0.3566453754901886-0.2341471016407013j)  ]])\n"
     ]
    }
   ],
   "source": [
    "A = unitary_random(num_qubits=1)\n",
    "B = unitary_random(num_qubits=1)\n",
    "print(f\"矩阵A为：\\n{A}\\n\")\n",
    "print(f\"矩阵B为：\\n{B}\")\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**矩阵乘法**：矩阵乘法A*B有两种实现方式，第一种是使用paddle内部的 `matmul` 函数，第二种是直接使用 `@` 符号代表矩阵乘法。\n",
    "\n",
    "**矩阵转置**：计算矩阵A的转置可以使用 `A.t()`。\n",
    "\n",
    "**矩阵共轭**：计算矩阵A的共轭可以使用 `A.conj()` 。\n",
    "\n",
    "**矩阵的共轭转置**：计算一个矩阵的共轭转置有两种方法，一种是将两步操作分开，先后求取其共轭和转置 `A.conj().t()` ；另一种是直接调用量桨内部的 `dagger` 函数求取共轭转置。\n",
    "\n",
    "**矩阵的张量积**：矩阵的张量积 $A\\otimes B$ 可以使用 `paddle.kron()` 来计算。\n",
    "\n",
    "**多个矩阵的张量积**：多个矩阵的张量积可以使用量桨 `linalg` 内的 `NKron()` 来计算。\n",
    "\n",
    "**矩阵的迹**：对矩阵求迹可以使用 `paddle.trace()` 函数。\n",
    "\n",
    "**矩阵的偏迹**：对矩阵求偏迹可以使用量桨 `qinfo` 内的 `partial_trace()` 函数。\n",
    "\n",
    "**矩阵的特征值与特征向量**：矩阵的特征值和特征向量可以使用 `eig` 函数计算，也可以使用 `eigvals` 来单独计算矩阵的特征值。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "multiplyAB_1 = paddle.matmul(A, B)  # 矩阵乘法\n",
    "multiplyAB_2 = A @ B  # 矩阵乘法\n",
    "A_transpose = A.t()  # 矩阵的转置\n",
    "A_conjugate = A.conj()  # 矩阵的共轭\n",
    "A_dagger = dagger(A)  # 矩阵的共轭转置\n",
    "tensor_product_AB = paddle.kron(A, B)  # 矩阵的张量积\n",
    "tensor_product_ABA = NKron(A, B, A)  # 多个矩阵的张量积\n",
    "A_trace = paddle.trace(A)  # 矩阵的迹\n",
    "partial_trace_matrix = partial_trace(tensor_product_AB, dim1=2, dim2=2, A_or_B=2)  # 矩阵的偏迹\n",
    "eigenvalue, eigenvector = eig(A)  # 矩阵的特征值与特征向量"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 量子信息中的常见函数"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们先使用 `linalg` 里的 `density_matrix_random` 创建两个随机单比特量子态。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "第一个量子态为：\n",
      " Tensor(shape=[2, 2], dtype=complex64, place=Place(cpu), stop_gradient=True,\n",
      "       [[ (0.8994134664535522+0j)                   ,\n",
      "         (-0.28608179092407227-0.09287679195404053j)],\n",
      "        [(-0.28608179092407227+0.09287679195404053j),\n",
      "          (0.10058653354644775+0j)                  ]])\n",
      "\n",
      "第二个量子态为：\n",
      " Tensor(shape=[2, 2], dtype=complex64, place=Place(cpu), stop_gradient=True,\n",
      "       [[(0.17564187943935394+0j)                 ,\n",
      "         (0.32964783906936646-0.1900634616613388j)],\n",
      "        [(0.32964783906936646+0.1900634616613388j),\n",
      "         (0.8243581056594849+0j)                  ]])\n"
     ]
    }
   ],
   "source": [
    "state1 = density_matrix_random(1)\n",
    "print(f\"第一个量子态为：\\n {state1}\\n\")\n",
    "state2 = density_matrix_random(1)\n",
    "print(f\"第二个量子态为：\\n {state2}\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们利用 `state_fidelity()` 可以获得两个量子态之间保真度。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "两个量子态之间的保真度为:\n",
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [0.29595008])\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "c:\\Users\\liugeng02\\Anaconda3\\envs\\2023q2\\lib\\site-packages\\paddle\\fluid\\framework.py:1104: DeprecationWarning: `np.bool` is a deprecated alias for the builtin `bool`. To silence this warning, use `bool` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.bool_` here.\n",
      "Deprecated in NumPy 1.20; for more details and guidance: https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations\n",
      "  elif dtype == np.bool:\n"
     ]
    }
   ],
   "source": [
    "fidelity = state_fidelity(state1, state2)\n",
    "print(f\"两个量子态之间的保真度为:\\n{fidelity}\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "值得注意的是，量桨有关量子态的函数兼容多种不同形式的输入，其形式可以是 `paddle.Tensor`, `numpy.ndarray`, 或是 `State`。同时，函数输出的格式与输入格式是匹配的，例如输入 `numpy.ndarray` 形式的数据将会同样获得 `numpy.ndarray` 形式的结果。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.29595009466850986\n"
     ]
    }
   ],
   "source": [
    "# 以numpy.ndarray形式输入\n",
    "state1 = state1.numpy()\n",
    "state2 = state2.numpy()\n",
    "print(state_fidelity(state1, state2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [0.29595008])\n"
     ]
    }
   ],
   "source": [
    "# 以State形式输入\n",
    "state1 = State(state1)\n",
    "state2 = State(state2)\n",
    "print(state_fidelity(state1, state2))"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "利用 `trace_distance()` 我们可以计算两个量子态之间的迹距离"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "两个量子态之间的迹距离为:\n",
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [0.95520324])\n"
     ]
    }
   ],
   "source": [
    "TraceDistance = trace_distance(state1, state2)\n",
    "print(f\"两个量子态之间的迹距离为:\\n{TraceDistance}\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "使用 `von_neumann_entropy()` 可以计算量子态的冯诺依曼熵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "量子态的冯诺依曼熵为:\n",
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [0.])\n"
     ]
    }
   ],
   "source": [
    "v_entropy = von_neumann_entropy(state2)\n",
    "print(f\"量子态的冯诺依曼熵为:\\n{v_entropy}\")"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "使用 `relative_entropy()` 可以计算两个量子态的相对熵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "logm result may be inaccurate, approximate err = 1.2365573141431553e-07\n",
      "logm result may be inaccurate, approximate err = 2.1966168634145265e-08\n",
      "两个量子态之间的相对熵为:\n",
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [23.84987831])\n"
     ]
    }
   ],
   "source": [
    "r_entropy = relative_entropy(state1, state2)\n",
    "print(f\"两个量子态之间的相对熵为:\\n{r_entropy}\")"
   ]
  }
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