PaddleQuantum_Linalg_and_Qinfo_EN.ipynb

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    "# Introduction to Linear Algebra and Quantum Information modules of Paddle Quantum\n",
    "*Copyright (c) 2023 Institute for Quantum Computing, Baidu Inc. All Rights Reserved.*\n",
    "\n",
    "`linalg` and `qinfo` module are Paddle Quantum modules that support operations of matrices and functions of quantum information respectively. `linalg` module provides many matrix operations includeing transpose, conjuagte, tensor product and trace. And `qinfo` module mainly includes partial trace, different quantum entropy, fidelity of quantum states and trace distance, which are functions that related to quantum information theory."
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   "source": [
    "## Common operations of matrix\n",
    "\n",
    "We know that quantum mechanics has a close connection with linear algebra. Here, we will introduce the common operations of matrix in `linalg` module with simple examples."
   ]
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      "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",
    "paddle_quantum.set_backend(\"density_matrix\")"
   ]
  },
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   "source": [
    "We first initialize two random unitary matrices through built-in function of Paddle Quantum. The matrix A and B here are both in `paddle.Tensor` format."
   ]
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     "text": [
      "Matrix A is:\n",
      "Tensor(shape=[2, 2], dtype=complex64, place=Place(cpu), stop_gradient=True,\n",
      "       [[(-0.008479282259941101-0.3407862186431885j),\n",
      "          (0.6076244115829468-0.7173460721969604j)  ],\n",
      "        [(-0.3203158378601074-0.8838498592376709j)  ,\n",
      "         (-0.12643268704414368+0.3165784478187561j) ]])\n",
      "\n",
      "Matrix B is:\n",
      "Tensor(shape=[2, 2], dtype=complex64, place=Place(cpu), stop_gradient=True,\n",
      "       [[(-0.6520524621009827-0.2549170255661011j) ,\n",
      "         (-0.5733906030654907-0.42552101612091064j)],\n",
      "        [ (0.5613574981689453+0.44127389788627625j),\n",
      "         (-0.4174516499042511-0.5620402097702026j) ]])\n"
     ]
    }
   ],
   "source": [
    "A = unitary_random(num_qubits=1)\n",
    "B = unitary_random(num_qubits=1)\n",
    "print(f\"Matrix A is:\\n{A}\\n\")\n",
    "print(f\"Matrix B is:\\n{B}\")\n"
   ]
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   "source": [
    "**matrix multiplication**：There are two ways to calculate matrix multiplication, the first way is using `matmul` method of Paddle, and the second way is directly using `@`.\n",
    "\n",
    "**transpose**：The transpose of matrix A can be calculated using `A.t()`.\n",
    "\n",
    "**conjugate**：The conjugate of matrix A can be calculated using `A.conjugate()`.\n",
    "\n",
    "**conjugate transpose**：There are two ways to calculate conjugate transpose of a matrix. The first way is calculating the conjugate of the matrix followed by calculating the transpose, using `A.conj().t()`; the second way is directly using `dagger` method of Paddle Quantum.\n",
    "\n",
    "**tensor product of two matrices**：The tensor product of two matrices can be calculated using `paddle.kron()`.\n",
    "\n",
    "**tensor product of multiple matrices**：The tensor product of multiple matrices can be calculated using `NKron()` of Paddle Quantum.\n",
    "\n",
    "**trace of matrix**：The trace of a matrix can be calculated using `paddle.trace()`.\n",
    "\n",
    "**partial trace of matrix**：The partial trace of a matrix can be calculated using `partial_trace()` of Paddle Quantum.\n",
    "\n",
    "**eigenvalue and eigenvector of matrix**：The eigenvalues and eigenvectors of a matrix can be calculated using `eig`, and the eigenvalues can be directly calculated using `eigvals`."
   ]
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   "cell_type": "code",
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   "source": [
    "multiplyAB_1 = paddle.matmul(A, B)  # matrix multiplication\n",
    "multiplyAB_2 = A @ B  # matrix multiplication\n",
    "A_transpose = A.t()  # transpose\n",
    "A_conjugate = A.conj()  # conjugate\n",
    "A_dagger = dagger(A)  # conjugate transpose\n",
    "tensor_product_AB = paddle.kron(A, B)  # tensor product of two matrices\n",
    "tensor_product_ABA = NKron(A, B, A)  # tensor product of multiple matrices\n",
    "A_trace = paddle.trace(A)  # trace of the matrix\n",
    "partial_trace_matrix = partial_trace(tensor_product_AB, dim1=2, dim2=2, A_or_B=2)  # partial trace of matrix\n",
    "eigenvalue, eigenvector = eig(A)  # eigenvalue and eigenvector of the matrix"
   ]
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   "source": [
    "## Common functions in quantum information"
   ]
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   "source": [
    "We first initialize two random single qubit quantum states with `density_matrix_random`."
   ]
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     "text": [
      "The first quantum state is:\n",
      " Tensor(shape=[2, 2], dtype=complex64, place=Place(cpu), stop_gradient=True,\n",
      "       [[(0.5706692337989807+0j)                  ,\n",
      "         (0.4849703907966614+0.09904339164495468j)],\n",
      "        [(0.4849703907966614-0.09904339164495468j),\n",
      "         (0.4293307662010193+0j)                  ]])\n",
      "\n",
      "The second quantum state is:\n",
      " Tensor(shape=[2, 2], dtype=complex64, place=Place(cpu), stop_gradient=True,\n",
      "       [[ (0.8634329438209534+0j)                   ,\n",
      "         (-0.04915342852473259+0.33985358476638794j)],\n",
      "        [(-0.04915342852473259-0.33985358476638794j),\n",
      "          (0.13656707108020782+0j)                  ]])\n"
     ]
    }
   ],
   "source": [
    "state1 = density_matrix_random(1)\n",
    "print(f\"The first quantum state is:\\n {state1}\\n\")\n",
    "state2 = density_matrix_random(1)\n",
    "print(f\"The second quantum state is:\\n {state2}\")"
   ]
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   "source": [
    "We can use `state_fidelity()` to calculate the fidelity of two quantum states."
   ]
  },
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   "cell_type": "code",
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     "text": [
      "The fidelity is:\n",
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [0.75565338])\n"
     ]
    },
    {
     "name": "stderr",
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     "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\"The fidelity is:\\n{fidelity}\")"
   ]
  },
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   "source": [
    "It is worth noting that the functions of Paddle Quantum that related to quantum states supports different format of input, including `paddle.Tensor`, `numpy.ndarray`, and `State`. At the same time, the output format is consistent with the input format. For example, the output will be `numpy.ndarray` format if the input is also `numpy.ndarray` format."
   ]
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   "cell_type": "code",
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     "text": [
      "0.7556533814816342\n"
     ]
    }
   ],
   "source": [
    "# Input with numpy.ndarray format\n",
    "state1 = state1.numpy()\n",
    "state2 = state2.numpy()\n",
    "print(state_fidelity(state1, state2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [0.75565338])\n"
     ]
    }
   ],
   "source": [
    "# Input with State format\n",
    "state1 = State(state1)\n",
    "state2 = State(state2)\n",
    "print(state_fidelity(state1, state2))"
   ]
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   "source": [
    "We can use `trace_distance` to calculate the trace distance between two quantum states."
   ]
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   "cell_type": "code",
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     "text": [
      "The trace distance is:\n",
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [0.65497208])\n"
     ]
    }
   ],
   "source": [
    "TraceDistance = trace_distance(state1, state2)\n",
    "print(f\"The trace distance is:\\n{TraceDistance}\")"
   ]
  },
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   "source": [
    "We can use `von_neumann_entropy()` to calculate the von neumann entropy of a quantum state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "The von veumann entropy of the quantum state is:\n",
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [0.00000026])\n"
     ]
    }
   ],
   "source": [
    "v_entropy = von_neumann_entropy(state2)\n",
    "print(f\"The von veumann entropy of the quantum state is:\\n{v_entropy}\")"
   ]
  },
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   "source": [
    "We can use `relative_entropy()` to calculate the relative entropy of two quantum states."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "logm result may be inaccurate, approximate err = 3.3328131580525864e-07\n",
      "logm result may be inaccurate, approximate err = 2.3121323283812061e-07\n",
      "The relative entropy is:\n",
      "Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,\n",
      "       [11.14819717])\n"
     ]
    }
   ],
   "source": [
    "r_entropy = relative_entropy(state1, state2)\n",
    "print(f\"The relative entropy is:\\n{r_entropy}\")"
   ]
  }
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