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    "## Project 3: Markovian reservoir engineering\n",
    "\n",
    "In Chapter 5, we introduced an example in which the interaction of a two-qubit system with its environment results in a maximally entangled state between the system qubits. In this project, we will verify them experimentally.\n",
    "\n",
    "The overall goal is to use the given circuits (appended again below) to drive the state of the system from maximally mixed to $|\\psi^-\\rangle$. In this project, we will not use the tomographic reconstruction of the two-qubit state; instead, we will measure the populations of the four Bell states by changing basis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from qiskit import QuantumRegister, QuantumCircuit\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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     "output_type": "stream",
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      "<ipython-input-2-5967a9e13278>:28: DeprecationWarning: The QuantumCircuit.cu3 method is deprecated as of 0.16.0. It will be removed no earlier than 3 months after the release date. You should use the QuantumCircuit.cu method instead, where cu3(ϴ,φ,λ) = cu(ϴ,φ,λ,0).\n  zz.cu3(theta, 0.0, 0.0, q[a_zz], q[system[1]])\n"
     ]
    },
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\n"
     },
     "metadata": {},
     "execution_count": 2
    }
   ],
   "source": [
    "#######################\n",
    "#  ZZ pump on IBMQX2  #\n",
    "#######################\n",
    "\n",
    "# Quantum register\n",
    "q = QuantumRegister(5, name='q')\n",
    "\n",
    "# Quantum circuit  \n",
    "zz = QuantumCircuit(q)\n",
    "\n",
    "# ZZ pump acting on system qubits\n",
    "## Qubit identification\n",
    "system = [2, 1]\n",
    "a_zz = 0\n",
    "\n",
    "## Define pump efficiency \n",
    "## and corresponding rotation\n",
    "p = 0.5\n",
    "theta = 2 * np.arcsin(np.sqrt(p))\n",
    "\n",
    "## Construct circuit\n",
    "### Map information to ancilla\n",
    "zz.cx(q[system[0]], q[system[1]])\n",
    "zz.x(q[a_zz])\n",
    "zz.cx(q[system[1]], q[a_zz])\n",
    "    \n",
    "### Conditional rotation\n",
    "zz.cu3(theta, 0.0, 0.0, q[a_zz], q[system[1]])\n",
    "    \n",
    "### Inverse mapping\n",
    "zz.cx(q[system[1]], q[a_zz])\n",
    "zz.cx(q[system[0]], q[system[1]])\n",
    "\n",
    "# Draw circuit\n",
    "zz.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 638.12x325.08 with 1 Axes>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#######################\n",
    "#  XX pump on IBMQX2  #\n",
    "#######################\n",
    "\n",
    "# Quantum register\n",
    "q = QuantumRegister(5, name='q')\n",
    "\n",
    "# Quantum circuit  \n",
    "xx = QuantumCircuit(q)\n",
    "\n",
    "# XX pump acting on system qubits\n",
    "## Qubit identification\n",
    "system = [2, 1]\n",
    "a_xx = 4\n",
    "\n",
    "## Define pump efficiency \n",
    "## and corresponding rotation\n",
    "p = 0.5\n",
    "theta = 2 * np.arcsin(np.sqrt(p))\n",
    "\n",
    "## Construct circuit\n",
    "### Map information to ancilla\n",
    "xx.cx(q[system[0]], q[system[1]])\n",
    "xx.h(q[system[0]])\n",
    "xx.x(q[a_xx])\n",
    "xx.cx(q[system[0]], q[a_xx])\n",
    "    \n",
    "### Conditional rotation\n",
    "xx.cu3(theta, 0.0, 0.0, q[a_xx], q[system[0]])\n",
    "    \n",
    "### Inverse mapping\n",
    "xx.cx(q[system[0]], q[a_xx])\n",
    "xx.h(q[system[0]])\n",
    "xx.cx(q[system[0]], q[system[1]])\n",
    "\n",
    "# Draw circuit\n",
    "xx.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "<Figure size 507.997x325.08 with 1 Axes>"
      ],
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\n"
     },
     "metadata": {},
     "execution_count": 3
    }
   ],
   "source": [
    "###########################\n",
    "#  ZZ-XX pumps on IBMQX2  #\n",
    "###########################\n",
    "\n",
    "# Quantum register\n",
    "q = QuantumRegister(5, name='q')\n",
    "\n",
    "# Quantum circuit  \n",
    "zz_xx = QuantumCircuit(q)\n",
    "\n",
    "# ZZ and XX pumps acting on system qubits\n",
    "## Qubit identification\n",
    "system = [2, 1]\n",
    "a_zz = 0\n",
    "a_xx = 4\n",
    "\n",
    "## Define pump efficiency \n",
    "## and corresponding rotation\n",
    "p = 0.5\n",
    "theta = 2 * np.arcsin(np.sqrt(p))\n",
    "\n",
    "## Construct circuit\n",
    "## ZZ pump\n",
    "### Map information to ancilla\n",
    "zz_xx.cx(q[system[0]], q[system[1]])\n",
    "zz_xx.x(q[a_zz])\n",
    "zz_xx.cx(q[system[1]], q[a_zz])\n",
    "    \n",
    "### Conditional rotation\n",
    "zz_xx.cu3(theta, 0.0, 0.0, q[a_zz], q[system[1]])\n",
    "    \n",
    "### Inverse mapping\n",
    "zz_xx.cx(q[system[1]], q[a_zz])\n",
    "#zz_xx.cx(q[system[0]], q[system[1]])\n",
    "\n",
    "## XX pump\n",
    "### Map information to ancilla\n",
    "#zz_xx.cx(q[system[0]], q[system[1]])\n",
    "zz_xx.h(q[system[0]])\n",
    "zz_xx.x(q[a_xx])\n",
    "zz_xx.cx(q[system[0]], q[a_xx])\n",
    "    \n",
    "### Conditional rotation\n",
    "zz_xx.cu3(theta, 0.0, 0.0, q[a_xx], q[system[0]])\n",
    "    \n",
    "### Inverse mapping\n",
    "zz_xx.cx(q[system[0]], q[a_xx])\n",
    "zz_xx.h(q[system[0]])\n",
    "zz_xx.cx(q[system[0]], q[system[1]])\n",
    "\n",
    "# Draw circuit\n",
    "zz_xx.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Task 1\n",
    "Write three functions, one for each channel (ZZ, XX, and their composition), returning a quantum circuit implementing the channel on the system qubits for a given value of the efficiency parameter $p$ (in the composition map, impose equal efficiencies for both). The circuits must include the **measurement of the system qubits in the Bell basis**. Tip: remove consecutive CNOT gates (and single-qubit ones) resulting in identity.\n",
    "\n",
    "Find the suggested structure for the functions below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def zz_pump(q, c, p, system, ancilla):\n",
    "    \"\"\"Returns a QuantumCircuit implementing the ZZ pump channel on the system qubits\n",
    "    \n",
    "    Args:\n",
    "        q (QuantumRegister): the register to use for the circuit\n",
    "        c (ClassicalRegister): the register to use for the measurement of the system qubits\n",
    "        p (float): the efficiency for the channel, between 0 and 1\n",
    "        system (list): list of indices for the system qubits\n",
    "        ancilla (int): index for the ancillary qubit\n",
    "    \n",
    "    Returns:\n",
    "        A QuantumCircuit object\n",
    "    \"\"\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def xx_pump(q, c, p, system, ancilla):\n",
    "    \"\"\"Returns a QuantumCircuit implementing the XX pump channel on the system qubits\n",
    "    \n",
    "    Args:\n",
    "        q (QuantumRegister): the register to use for the circuit\n",
    "        c (ClassicalRegister): the register to use for the measurement of the system qubits\n",
    "        p (float): the efficiency for the channel, between 0 and 1\n",
    "        system (list): list of indices for the system qubits\n",
    "        ancilla (int): index for the ancillary qubit\n",
    "    \n",
    "    Returns:\n",
    "        A QuantumCircuit object\n",
    "    \"\"\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def zz_xx_pump(q, c, p, system, ancillae):\n",
    "    \"\"\"Returns a QuantumCircuit implementing the composition channel on the system qubits\n",
    "    \n",
    "    Args:\n",
    "        q (QuantumRegister): the register to use for the circuit\n",
    "        c (ClassicalRegister): the register to use for the measurement of the system qubits\n",
    "        p (float): the efficiency for both channels, between 0 and 1\n",
    "        system (list): list of indices for the system qubits\n",
    "        ancillae (list): list of indices for the ancillary qubits\n",
    "    \n",
    "    Returns:\n",
    "        A QuantumCircuit object\n",
    "    \"\"\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Task 2\n",
    "\n",
    "We need to set the two-qubit system initially in the maximally mixed state $\\rho = I_4/4$, where $I_4$ is the $4\\times4$ identity matrix. In principle, this could be done by entangling the system with other ancillary qubits, but that would require two extra qubits in our simulation. Instead, we can create a *proper statistical mixture*. This means that we can obtain $\\rho$ by mixing four initially pure states, e.g. the two-qubit computational basis states.\n",
    "\n",
    "For each initial state of the qubits $|00\\rangle$, $|01\\rangle$, $|10\\rangle$ and $|11\\rangle$, apply the three channels for different values of $p \\in [0,1]$ and save the resulting Bell populations for the next task."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Task 3\n",
    "\n",
    "To finally simulate the effect of the different channels on the maximally mixed state, average all the results over the four initial states. Plot the results as a function of the channel efficiency $p$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Homework\n",
    "\n",
    "Run the circuits on the IBM Q Experience with noise mitigation and plot the results."
   ]
  }
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