start working on the code

Author: DavitKhach

Hamiltonian_simulation.ipynb

@@ -196,7 +196,50 @@
  "The code\n",
  "</h3>\n",
  "\n",
- "Before going into details"
+ "Before going into Qiskit code, note that:\n",
+ "\n",
+ "\\begin{equation*}\n",
+ "H_y = u2(\\frac{\\pi}{2}, \\frac{\\pi}{2})\n",
+ "\\qquad \\qquad\n",
+ "H = u2(0, \\pi)\n",
+ "\\\\\n",
+ "u2(\\varphi, \\lambda) = \\frac{1}{\\sqrt{2}}\n",
+ "\\begin{pmatrix}\n",
+ "1 & -e^{i\\lambda} \\\\\n",
+ "e^{i\\varphi} & e^{i(\\varphi + \\lambda)}\n",
+ "\\end{pmatrix}\n",
+ "\\end{equation*}\n",
+ "\n",
+ "So, in the code we will use $u2(\\frac{\\pi}{2}, \\frac{\\pi}{2})$ gate for implementing $H_y$. For $H$ gate we can use either `.h()` or `.u2()` and for better readability we will use `.h()` in the code.\n",
+ "\n",
+ "Firstly we should define what Hamiltonian we want to simulate. Let's do simulation for this Hamiltonian:\n",
+ "\n",
+ "$$H = 2 \\cdot \\sigma_x \\otimes \\sigma_z \\otimes \\sigma_x + 5 \\cdot \\sigma_z \\otimes \\sigma_y \\otimes I + 7 \\cdot I \\otimes \\sigma_y \\otimes \\sigma_z$$\n",
+ "\n",
+ "It can be shown that all three terms commute to each other, thus we can write $e^{iHt}$ in this form:\n",
+ "\n",
+ "$$e^{iHt} = e^{i2 \\cdot \\sigma_x \\otimes \\sigma_z \\otimes \\sigma_x t} \\; \\cdot \\; e^{i5 \\cdot \\sigma_z \\otimes \\sigma_y \\otimes I t} \\; \\cdot \\; e^{i7 \\cdot I \\otimes \\sigma_y \\otimes \\sigma_z t}$$\n",
+ "\n",
+ "This will help us to not worry about the Trotterization procidure and will let us to draw simpler circuit at the end. Here we start coding:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "from qiskit import *\n",
+ "from numpy import pi as PI\n",
+ "\n",
+ "# put the Hamiltonian in the dictionary\n",
+ "\n",
+ "hamiltonian = {\"XZX\": 2, \"ZYI\": 5, \"IYZ\": 7}\n",
+ "\n",
+ "# define function for ZZ..Z exponantiation\n",
+ "def exp_all_z(quantum_register, qubit_list, t=1):\n",
+ " \n",
+ " "
  ]
  },
  {