SecondQuantizedOp

FermionicSumOp.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "neural-latex",
   "metadata": {},
   "source": [
    "# FermionicOp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "handed-examination",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "from qiskit_nature.operators.second_quantization import FermionicOp"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "affiliated-arrangement",
   "metadata": {},
   "source": [
    "## init"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "solved-democrat",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "I * 1\n",
      "\n",
      "+ * 1\n",
      "\n",
      "- * 1\n",
      "\n",
      "N * 1\n",
      "\n",
      "E * 1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "for l in ['I', '+', '-', 'N', 'E']:\n",
    "    print(FermionicOp(l))\n",
    "    print()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "available-aquatic",
   "metadata": {},
   "outputs": [],
   "source": [
    "# print(FermionicOp(\"N\", coeff=2)) \n",
    "# this initialization is deprecated. Use 2 * FermionicOp(\"N\") alternatevely."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "buried-floating",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  N * 1\n",
      "+ E * 1\n"
     ]
    }
   ],
   "source": [
    "print(FermionicOp([(\"N\", 1), (\"E\", 1)]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "possible-guess",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp(('II', 0))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FermionicOp([], register_length=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "headed-garbage",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp([('N', 1), ('E', 3.14)])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FermionicOp([(\"N\", 1), (\"E\", 3.14)]) # __repr__ is evaluatable"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "superb-workplace",
   "metadata": {},
   "source": [
    "## neg, mul, rmul, truediv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "clinical-vanilla",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp(('+N-EII', -1))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-FermionicOp(\"+N-EII\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "nearby-resident",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp(('+-', 2))"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FermionicOp(\"+-\") * 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "professional-height",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp([('+N', (6+3j)), ('E-', (2+1j))])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2 + 1j) * FermionicOp([(\"+N\", 3), (\"E-\", 1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "complimentary-mercury",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp([('+N', 1.0), ('E-', 0.3333333333333333)])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FermionicOp([(\"+N\", 3), (\"E-\", 1)]) / 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "banned-freeware",
   "metadata": {},
   "source": [
    "## add, sub"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "saving-makeup",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp([('+N', 3), ('E-', 1)])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "3 * FermionicOp(\"+N\") + FermionicOp(\"E-\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "royal-generic",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp([('++', 3), ('--', -2)])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "3 * FermionicOp(\"++\") - 2 * FermionicOp(\"--\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "decent-belarus",
   "metadata": {},
   "source": [
    "## matmul"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "periodic-conflict",
   "metadata": {},
   "outputs": [],
   "source": [
    "from itertools import product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "linear-delight",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "I * I\n",
      "== I * 1\n",
      "\n",
      "I * +\n",
      "== + * 1\n",
      "\n",
      "I * -\n",
      "== - * 1\n",
      "\n",
      "I * N\n",
      "== N * 1\n",
      "\n",
      "I * E\n",
      "== E * 1\n",
      "\n",
      "+ * I\n",
      "== + * 1\n",
      "\n",
      "+ * +\n",
      "== I * 0\n",
      "\n",
      "+ * -\n",
      "== N * 1\n",
      "\n",
      "+ * N\n",
      "== I * 0\n",
      "\n",
      "+ * E\n",
      "== + * 1\n",
      "\n",
      "- * I\n",
      "== - * 1\n",
      "\n",
      "- * +\n",
      "== E * 1\n",
      "\n",
      "- * -\n",
      "== I * 0\n",
      "\n",
      "- * N\n",
      "== - * 1\n",
      "\n",
      "- * E\n",
      "== I * 0\n",
      "\n",
      "N * I\n",
      "== N * 1\n",
      "\n",
      "N * +\n",
      "== + * 1\n",
      "\n",
      "N * -\n",
      "== I * 0\n",
      "\n",
      "N * N\n",
      "== N * 1\n",
      "\n",
      "N * E\n",
      "== I * 0\n",
      "\n",
      "E * I\n",
      "== E * 1\n",
      "\n",
      "E * +\n",
      "== I * 0\n",
      "\n",
      "E * -\n",
      "== - * 1\n",
      "\n",
      "E * N\n",
      "== I * 0\n",
      "\n",
      "E * E\n",
      "== E * 1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "for l1, l2 in product(['I', '+', '-', 'N', 'E'], repeat=2):\n",
    "    print(l1, '*', l2)\n",
    "    print(\"==\", FermionicOp(l1) @ FermionicOp(l2))\n",
    "    print()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "shared-library",
   "metadata": {},
   "source": [
    "$$\n",
    "c_1^\\dagger c_0c_1 = - c_1^\\dagger c_1c_0 = -n_1 c_0\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "chemical-spanish",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N- * -1\n"
     ]
    }
   ],
   "source": [
    "print(FermionicOp(\"+-\") @ FermionicOp(\"-I\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "overhead-mustang",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp([('+N', 3), ('N+', 3), ('E-', 1), ('-E', -1)])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(3 * FermionicOp(\"+N\") + FermionicOp(\"E-\")) @ (FermionicOp(\"II\") + FermionicOp(\"-+\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "american-camel",
   "metadata": {},
   "source": [
    "## pow"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "technical-atlantic",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp(('II', 0))"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FermionicOp([(\"+N\", 3), (\"E-\", 1)]) ** 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "assured-tumor",
   "metadata": {},
   "source": [
    "$$\n",
    "(3c_1^\\dagger c_0^\\dagger c_0 + c_1^\\dagger c_1 c_0)^2\n",
    "= 9c_1^\\dagger c_0^\\dagger c_0 c_1^\\dagger c_0^\\dagger c_0 + 3c_1^\\dagger c_0^\\dagger c_0 c_1^\\dagger c_1 c_0 + 3c_1^\\dagger c_1 c_0 c_1^\\dagger c_0^\\dagger c_0 + c_1^\\dagger c_1 c_0 c_1^\\dagger c_1 c_0\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "interim-knock",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp(('+-', -3))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FermionicOp([(\"+N\", 3), (\"N-\", 1)]) ** 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "civil-cursor",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp(('IIII', 27))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(3 * FermionicOp(\"IIII\")) ** 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "nearby-given",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp(('II', 1))"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FermionicOp([(\"+N\", 3), (\"E-\", 1)]) ** 0"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "personalized-plumbing",
   "metadata": {},
   "source": [
    "## dagger"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "lined-spare",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "FermionicOp([('-N', 3), ('N+', 1), ('++', (-2+4j))])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FermionicOp([(\"+N\", 3), (\"N-\", 1), (\"--\", 2 + 4j)]).dagger"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "racial-message",
   "metadata": {},
   "source": [
    "Note that $$\n",
    "(c_1c_0)^\\dagger = c_0^\\dagger c_1^\\dagger= - c_1^\\dagger c_0^\\dagger.\n",
    "$$"
   ]
  }
 ],
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Jordan_Wigner_example.ipynb