{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c, d = sympy.symbols(\"a b c d\")\n",
    "i = sympy.I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = sympy.Matrix([[0, 1], [1, 0]])\n",
    "Z = sympy.Matrix([[1, 0], [0, -1]])\n",
    "\n",
    "M = sympy.Matrix([[a, b], [c, d]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\1 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0  1⎤\n",
       "⎢    ⎥\n",
       "⎣1  0⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  0 ⎤\n",
       "⎢     ⎥\n",
       "⎣0  -1⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡a  b⎤\n",
       "⎢    ⎥\n",
       "⎣c  d⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "term_X = M*M + i*X\n",
    "term_Z = M*M - i*Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a^{2} + b c & a b + b d + i\\\\a c + c d + i & b c + d^{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡   2                        ⎤\n",
       "⎢  a  + b⋅c     a⋅b + b⋅d + ⅈ⎥\n",
       "⎢                            ⎥\n",
       "⎢                        2   ⎥\n",
       "⎣a⋅c + c⋅d + ⅈ    b⋅c + d    ⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term_X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a^{2} + b c - i & a b + b d\\\\a c + c d & b c + d^{2} + i\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ 2                        ⎤\n",
       "⎢a  + b⋅c - ⅈ   a⋅b + b⋅d  ⎥\n",
       "⎢                          ⎥\n",
       "⎢                     2    ⎥\n",
       "⎣ a⋅c + c⋅d    b⋅c + d  + ⅈ⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "term_Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left\\{ a : - \\frac{\\sqrt{2}}{2}, \\  b : \\frac{\\sqrt{2} i}{2}, \\  c : \\frac{\\sqrt{2} i}{2}, \\  d : - \\frac{\\sqrt{2}}{2}\\right\\}, \\  \\left\\{ a : \\frac{\\sqrt{2}}{2}, \\  b : - \\frac{\\sqrt{2} i}{2}, \\  c : - \\frac{\\sqrt{2} i}{2}, \\  d : \\frac{\\sqrt{2}}{2}\\right\\}, \\  \\left\\{ a : - \\frac{\\sqrt{2} i}{2}, \\  b : \\frac{\\sqrt{2}}{2}, \\  c : \\frac{\\sqrt{2}}{2}, \\  d : - \\frac{\\sqrt{2} i}{2}\\right\\}, \\  \\left\\{ a : \\frac{\\sqrt{2} i}{2}, \\  b : - \\frac{\\sqrt{2}}{2}, \\  c : - \\frac{\\sqrt{2}}{2}, \\  d : \\frac{\\sqrt{2} i}{2}\\right\\}\\right]$"
      ],
      "text/plain": [
       "⎡⎧   -√2      √2⋅ⅈ     √2⋅ⅈ     -√2 ⎫  ⎧   √2     -√2⋅ⅈ      -√2⋅ⅈ      √2⎫  ⎧\n",
       "⎢⎨a: ────, b: ────, c: ────, d: ────⎬, ⎨a: ──, b: ──────, c: ──────, d: ──⎬, ⎨\n",
       "⎣⎩    2        2        2        2  ⎭  ⎩   2        2          2        2 ⎭  ⎩\n",
       "\n",
       "   -√2⋅ⅈ      √2     √2     -√2⋅ⅈ ⎫  ⎧   √2⋅ⅈ     -√2      -√2      √2⋅ⅈ⎫⎤\n",
       "a: ──────, b: ──, c: ──, d: ──────⎬, ⎨a: ────, b: ────, c: ────, d: ────⎬⎥\n",
       "     2        2      2        2   ⎭  ⎩    2        2        2        2  ⎭⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(term_X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left\\{ a : - \\sqrt{i}, \\  b : 0, \\  c : 0, \\  d : - \\sqrt{- i}\\right\\}, \\  \\left\\{ a : - \\sqrt{i}, \\  b : 0, \\  c : 0, \\  d : \\sqrt{- i}\\right\\}, \\  \\left\\{ a : \\sqrt{i}, \\  b : 0, \\  c : 0, \\  d : - \\sqrt{- i}\\right\\}, \\  \\left\\{ a : \\sqrt{i}, \\  b : 0, \\  c : 0, \\  d : \\sqrt{- i}\\right\\}\\right]$"
      ],
      "text/plain": [
       "⎡⎧                          ____⎫  ⎧                         ____⎫  ⎧         \n",
       "⎢⎨a: -√ⅈ, b: 0, c: 0, d: -╲╱ -ⅈ ⎬, ⎨a: -√ⅈ, b: 0, c: 0, d: ╲╱ -ⅈ ⎬, ⎨a: √ⅈ, b:\n",
       "⎣⎩                              ⎭  ⎩                             ⎭  ⎩         \n",
       "\n",
       "                ____⎫  ⎧                        ____⎫⎤\n",
       " 0, c: 0, d: -╲╱ -ⅈ ⎬, ⎨a: √ⅈ, b: 0, c: 0, d: ╲╱ -ⅈ ⎬⎥\n",
       "                    ⎭  ⎩                            ⎭⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(term_Z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2} i}{2}\\\\\\frac{\\sqrt{2} i}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-√2   √2⋅ⅈ⎤\n",
       "⎢────  ────⎥\n",
       "⎢ 2     2  ⎥\n",
       "⎢          ⎥\n",
       "⎢√2⋅ⅈ  -√2 ⎥\n",
       "⎢────  ────⎥\n",
       "⎣ 2     2  ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt_miX = sympy.Matrix([[-sympy.sqrt(2)/2, i*sympy.sqrt(2)/2], [i*sympy.sqrt(2)/2, -sympy.sqrt(2)/2]])\n",
    "sqrt_miX"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAzCAYAAABbldAtAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFn0lEQVR4Ae1ci1EbMRC1MymASTpwOgBSQUwHkFQQ6IAMFTCkA+iAgQ5CKkigA+ggxB047120Hvm8ts/2rVayTzPCd5JO2s/b1edu6fd6vT3kmTQej0czhV1B8RLo9/uqvt+As0fkv7V85c0xCN5HHi/K3jQWOv5dTdfU/R2BwHQED9CP8tn/Yte/BONBRFNMX3XtSl2hg0OeU7oGG5WuBQhZsUVvQIJA9FNWhG0xMVkCAfK+QHafnrZY7zOsvZ0pcS6ANxiAhAG8wUOdFNSdo+wD6nKYuurkNb4HHwLyP+QH+Qo8vTTuwKBhdkAAj99C1tglAO61ilLKAAIuzi+h+IoP3HMV/4hfzt1uYMgKCEEoh/MsHuW0nmIT+DsF8XsCAjKC6xHKCYpr5COWeaTc1gh0mZcegkg05gnG0RbAv1A+DIaQiJTpYbIBQhDCMLaWaVK34m4ILl4VTuTwjvUuKaepQd0pBIDQpXJaeAZQvrtIasNBG1r7uw2HWfvxJB4BQjhGXob2Yyj5RuHkIiifJ2IES6lJlCzWH/MhXkI9/o0bWl2bewQAgFs+KpAr4gONkdCGi6WphHJuJTl/MnF+dVtVVxTY/3lvP4Q+wkIgQBFE6E/kVZB6AguuFkRBkbRyMniO+32pw32czlCu7QheUS7bRU4P3FqWmsTqNfrFW/BcwSUtBAKUQDemWnETavF8ZcEAAHcC4hlo2ZOEOip4xhuwQRi/hzbH4QFt6ghVef+QF/BBIjWjkjI3j5dkjRAUSiVyrUB3Hyda+TIFc2q5F2DEDxd2zdPSOv9kQTzCzGlqKv6SACEwI8eqE/ceLH2hgtGG1rKPfMt+cC/98La0xAXvoUI0ve6TJ9CTAQFM0u0R8adBuZQHLX3ZAVIlODx/j+doTc98sMQEHuj5XsGHTHUENoH+GfmrJ08L1wgGhNGah8gEA4Hxe5kVoP4BbZm5xhgFYeKy2ETrvwA/MkV8xP0n8FUtsL24SgqEoFQyTE9AIEwtHOcJAc+5ncHPo2ndcvAywrOT6XHdftp+LtnUEBHOqYDu8AVCcVslR/R0l5BAciBA+TwXIACys4pdRkTSqUEEDTBoh0dS3f06SCC5R3DgsRuygQQ6IDQQ0i406YCwC1puwGMHhAZC2oUmLovFXRDsOjzikIlH6V+QeXhmlrBYr95+xQN0HiGWhv81T15vqSjLrLHZAUGTikNZ8AZu0V0dEByUPmdIHru7vVk1XyMA6cJcNlE9cxThVhxeQKnRXW0Qhf655lgYIWYKBBCQZVRPG8JtuQ8et1seuZ+hf/nkTyXdDAgAQbZRPaoknAohJ76Amxvd1QZZTY70LdcIfMWsvWN3j+ppQ7gt9sGpc9nHOS0Op3dlCYQhhtS+3OX7eCbW73QK3iCL6C6TqSEwuEzJ8sHmsnbbXG+2Uwg6aBwhZuURRMli/bEyxUtwbtzaBEVsEt3VhlxWihAz8QgNuWDQy1YmgIDbNfkcT40LCW2uLQSAvgfod6UIMSsgiNVrfIq3cIvq0YiSMgiRnsozuqsiZUM6Vo4QMwECtitZR/WI0rVf0o5y1Yq19vUyPF99hwlFcicgnmHqI13UzY3ukv42oSM8u1KEmNUagfxkG9Ujwrb8DcrYJLqrDfI4PS0MIJJBLIGQbVSPMJ/gV47XJ6eG8AYMbmmknE3oC1MLX2vfsh/cCy1qt2ZAgEVkG9WjSsKgEDLgNLFOdFcb1KwUIWayRoi4yDKqJ6IvxSUtcYjcOLqrDaIAwpUixEyBEObJiVtsg8HS+ggK4VG7bCenFo6W/GDsxhFiZlODJYMF9p19dFcHhASogmVmH91lOjUkkHExQwAMWUd3CRAGYbsxESwI194TTOq7izIlUNczuKje+fCzZv7jiUGNrRsAgV+1dGnLJAAg/ABL3MXE6eEf3zdR3upFKMEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\sqrt{i} & 0\\\\0 & - \\sqrt{- i}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-√ⅈ     0   ⎤\n",
       "⎢            ⎥\n",
       "⎢        ____⎥\n",
       "⎣ 0   -╲╱ -ⅈ ⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt_iZ = sympy.Matrix([[-sympy.sqrt(i), 0], [0, -sympy.sqrt(-i)]])\n",
    "sqrt_iZ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & - i\\\\- i & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0   -ⅈ⎤\n",
       "⎢      ⎥\n",
       "⎣-ⅈ  0 ⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt_miX**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}i & 0\\\\0 & - i\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ⅈ  0 ⎤\n",
       "⎢     ⎥\n",
       "⎣0  -ⅈ⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt_iZ**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{i \\sqrt{- i}}{2} - \\frac{\\sqrt{i} i}{2} & - \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2}\\\\- \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2} & \\frac{\\sqrt{i} i}{2} + \\frac{i \\sqrt{- i}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡      ____                   ____ ⎤\n",
       "⎢  ⅈ⋅╲╱ -ⅈ    √ⅈ⋅ⅈ     √ⅈ   ╲╱ -ⅈ  ⎥\n",
       "⎢- ──────── - ────   - ── + ────── ⎥\n",
       "⎢     2        2       2      2    ⎥\n",
       "⎢                                  ⎥\n",
       "⎢           ____               ____⎥\n",
       "⎢    √ⅈ   ╲╱ -ⅈ     √ⅈ⋅ⅈ   ⅈ⋅╲╱ -ⅈ ⎥\n",
       "⎢  - ── + ──────    ──── + ────────⎥\n",
       "⎣    2      2        2        2    ⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H_ = sqrt_miX * sqrt_iZ**3 * sqrt_miX\n",
    "H_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{i \\left(- \\sqrt{i} + \\left(- i\\right)^{\\frac{5}{2}}\\right)}{2} & - \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2}\\\\- \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2} & \\frac{i \\left(\\sqrt{- i} + \\sqrt{i}\\right)}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡  ⎛          5/2⎞            ____ ⎤\n",
       "⎢ⅈ⋅⎝-√ⅈ + (-ⅈ)   ⎠     √ⅈ   ╲╱ -ⅈ  ⎥\n",
       "⎢─────────────────   - ── + ────── ⎥\n",
       "⎢        2             2      2    ⎥\n",
       "⎢                                  ⎥\n",
       "⎢           ____      ⎛  ____     ⎞⎥\n",
       "⎢    √ⅈ   ╲╱ -ⅈ     ⅈ⋅⎝╲╱ -ⅈ  + √ⅈ⎠⎥\n",
       "⎢  - ── + ──────    ───────────────⎥\n",
       "⎣    2      2              2       ⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H_.simplify()\n",
    "H_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\sqrt{2}}{2} & \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2} & - \\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡√2   √2 ⎤\n",
       "⎢──   ── ⎥\n",
       "⎢2    2  ⎥\n",
       "⎢        ⎥\n",
       "⎢√2  -√2 ⎥\n",
       "⎢──  ────⎥\n",
       "⎣2    2  ⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H = sympy.Matrix([[1, 1], [1, -1]]) / sympy.sqrt(2)\n",
    "H"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "phi = sympy.symbols(\"phi\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAVCAYAAAB7R6/OAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAARUlEQVQoFWNgYGAoA+K7UGyEhc/QCRR0+f//PwMyBoqBxZmADLxgVAEkeEbDgchwYIEmp5mMjIwgZigQuwBxOhALAfFuAP68FOR8/BSoAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\right]$"
      ],
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.exp(i*phi)*H_ - H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{i \\left(\\left(- i\\right)^{\\frac{9}{2}} + \\sqrt{i}\\right)}{2} & \\frac{\\left(- i\\right)^{\\frac{5}{2}}}{2} + \\frac{\\sqrt{i}}{2}\\\\\\frac{\\left(- i\\right)^{\\frac{5}{2}}}{2} + \\frac{\\sqrt{i}}{2} & \\frac{i \\left(- \\sqrt{i} + \\left(- i\\right)^{\\frac{5}{2}}\\right)}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡  ⎛    9/2     ⎞        5/2        ⎤\n",
       "⎢ⅈ⋅⎝(-ⅈ)    + √ⅈ⎠    (-ⅈ)      √ⅈ   ⎥\n",
       "⎢────────────────    ─────── + ──   ⎥\n",
       "⎢       2               2      2    ⎥\n",
       "⎢                                   ⎥\n",
       "⎢      5/2           ⎛          5/2⎞⎥\n",
       "⎢  (-ⅈ)      √ⅈ    ⅈ⋅⎝-√ⅈ + (-ⅈ)   ⎠⎥\n",
       "⎢  ─────── + ──    ─────────────────⎥\n",
       "⎣     2      2             2        ⎦"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H__ = H_ * sympy.exp(i * sympy.pi )\n",
    "H__.simplify()\n",
    "H__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some other calculatuions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAcAAAAOCAYAAADjXQYbAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAo0lEQVQYGX2QARGCQBBF9xgDkAErGMEIWEEb0AEaaAVtYATHCFhBGpzv3+ydDCPuzGd3///HfbAYo81hZh04i6sYlnWCmEQGOdbq18ni/StuZAsh1LQj2IKRqwa6paS03u/eM781CxWnGogHUB3AK03+qLOTXdG7spfBrHXxa56JT8Rr3tVzIKXVK9tEesD8nTsEGW8ecNRefh/knV2YMF0kfgByKG0w9dMLeQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle i$"
      ],
      "text/plain": [
       "ⅈ"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp(i * sympy.pi/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{\\sqrt{i}}{2} + \\frac{\\sqrt{- i}}{2} & \\frac{\\sqrt{i} i}{2} + \\frac{i \\sqrt{- i}}{2}\\\\\\frac{\\sqrt{i} i}{2} + \\frac{i \\sqrt{- i}}{2} & - \\frac{\\sqrt{- i}}{2} + \\frac{\\sqrt{i}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡          ____              ____⎤\n",
       "⎢   √ⅈ   ╲╱ -ⅈ    √ⅈ⋅ⅈ   ⅈ⋅╲╱ -ⅈ ⎥\n",
       "⎢ - ── + ──────   ──── + ────────⎥\n",
       "⎢   2      2       2        2    ⎥\n",
       "⎢                                ⎥\n",
       "⎢           ____       ____      ⎥\n",
       "⎢√ⅈ⋅ⅈ   ⅈ⋅╲╱ -ⅈ      ╲╱ -ⅈ    √ⅈ ⎥\n",
       "⎢──── + ────────   - ────── + ── ⎥\n",
       "⎣ 2        2           2      2  ⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt_miX * sqrt_iZ * sqrt_miX"
   ]
  }
 ],
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