Compare qsharp-integrations translated U gate (in Q#) with Qiskit's U gate (QASM)

U_gate_Q#_parser.ipynb

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 "cells": [
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   "cell_type": "code",
   "execution_count": 1,
   "id": "f9c5010e-ab54-4ffa-9415-817133675953",
   "metadata": {},
   "outputs": [],
   "source": [
    "import functools\n",
    "from sympy import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "3f78a76f-d97c-4c0c-a391-33476737da41",
   "metadata": {},
   "outputs": [],
   "source": [
    "theta, phi, lmd = symbols(r\"\\theta \\phi \\lambda\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "78a425f0-ab02-4f9f-8698-47e144342ce5",
   "metadata": {},
   "outputs": [],
   "source": [
    "Id = Matrix([\n",
    "    [1, 0],\n",
    "    [0, 1]\n",
    "])\n",
    "\n",
    "X = Matrix([\n",
    "    [0, 1],\n",
    "    [1, 0]\n",
    "])\n",
    "\n",
    "Y = Matrix([\n",
    "    [0 , -1j],\n",
    "    [1j,   0]\n",
    "])\n",
    "\n",
    "Z = Matrix([\n",
    "    [1,  0],\n",
    "    [0, -1]\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "a86cd12c-456b-4ebe-9d91-57b8fae2591b",
   "metadata": {},
   "outputs": [],
   "source": [
    "def R(angle, op):\n",
    "    return exp(-I * angle * op / 2)\n",
    "\n",
    "Rx = functools.partial(R, op=X)\n",
    "Ry = functools.partial(R, op=Y)\n",
    "Rz = functools.partial(R, op=Z)\n",
    "gp = functools.partial(R, op=Id)  # global phase"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "085bc918-722d-4346-8175-0a7c96a24dd1",
   "metadata": {},
   "outputs": [],
   "source": [
    "def U_qiskit(theta, phi, lmd):\n",
    "    \"\"\"\n",
    "    https://qiskit.org/documentation/stubs/qiskit.circuit.library.UGate.html\n",
    "    Matrix Representation\n",
    "    \"\"\"\n",
    "    return Matrix([\n",
    "        [cos(theta / 2), -exp(I * lmd) * sin(theta / 2)],\n",
    "        [exp(I * phi) * sin(theta / 2), exp(I * (phi + lmd)) * cos(theta / 2)]\n",
    "    ])\n",
    "\n",
    "\n",
    "def U_qs_parser(theta, phi, lmd):\n",
    "    \"\"\"https://github.com/qsharp-community/qsharp-integrations/blob/main/src/OpenQasmReader/Parser.cs#L740-L778\"\"\"\n",
    "    return simplify(Rx(lmd) * Ry(phi) * Rz(theta))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "7f56c103-629a-4080-8cc8-7e0174ff3524",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\frac{\\theta}{2} \\right)} & - e^{i \\lambda} \\sin{\\left(\\frac{\\theta}{2} \\right)}\\\\e^{i \\phi} \\sin{\\left(\\frac{\\theta}{2} \\right)} & e^{i \\left(\\lambda + \\phi\\right)} \\cos{\\left(\\frac{\\theta}{2} \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[            cos(\\theta/2),         -exp(I*\\lambda)*sin(\\theta/2)],\n",
       "[exp(I*\\phi)*sin(\\theta/2), exp(I*(\\lambda + \\phi))*cos(\\theta/2)]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_qasm = U_qiskit(theta, phi, lmd)\n",
    "U_qasm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "539fcc6a-d6ab-43b4-b6cc-5e52d39f8c5d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.25 \\left(i \\left(1 - e^{i \\lambda}\\right) \\left(1 - e^{1.0 i \\phi}\\right) + \\left(e^{i \\lambda} + 1\\right) \\left(e^{1.0 i \\phi} + 1\\right)\\right) e^{\\frac{i \\left(- \\lambda - 1.0 \\phi - \\theta\\right)}{2}} & 0.25 \\left(\\left(1 - e^{i \\lambda}\\right) \\left(e^{1.0 i \\phi} + 1\\right) - i \\left(1 - e^{1.0 i \\phi}\\right) \\left(e^{i \\lambda} + 1\\right)\\right) e^{\\frac{i \\left(- \\lambda - 1.0 \\phi + \\theta\\right)}{2}}\\\\0.25 \\left(\\left(1 - e^{i \\lambda}\\right) \\left(e^{1.0 i \\phi} + 1\\right) + i \\left(1 - e^{1.0 i \\phi}\\right) \\left(e^{i \\lambda} + 1\\right)\\right) e^{\\frac{i \\left(- \\lambda - 1.0 \\phi - \\theta\\right)}{2}} & 0.25 \\left(- i \\left(1 - e^{i \\lambda}\\right) \\left(1 - e^{1.0 i \\phi}\\right) + \\left(e^{i \\lambda} + 1\\right) \\left(e^{1.0 i \\phi} + 1\\right)\\right) e^{\\frac{i \\left(- \\lambda - 1.0 \\phi + \\theta\\right)}{2}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.25*(I*(1 - exp(I*\\lambda))*(1 - exp(1.0*I*\\phi)) + (exp(I*\\lambda) + 1)*(exp(1.0*I*\\phi) + 1))*exp(I*(-\\lambda - 1.0*\\phi - \\theta)/2),  0.25*((1 - exp(I*\\lambda))*(exp(1.0*I*\\phi) + 1) - I*(1 - exp(1.0*I*\\phi))*(exp(I*\\lambda) + 1))*exp(I*(-\\lambda - 1.0*\\phi + \\theta)/2)],\n",
       "[0.25*((1 - exp(I*\\lambda))*(exp(1.0*I*\\phi) + 1) + I*(1 - exp(1.0*I*\\phi))*(exp(I*\\lambda) + 1))*exp(I*(-\\lambda - 1.0*\\phi - \\theta)/2), 0.25*(-I*(1 - exp(I*\\lambda))*(1 - exp(1.0*I*\\phi)) + (exp(I*\\lambda) + 1)*(exp(1.0*I*\\phi) + 1))*exp(I*(-\\lambda - 1.0*\\phi + \\theta)/2)]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_qs = U_qs_parser(theta, phi, lmd)\n",
    "U_qs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "be0b833a-26c9-4b2c-844d-adc834d06665",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_qs == U_qasm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "3d2b475f-dc30-44f1-a5a0-e86e4a6ad1ab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & -1\\\\1 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0, -1],\n",
       "[1,  0]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_qasm.subs([(theta, pi), (phi, 0), (lmd, 0)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "75ad10cd-4d3e-46d8-a032-9f1efa1c3c3c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- 1.0 i & 0\\\\0 & 1.0 i\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-1.0*I,     0],\n",
       "[     0, 1.0*I]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_qs.subs([(theta, pi), (phi, 0), (lmd, 0)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c6a8eaf0-8b20-458d-8d46-553ee82f35c1",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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