{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Quick Guide\n",
    "To introduce the main functionality of Haarpy consider the following problem: imagine that you can generate unitary matrices at random (from the Haar measure); you would like to estimate the average of $|U_{i,j}|^2$ which we write mathematically as $\\int dU |U_{i,j}|^2 = \\int dU U_{i,j} U_{i,j}^*$. We could obtain this average by using the random matrix functionality of SciPy as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import unitary_group"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.4964599 , 0.32742463, 0.25429793])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.seed(137)\n",
    "shots = 1000\n",
    "\n",
    "# For unitary matrices of size between 2 and 4 produce 1000 shots and calculate the average\n",
    "# of the absolute value squared of the 0,1 entry\n",
    "np.array(\n",
    "    [\n",
    "        np.mean([np.abs(unitary_group.rvs(dim=dim)[0, 1]) ** 2 for _ in range(shots)])\n",
    "        for dim in range(2, 5)\n",
    "    ]\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Haarpy allows you to obtain this average (and many others!) analytically. We first recall that the expression we are trying to calculate is $\\int dU \\left|U_{i,j}\\right|^2 = \\int dU U_{i,j} U_{i,j}^*$. With this expression in mind we can use Sympy to create a symbolic variable $d$ for the dimension of the unitary and write"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Symbol\n",
    "from haarpy import haar_integral_unitary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{d}$"
      ],
      "text/plain": [
       "1/d"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = Symbol(\"d\")\n",
    "haar_integral_unitary((\"i\", \"j\", \"i\", \"j\"), d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also put integers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Fraction(1, 3)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "haar_integral_unitary((\"i\", \"j\", \"i\", \"j\"), 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice the order of the indices! The first `\"i\"` and `\"j\"` are the indices of $U$ while the second pair of `\"i\"` and `\"j\"` are the indices of $U^*$.\n",
    "\n",
    "Imagine that now we want to calculate something like $\\int dU U_{i,m} U_{j,n} U_{k,o} U_{i,m}^* U_{j,n}^* U_{k,p}^*$ $= \\int dU \\left|U_{i,m} U_{j,n} U_{k,o}\\right|^2$ we simply do"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d^{2} - 2}{d \\left(d - 2\\right) \\left(d - 1\\right) \\left(d + 1\\right) \\left(d + 2\\right)}$"
      ],
      "text/plain": [
       "(d**2 - 2)/(d*(d - 2)*(d - 1)*(d + 1)*(d + 2))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "haar_integral_unitary((\"ijk\", \"mno\", \"ijk\", \"mno\"), d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The averages we are calculating are obtained by using so-called [Weingarten calculus](https://doi.org/10.1155/S107379280320917X)."
   ]
  }
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