cgranade/Animations.ipynb

## Animations.ipynb

      
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## QInfer Tomography Tutorial.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# QInfer Tomography Tutorial #\n",
    "\n",
    "[Christopher Granade](http://www.cgranade.com/), Joshua Combes and D. G. Cory. <br>\n",
    "Supplemental material to *Practical Bayesian Tomography*.\n",
    "$\n",
    "\\newcommand{\\id}{\\unicode{x1D7D9}}\n",
    "\\newcommand{\\sket}[1]{|#1\\rangle\\!\\rangle}\n",
    "\\newcommand{\\sbraket}[1]{\\langle\\!\\langle#1\\rangle\\!\\rangle}\n",
    "\\newcommand{\\sbra}[1]{\\langle\\!\\langle#1|}\n",
    "\\newcommand{\\Cov}{\\operatorname{Cov}}\n",
    "\\newcommand{\\Tr}{\\operatorname{Tr}}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preamble"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we take care of importing the various libraries that we need, as well as configuring settings needed to export this tutorial in a nice way."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from __future__ import division\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_formats = ['png', 'pdf']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# This is redundant with the actual tutorial, but we use it to force\n",
    "# grayscale images where supported.\n",
    "import qutip as qt\n",
    "qt.settings.colorblind_safe = True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from IPython.display import display, Latex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('ggplot')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# We suppress warnings in this tutorial for ease of exporting to LaTeX.\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this Notebook, we demonstrate the use of [QInfer](https://github.com/csferrie/python-qinfer) for Bayesian state and process tomography. In particular, we show how to estimate states and channels given data synthesized from a description of a true state, and discuss how to obtain region estimates, covariance superoperators and other useful functions of tomography posteriors. We then discuss how to apply these techniques in experimental systems."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The tomography implementation in QInfer is based on [QuTiP](http://qutip.org/) and [NumPy](numpy.org), so we start by importing everything here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import qutip as qt\n",
    "import qinfer as qi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a first step, we define a *basis* for performing tomography; the choice of basis is largely arbitrary, but depending on the experiment, some bases may be more or less convienent. Here, we focus on the example of the single-qubit Pauli basis $B = \\{\\id, \\sigma_x, \\sigma_y, \\sigma_z\\}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "            <strong>TomographyBasis:</strong>\n",
       "                dims=$2$\n",
       "            <p>\n",
       "                \\begin{equation}\n",
       "                    \n",
       "                𝟙 =                   \n",
       "                \\left(\\begin{matrix}\n",
       "                    0.707&0\\\\0&0.707\n",
       "                \\end{matrix}\\right)\n",
       "                ,\n",
       "                \\sigma_x =                   \n",
       "                \\left(\\begin{matrix}\n",
       "                    0&0.707\\\\0.707&0\n",
       "                \\end{matrix}\\right)\n",
       "                ,\n",
       "                \\sigma_y =                   \n",
       "                \\left(\\begin{matrix}\n",
       "                    0&-0.707\\mathrm{i}\\\\0.707\\mathrm{i}&0\n",
       "                \\end{matrix}\\right)\n",
       "                ,\n",
       "                \\sigma_z =                   \n",
       "                \\left(\\begin{matrix}\n",
       "                    0.707&0\\\\0&-0.707\n",
       "                \\end{matrix}\\right)\n",
       "                \n",
       "                \\end{equation}\n",
       "            </p>\n",
       "            "
      ],
      "text/plain": [
       "<TomographyBasis dims=[2] at 140700690071184>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "basis = qi.tomography.pauli_basis(1)\n",
    "display(basis)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will get a lot of use out of the Pauli basis, so we also define some useful shorthand."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "I, X, Y, Z = qt.qeye(2), qt.sigmax(), qt.sigmay(), qt.sigmaz()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Basis objects are responsible for converting between QuTiP's rich ``Qobj`` format and the unstructured model parameter representation used by QInfer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.70710678,  0.70710678,  0.        ,  0.        ])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(basis.state_to_modelparams(I / 2 + X / 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\\begin{equation*}\\left(\\begin{array}{*{11}c}1.000 & 0.0\\\\0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
      ],
      "text/plain": [
       "Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\n",
       "Qobj data =\n",
       "[[ 1.  0.]\n",
       " [ 0.  0.]]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(basis.modelparams_to_state(np.array([1, 0, 0, 1]) / np.sqrt(2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having defined a basis, we then define the core object describing a tomography experiment, the *model*. In QInfer, models encapsulate the likelihood function, experimental parameters and other useful metadata about the experimental properties being estimated. In our case, we use ``TomographyModel`` to describe the single-shot experiment, and ``BinomialModel`` to describe batches of the single-shot experiment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "            <strong>BinomialModel</strong> at 0x7ff76ea81050: 4 model parameters\n",
       "        <br>\n",
       "            <p>Model chain:</p>\n",
       "            <ul><li>\n",
       "            <strong>TomographyModel</strong> at 0x7ff76ea81290: 4 model parameters\n",
       "        </li>\n",
       "            </ul>\n",
       "            "
      ],
      "text/plain": [
       "<qinfer.derived_models.BinomialModel at 0x7ff76ea81050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "model = qi.BinomialModel(qi.tomography.TomographyModel(basis))\n",
    "display(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A ``Model`` defines a vector of model parameters; for a single qubit ``TomographyModel``, this is a vector of length 4, each describing a different element of the Hermitian operator basis. Each ``Model`` also defines experiment parameters as a NumPy record array. A record then describes a single measurement of the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('meas', float, 4), ('n_meas', 'uint')]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(model.expparams_dtype)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, the experiment parameters record has two *fields*: ``meas`` and ``n_meas``. The first is a vector of four floats corresponding to $\\sket{M} = (\\sbraket{B_0 | M}, \\sbraket{B_1 | M}, \\sbraket{B_2 | M}, \\sbraket{B_3 | M)}$. The second is an unsigned integer (``uint``) describing how many times that measurement is performed. For instance, measuring $(\\id + \\sigma_z) / 2$ 40 times is given by the array:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([([0.7071067811865475, 0.0, 0.0, 0.7071067811865475], 40L)], \n",
       "      dtype=[('meas', '<f8', (4,)), ('n_meas', '<u8')])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "expparams = np.array([\n",
    "    # Each tuple, marked with (), defines a single record.\n",
    "    (\n",
    "        # Within each tuple, fields are separated by commas.\n",
    "        # The fields follow in the order given by the model,\n",
    "        # so the first field is meas, a length-4 vector.\n",
    "        [1 / np.sqrt(2), 0, 0, 1 / np.sqrt(2)],\n",
    "        # The second field is then the number of measurements.\n",
    "        40\n",
    "    )\n",
    "],\n",
    "# We finish building the array by passing along the right data\\\n",
    "# type to NumPy. This is somwhat of a QInfer idiom.\n",
    "dtype=model.expparams_dtype)\n",
    "display(expparams)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The fields of a record array can be obtained by indexing. For instance, the ``['meas']`` field is then a $1 \\times 4$ array, with the first index allowing for a sequence of measurements to be described at once."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.70710678,  0.        ,  0.        ,  0.70710678]])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(expparams['meas'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that  by convention, ``meas`` is normalized to $1 / \\sqrt{d}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Often, we will not construct experiments directly, but will instead rely on QInfer's heuristics (described below). In any case, once we have a model, the next step is to create a prior. QInfer comes with several useful fiducial priors, as well as insightful priors constructed from amplitude damping channels. For instance, to create a Hilbert-Schmidt uniform prior constrained to rebits, we use the ``GinibreReditDistribution``:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "fiducial_prior = qi.tomography.GinibreReditDistribution(basis)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "application/pdf": 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E70DJ09Ql0uFwtnG5XJEA0OvsKv/3aeoAfaTxEi6jlvhh0FyhcmVDp0mAdEPk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\nPhF6wKzdnSEg2IgEhEmWh+ATFpgxskg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XOswEPxumMHYk6sV65N0YcohWiA4QJZUVK8iH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\nKAL0WKTHJP9i+pCNZBUKXnBli1c+trR