cgranade/tensor_contract_example

## tensor_contract_example
{
 "metadata": {
  "name": "",
  "signature": "sha256:d7454350e8574bbb2d317221e2bcdea00fd12b8579a7168fea3b30a93785dc20"
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 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "QuTiP Example: Superoperators, Pauli Basis and Channel Contraction"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "[Christopher Granade](http://www.cgranade.com/) <br>\n",
      "Institute for Quantum Computing"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Preamble"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Features"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import print_function"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Imports"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "\n",
      "import qutip as qt\n",
      "from qutip.ipynbtools import version_table\n",
      "from qutip.superop_reps import _pauli_basis\n",
      "\n",
      "import itertools as it"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Plotting Support"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import matplotlib.pyplot as plt\n",
      "from mpltools.special import hinton"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "TeX Macros"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\newcommand{\\ket}[1]{\\left|#1\\right\\rangle}$\n",
      "$\\newcommand{\\bra}[1]{\\left\\langle#1\\right|}$\n",
      "$\\newcommand{\\cnot}{{\\scriptstyle \\rm CNOT}}$\n",
      "$\\newcommand{\\Tr}{\\operatorname{Tr}}$"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Superoperator Representations and Plotting"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For this example, we show superoperators as matrices in the *Pauli basis*, such that any Hermicity-preserving map is represented by a real-valued matrix."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def to_super_pauli(qobj):\n",
      "    nq = int(np.log2(qobj.shape[0]) / 2)\n",
      "    if 4**nq != qobj.shape[0]:\n",
      "        raise ValueError(\"Wrong dimension; not a power of 2.\")\n",
      "\n",
      "    B = _pauli_basis(nq) / np.sqrt(2**nq)\n",
      "    # Try and force the dims.\n",
      "    if B.shape == qobj.shape:\n",
      "        B.dims = qobj.dims\n",
      "\n",
      "    return B.dag() * qt.to_super(qobj) * B"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This representation is especially useful, as it allows us to effectively plot maps using *Hinton diagrams*."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot_paulisuper_hinton(qobj):\n",
      "    nq = int(np.log2(qobj.shape[0]) / 2)\n",
      "    plot_data = qobj.data.todense()\n",
      "    f = hinton(plot_data)\n",
      "    labels = list(it.imap(\"\".join, it.product('IXYZ', repeat=nq)))\n",
      "    plt.xticks(range(len(labels)), labels)\n",
      "    plt.gca().xaxis.tick_top()\n",
      "    plt.yticks(range(len(labels)), labels)\n",
      "    return f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As an example, conjugation by $\\sigma_z$ leaves $\\mathbb{1}$ and $\\sigma_z$ invariant, but flips the sign of $\\sigma_x$ and $\\sigma_y$. This is indicated in Hinton diagrams by a black square for the sign change and a white square for a +1 sign."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_paulisuper_hinton(to_super_pauli(qt.to_super(qt.sigmaz())))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPcAAAD7CAYAAAC2TgIoAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nRK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fbe05087f10>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As a couple more examples, we also consider the supermatrix for a Hadamard transform and for $\\sigma_z \\otimes H$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_paulisuper_hinton(to_super_pauli(qt.to_super(qt.hadamard_transform())))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPcAAAD7CAYAAAC2TgIoAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n7nBWqssAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fbe05068050>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_paulisuper_hinton(to_super_pauli(qt.to_super(qt.tensor(qt.sigmaz(), qt.hadamard_transform()))))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7fbe04f0b690>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Reduced Channels"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As an example of tensor contraction, we now consider the map $S(\\rho) = \\Tr_2[\\cnot (\\rho \\otimes \\ket{0}\\bra{0}) \\cnot^\\dagger]$.\n",
      "This should give us a completely dephasing map. The channel on both qubits is just the superunitary $\\cnot$ channel:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_paulisuper_hinton(to_super_pauli(qt.to_super(qt.cnot())))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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a0SAsEvTQnqw+p/pSD6UFfiwT7yHgdcJluwBXAm8RquxIarm0wI9l4j0MHAbuH7bbC+wA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\nAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fbe04dd4050>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The following is a bit of a hack to dot the tensor by the vectorization of $\\ket{0}\\bra{0}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s_prep = qt.Qobj(reduce(np.kron, [np.eye(2), [[1], [0]]]*2), dims=[[[2, 2], [2, 2]], [[2], [2]]])\n",
      "s_prep"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "Quantum object: dims = [[[2, 2], [2, 2]], [[2], [2]]], shape = [16, 4], type = other\\begin{equation*}\\left(\\begin{array}{*{11}c}1.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 1.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 1.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 1.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\0.0 & 0.0 & 0.0 & 0.0\\\\\\end{array}\\right)\\end{equation*}"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "Quantum object: dims = [[[2, 2], [2, 2]], [[2], [2]]], shape = [16, 4], type = other\n",
        "Qobj data =\n",
        "[[ 1.  0.  0.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  1.  0.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  1.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  0.  1.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  0.  0.]\n",
        " [ 0.  0.  0.  0.]]"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We now complete by multiplying the superunitary $\\cnot$ by the preparation channel above, then applying the partial trace channel by contracting the second and fourth index indices. As expected, this gives us a dephasing map."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_paulisuper_hinton(qt.tensor_contract(qt.to_super(qt.cnot()), (1, 3)) * s_prep)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPcAAAD7CAYAAAC2TgIoAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\npG0Bh7Od2Sm/AAAAAACAa/wPKsawRXbUKdcAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7fbe04d74310>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Epilouge"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "version_table()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
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