jupyter_notebooks/Examples/2Q-GST: creating gate sets.ipynb

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    "# Creating a 2-qubit gate set with a custom 2-qubit gate\n",
    "While pyGSTi is able to support several common types of 2-qubit gates, the space of all possible 2-qubit gates is so large that some users will need to construct their own particular 2-qubit gate \"from scratch\".  In this example, we look at how to construct a 2-qubit gateset manually.  We'll use `pygsti.construction.build_gateset` to construct the single-qubit gates and we'll create a \"non-standard\" 2-qubit gate by specifying the unitary operation it performs as a $4\\times 4$ matrix. \n",
    "\n",
    "To perform GST on such a gate set, one may need to compute new sets of fiducials and/or germ sequences.  Once these are obtained, 2-qubit GST can be run just as in the example which uses one of pyGSTi's built-in 2Q gate sets."
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   "source": [
    "import pygsti"
   ]
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    "## Step 1: Create a gateset with only single-qubit gates\n",
    "Since the space of single-qubit gates is relatively small, we'll assume that the single-qubit gates in our gateset are able to be specified using `pygsti.construction.build_gateset`.  So we'll start by construct a `GateSet` object containing all but the two-qubit gate(s) using `build_gateset`. "
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   "source": [
    "gs_target = pygsti.construction.build_gateset( \n",
    "            [4], [('Q0','Q1')],['Gii','Gix','Giy','Gxi','Gyi'], \n",
    "            [ \"I(Q0):I(Q1)\", \"X(pi/2,Q1)\", \"Y(pi/2,Q1)\", \"X(pi/2,Q0)\", \"Y(pi/2,Q0)\" ],\n",
    "            effectLabels=['00','01','10','11'], effectExpressions=[\"0\",\"1\",\"2\",\"3\"])"
   ]
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    "There are lots of arguments to this function, so let's review what they mean:\n",
    "- `[4]` = a 4-dimensional Hilbert (state) space\n",
    "- `[('Q0','Q1')]` = interpret this 4-d space as that of two qubits 'Q0', and 'Q1' (note these labels *must* begin with 'Q'!)\n",
    "- `\"Gix\"` = gate label; can be anything that begins with 'G' and is followed by lowercase letters\n",
    "- `\"X(pi/2,Q1)\"` = pi/2 single-qubit x-rotation gate on the qubit labeled Q1\n",
    "- `\"rho0\"` = prep label; can be anything that begins with \"rho\"\n",
    "- `'10'` = designates a POVM effect label whose corresponding vector is given by the `effectExpressions` argument.\n",
    "- `\"2\"` = a prep or effect expression indicating a projection/preparation of the 3rd (b/c 0-based) computational basis element\n",
    "\n",
    "You can also explicity add identity operations, e.g. `\"I(Q0)\"`, to the rotation gates to get the same gate set (see `gs_targetB` below),  and this same syntax can be used for non-entangling 2-qubit gates, e.g. `\"X(pi/2,Q0):X(pi/2,Q1)\"`."
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   "source": [
    "gs_targetB = pygsti.construction.build_gateset( \n",
    "            [4], [('Q0','Q1')],['Gii','Gix','Giy','Gxi','Gyi','Gcnot'], \n",
    "            [ \"I(Q0):I(Q1)\", \"I(Q0):X(pi/2,Q1)\", \"I(Q0):Y(pi/2,Q1)\", \"X(pi/2,Q0):I(Q1)\", \"Y(pi/2,Q0):I(Q1)\", \"CNOT(Q0,Q1)\" ],\n",
    "            effectLabels=['00','01','10','11'], effectExpressions=[\"0\",\"1\",\"2\",\"3\"])\n",
    "assert(abs(gs_target.frobeniusdist(gs_targetB)) < 1e-6)"
   ]
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   "source": [
    "If our 2-qubit gate happens to be one that *can* be specified using `build_gateset` then we can just use it to construct the entire `GateSet` and be done.  Currently, `build_gateset` can create any controlled $X$, $Y$, or $Z$ rotation using `CX`, `CY` and `CZ`, as well as the standard `CNOT` and `CPHASE` gates.  Below we demonstrate creation with the CNOT gate.  The resulting `GateSet` is then identical to `pygsti.construction.std2Q_XYCNOT.gs_target`."
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   "source": [
    "gs_withCNOT = pygsti.construction.build_gateset( \n",
    "            [4], [('Q0','Q1')],['Gii','Gix','Giy','Gxi','Gyi','Gcnot'], \n",
    "            [ \"I(Q0):I(Q1)\", \"I(Q0):X(pi/2,Q1)\", \"I(Q0):Y(pi/2,Q1)\", \"X(pi/2,Q0):I(Q1)\", \"Y(pi/2,Q0):I(Q1)\", \"CNOT(Q0,Q1)\" ],\n",
    "            effectLabels=['00','01','10','11'], effectExpressions=[\"0\",\"1\",\"2\",\"3\"])\n",
    "\n",
    "#Note this is the same gateset as one of pyGSTi's standard gate sets:\n",
    "from pygsti.construction import std2Q_XYICNOT\n",
    "assert(abs(gs_withCNOT.frobeniusdist(std2Q_XYICNOT.gs_target)) < 1e-6)"
   ]
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   "source": [
    "Thus, since our `gs_target` just contains the 1-qubit gates of `std2Q_XYCNOT.gs_target`, we could also create has all the 1-qubit gates are the same a third way to obtain `gs_target` is to just remove `Gcnot` from the standard gate set:"
   ]
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   "source": [
    "gs_targetC = std2Q_XYICNOT.gs_target.copy()\n",
    "del gs_targetC.gates['Gcnot']\n",
    "assert(abs(gs_target.frobeniusdist(gs_targetC)) < 1e-6)"
   ]
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   "source": [
    "## Step 2: Create a custom 2-qubit gate\n",
    "We're assuming that `build_gateset` can't make the 2-qubit gate we want, so we'll need to create our own. Below we demonstrate how to create a 2-qubit gate from a given unitary which acts on the 2-qubit, 4-dimensional, state space."
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   "source": [
    "import numpy as np\n",
    "\n",
    "#Unitary in acting on the state-space { |A>, |B>, |C>, |D> } == { |00>, |01>, |10>, |11> }.\n",
    "# This unitary rotates the second qubit by pi/2 in either the (+) or (-) direction based on \n",
    "# the state of the first qubit.\n",
    "myUnitary = 1./np.sqrt(2) * np.array([[1,-1j,0,0],\n",
    "                                      [-1j,1,0,0],\n",
    "                                      [0,0,1,1j],\n",
    "                                      [0,0,1j,1]])\n",
    "\n",
    "#Convert this unitary into a \"superoperator\", which acts on the \n",
    "# space of vectorized density matrices instead of just the state space.\n",
    "# These superoperators are what GST calls \"gates\".\n",
    "mySuperOp_stdbasis = pygsti.unitary_to_process_mx(myUnitary)\n",
    "\n",
    "#After the call to unitary_to_process_mx, the superoperator is a complex matrix\n",
    "# in the \"standard\" or \"matrix unit\" basis given by { |A><A|, |A><B|, etc }.\n",
    "# For use in GST, we want to work with a *real* matrix in either the \n",
    "# Gell-Mann or Pauli-product basis. Here we choose the Pauli-product basis,\n",
    "# which is typically more intuitive when working with 2 qubits.\n",
    "mySuperOp_ppbasis = pygsti.change_basis(mySuperOp_stdbasis, \"std\", \"pp\")\n",
    "\n",
    "#The resulting superoperator in the Pauli-product basis is exactly\n",
    "# what goes into the GateSet object, which can be set using \n",
    "# dictionary syntax.  The line below names our two-qubit gate 'Gtq'\n",
    "gs_target['Gtq'] = mySuperOp_ppbasis"
   ]
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   "source": [
    "## That's it!\n",
    "We're done creating our 2-qubit gateset, `gs_target`, printed below.  To run 2-qubit GST with this gate set we ideally would generate fiducials and germs specifically for it.  Actually, since the 1-qubit gates are the same as other standard 2Q gate sets, and the fiducial seqeuences for these standard sets only contain 1-qubit gates, we can just use the fiducial sets from a standard set (e.g. `std2Q_XYCNOT`).  The germs should be computed, though typically you can get away with just using the germ set of a standard gate set and replacing it's 2-qubit gate with the custom one - so it can be worth checking whether such a set is complete before running a full germ-selection procedure. "
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     "text": [
      "rho0 =    0.5000        0        0   0.5000        0        0        0        0        0        0        0        0   0.5000        0        0   0.5000\n",
      "\n",
      "\n",
      "Mdefault = Unconstrained POVM with effect vectors:\n",
      "00:\n",
      " 0.50\n",
      "   0\n",
      "   0\n",
      " 0.50\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      " 0.50\n",
      "   0\n",
      "   0\n",
      " 0.50\n",
      "\n",
      "01:\n",
      " 0.50\n",
      "   0\n",
      "   0\n",
      "-0.50\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      " 0.50\n",
      "   0\n",
      "   0\n",
      "-0.50\n",
      "\n",
      "10:\n",
      " 0.50\n",
      "   0\n",
      "   0\n",
      " 0.50\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "-0.50\n",
      "   0\n",
      "   0\n",
      "-0.50\n",
      "\n",
      "11:\n",
      " 0.50\n",
      "   0\n",
      "   0\n",
      "-0.50\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "   0\n",
      "-0.50\n",
      "   0\n",
      "   0\n",
      " 0.50\n",
      "\n",
      "\n",
      "\n",
      "Gii = \n",
      "   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000\n",
      "\n",
      "\n",
      "Gix = \n",
      "   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0  -1.0000        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0  -1.0000        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0  -1.0000        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0  -1.0000\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0\n",
      "\n",
      "\n",
      "Giy = \n",
      "   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0  -1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0  -1.0000        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0  -1.0000        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0  -1.0000        0        0\n",
      "\n",
      "\n",
      "Gxi = \n",
      "   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0  -1.0000        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0  -1.0000        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0  -1.0000        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0  -1.0000\n",
      "        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0\n",
      "\n",
      "\n",
      "Gyi = \n",
      "   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000\n",
      "        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0\n",
      "        0        0        0        0  -1.0000        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0  -1.0000        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0  -1.0000        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0  -1.0000        0        0        0        0        0        0        0        0\n",
      "\n",
      "\n",
      "Gtq = \n",
      "   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0        0  -1.0000\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0\n",
      "        0        0        0        0        0        0        0        0        0  -1.0000        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0  -1.0000        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0        0\n",
      "        0        0        0        0        0        0        0        0        0        0        0        0        0   1.0000        0        0\n",
      "        0        0        0  -1.0000        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "        0        0   1.0000        0        0        0        0        0        0        0        0        0        0        0        0        0\n",
      "\n",
      "\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(gs_target)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": []
  }
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