qiskit/advanced/aer/6_extended_stabilizer_tutorial.ipynb

{
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![qiskit_header.png](attachment:qiskit_header.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Qiskit Aer: The Extended Stabilizer Simulator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Extended Simulator is a new method for classically simulating quantum circuits available in the latest release of [Qiskit-Aer](https://github.com/qiskit/qiskit-aer). \n",
    "\n",
    "This method is an implementation of the ideas published in the paper _Simulation of quantum circuits by low-rank stabilizer decompositions_ by Bravyi, Browne, Calpin, Campbell, Gosset & Howard, 2018, [**arXiv:1808.00128**](https://arxiv.org/abs/1808.00128).\n",
    "\n",
    "It uses a different representation of a quantum circuit, that gives it some unique capabilities. This notebook will give some examples of what the extended stabilizer method can do.\n",
    "\n",
    "For example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:17:32.305227Z",
     "start_time": "2019-08-19T17:17:32.300959Z"
    }
   },
   "outputs": [],
   "source": [
    "from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister\n",
    "from qiskit.compiler import assemble\n",
    "from qiskit.providers.aer import AerError, QasmSimulator\n",
    "from qiskit.tools.visualization import plot_histogram\n",
    "\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:21:43.756467Z",
     "start_time": "2019-08-19T17:21:43.742436Z"
    }
   },
   "outputs": [],
   "source": [
    "circ = QuantumCircuit(40, 40)\n",
    "\n",
    "# Initialise with a Hadamard layer\n",
    "circ.h(range(40))\n",
    "# Apply some random CNOT and T gates\n",
    "qubit_indices = [i for i in range(40)]\n",
    "for i in range(10):\n",
    "    control, target, t = random.sample(qubit_indices, 3)\n",
    "    circ.cx(control, target)\n",
    "    circ.t(t)\n",
    "circ.measure(range(40), range(40))\n",
    "\n",
    "qobj = assemble(circ, backend=QasmSimulator(), shots=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We've created a random circuit with just 60 gates, that acts on 40 qubits. But, because of the number of qubits, if we wanted to run this with say the statevector simulator then I hope you have access to terabytes of RAM!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:21:58.295935Z",
     "start_time": "2019-08-19T17:21:58.246239Z"
    }
   },
   "outputs": [],
   "source": [
    "statevector_job = QasmSimulator().run(qobj, backend_options={\n",
    "    'method': 'statevector'\n",
    "})\n",
    "# This should error!\n",
    "try:\n",
    "    statevector_job.result()\n",
    "except AerError as err:\n",
    "    print(err)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Extended Stabilizer method, in contrast, handles this circuit just fine. (Though it needs a couple of minutes!)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:22:26.158636Z",
     "start_time": "2019-08-19T17:22:22.534910Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This succeeded?: True\n"
     ]
    }
   ],
   "source": [
    "es_job = QasmSimulator().run(qobj, backend_options={\n",
    "    'method': 'extended_stabilizer'\n",
    "})\n",
    "result = es_job.result()\n",
    "print('This succeeded?: {}'.format(result.success))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How does this work?\n",
    "If you're interested in how exactly we can handle such large circuits, then for a detailed explanation you can [read the paper!](https://arxiv.org/abs/1808.00128)\n",
    "\n",
    "For running circuits, however, it's important to just understand the basics.\n",
    "\n",
    "The Extended Stabilizer method is made up of two parts. The first is a method of decomposing quantum circuits into _stabilizer circuits_, a special class of circuit that can be efficiently simulated classically. The second is then a way of combining these circuits to perform measurements.\n",
    "\n",
    "The number of terms you need scales with the number of what we call _non-Clifford Gates_. At the moment, the method knows how to handle the following methods:\n",
    "```python\n",
    "circ.t(qr[qubit])\n",
    "circ.tdg(qr[qubit])\n",
    "circ.ccx(qr[control_1], qr[control_2], qr[target])\n",
    "circ.u1(rotation_angle, qr[qubit])\n",
    "```\n",
    "\n",
    "The simulator is also able to handle circuits of up to 63 qubits.\n",
    "     \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One thing that's important to note is these decompositions are approximate. This means that the results aren't exactly the same as with the State Vector simulator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:23:52.873028Z",
     "start_time": "2019-08-19T17:23:52.863305Z"
    }
   },
   "outputs": [],
   "source": [
    "small_qr = QuantumRegister(2)\n",
    "small_cr = ClassicalRegister(2)\n",
    "small_circ = QuantumCircuit(small_qr, small_cr)\n",
    "small_circ.h(small_qr[0])\n",
    "small_circ.cx(small_qr[0], small_qr[1])\n",
    "small_circ.t(small_qr[0])\n",
    "small_circ.measure(small_qr, small_cr)\n",
    "small_qobj = assemble(small_circ, backend=QasmSimulator(), shots=100)\n",
    "# This circuit should give 00 or 11 with equal probability...\n",
    "expected_results ={'00': 50, '11': 50}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:23:54.879606Z",
     "start_time": "2019-08-19T17:23:53.249255Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 shots in 3.9687795639038086s\n"
     ]
    }
   ],
   "source": [
    "es_job = QasmSimulator().run(small_qobj, backend_options={'method':'extended_stabilizer'})\n",
    "es_result = es_job.result()\n",
    "es_counts = es_result.get_counts(small_circ)\n",
    "print('100 shots in {}s'.format(es_result.time_taken))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:23:55.222602Z",
     "start_time": "2019-08-19T17:23:55.101958Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot_histogram([expected_results, es_counts], legend=['Expected', 'Extended Stabilizer'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can control this approximation error using the `backend_options` in Qiskit Aer. The default error is 0.05. The smaller the error, the more precise the results, but also the longer your simulation will take and the more memory it will require."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:24:01.638958Z",
     "start_time": "2019-08-19T17:23:57.206364Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 shots in 10.484436273574829s\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x360 with 1 Axes>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "es_job = QasmSimulator().run(small_qobj, backend_options={\n",
    "    'method':'extended_stabilizer',\n",
    "    'extended_stabilizer_approximation_error': 0.03\n",
    "})\n",
    "es_result = es_job.result()\n",
    "es_counts = es_result.get_counts(small_circ)\n",
    "print('100 shots in {}s'.format(es_result.time_taken))\n",
    "plot_histogram([expected_results, es_counts], legend=['Expected', 'Extended Stabilizer'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulator Options"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are several other options you can tweak to control how the extended stabilizer method performs. What these options are and their explanation can all be found in the Qiskit Aer documentation. However, I want to highlight two important ones that can help to optimize your simulations.\n",
    "\n",
    "To perform measurements, the extended stabilizer method uses a Markov chain method to sample outcomes at random. This Markov chain has to be run for some time we call the 'mixing time' before it will start sampling, and has to be re-mixed for every circuit shot.\n",
    "\n",
    "If you expect your circuit output to be concentrated on just a few output states, then you can likely optimize your simulations by reducing the `extended_stabilizer_mixing_time` option.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:24:03.270544Z",
     "start_time": "2019-08-19T17:24:03.132881Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The circuit above, with 100 shots at precision 0.03 and default mixing time, needed 10s\n",
      "Dialing down the mixing time, we completed in just 0.26561832427978516s\n"
     ]
    }
   ],
   "source": [
    "print(\"The circuit above, with 100 shots at precision 0.03 \"\n",
    "      \"and default mixing time, needed {}s\".format(int(es_result.time_taken)))\n",
    "optimised_job = QasmSimulator().run(small_qobj, backend_options={\n",
    "    'method': 'extended_stabilizer',\n",
    "    'extended_stabilizer_approximation_error': 0.03,\n",
    "    'extended_stabilizer_mixing_time': 100\n",
    "})\n",
    "optimised_result = optimised_job.result()\n",
    "print('Dialing down the mixing time, we completed in just {}s'.format(optimised_result.time_taken))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Similarly, if your circuit has some non-zero probability on all amplitudes (e.g. if it's a random circuit), then you can avoid this expensive re-mixing step to take multiple shots from the output at once. This can be enabled by setting `extended_stabilizer_measure_sampling=True` in the `backend_options`.\n",
    "\n",
    "For example, let's look again at the random circuit from the start of the tutorial, running for 100 shots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:24:03.961088Z",
     "start_time": "2019-08-19T17:24:03.951284Z"
    }
   },
   "outputs": [],
   "source": [
    "multishot_qobj = assemble(circ, backend=QasmSimulator(), shots=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:24:20.656799Z",
     "start_time": "2019-08-19T17:24:04.332417Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 shots took 31.812614917755127 s\n"
     ]
    }
   ],
   "source": [
    "# It's a 40 qubit circuit so we use Extended Stabilizer automatically\n",
    "multishot_job = QasmSimulator().run(multishot_qobj, backend_options={\n",
    "    'extended_stabilizer_mixing_time': 100 # We set this here only to make the example run more quickly.\n",
    "})\n",
    "result = multishot_job.result()\n",
    "print(\"100 shots took {} s\".format(result.time_taken))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:24:21.070233Z",
     "start_time": "2019-08-19T17:24:20.918300Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "With the optimization, 50 shots took 0.15625762939453125 s\n"
     ]
    }
   ],
   "source": [
    "multishot_job = QasmSimulator().run(multishot_qobj, backend_options={\n",
    "    'extended_stabilizer_measure_sampling': True,\n",
    "    'extended_stabilizer_mixing_time': 100 # We set this here only to make the example run more quickly.\n",
    "})\n",
    "result = multishot_job.result()\n",
    "print(\"With the optimization, 50 shots took {} s\".format(result.time_taken))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How can I use it?\n",
    "\n",
    "In the latest release of Qiskit Aer, if you try to run a circuit on more qubits than can be handled by the state vector simulator, then Aer will automatically switch to using the extended stabilizer method. You can also force circuits to run on this method using the `backend_options` keyword argument to `QasmSimulator().run`. \n",
    "\n",
    "It's worth bearing in mind: if you have smaller circuits with lots of non-Clifford gates, then the statevector method will likely perform better than the extended stabilizer. If however you want to look at circuits on many qubits, without needing access to high performance computation, then give this method a try!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-08-19T17:24:45.930560Z",
     "start_time": "2019-08-19T17:24:45.920568Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<h3>Version Information</h3><table><tr><th>Qiskit Software</th><th>Version</th></tr><tr><td>Qiskit</td><td>0.14.0</td></tr><tr><td>Terra</td><td>0.11.0</td></tr><tr><td>Aer</td><td>0.3.4</td></tr><tr><td>Ignis</td><td>0.2.0</td></tr><tr><td>Aqua</td><td>0.6.1</td></tr><tr><td>IBM Q Provider</td><td>0.4.4</td></tr><tr><th>System information</th></tr><tr><td>Python</td><td>3.7.4 (default, Aug  9 2019, 18:34:13) [MSC v.1915 64 bit (AMD64)]</td></tr><tr><td>OS</td><td>Windows</td></tr><tr><td>CPUs</td><td>2</td></tr><tr><td>Memory (Gb)</td><td>7.9987335205078125</td></tr><tr><td colspan='2'>Tue Dec 10 16:02:44 2019 Eastern Standard Time</td></tr></table>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
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     "metadata": {},
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      "text/html": [
       "<div style='width: 100%; background-color:#d5d9e0;padding-left: 10px; padding-bottom: 10px; padding-right: 10px; padding-top: 5px'><h3>This code is a part of Qiskit</h3><p>&copy; Copyright IBM 2017, 2019.</p><p>This code is licensed under the Apache License, Version 2.0. You may<br>obtain a copy of this license in the LICENSE.txt file in the root directory<br> of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.<p>Any modifications or derivative works of this code must retain this<br>copyright notice, and modified files need to carry a notice indicating<br>that they have been altered from the originals.</p></div>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
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   ],
   "source": [
    "import qiskit.tools.jupyter\n",
    "%qiskit_version_table\n",
    "%qiskit_copyright"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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