{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "b10488aa-dfe0-4594-939c-14a215a3c793",
   "metadata": {},
   "source": [
    "<h1><center>Introduction to quantum science and technology - QUANT 400</center></h1>\n",
    "\n",
    "<p><center> <b>Lecturer:</b> <i>Prof. G. Carleo</i> </center><p>\n",
    "    \n",
    "<p><center> <b>Assistant: </b> <i>friederike.metz@epfl.ch"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46b0c9b1-3617-43e1-bfe5-2a76237204f9",
   "metadata": {},
   "source": [
    "## Exercise 2 - Quantum computing"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "caee92ab-871c-4b84-964f-4d1c2e75a463",
   "metadata": {},
   "source": [
    "### Very brief introduction to Qiskit\n",
    "\n",
    "[Qiskit](https://www.ibm.com/quantum/qiskit) is an open-source software for working with quantum computers at the level of quantum circuits, operators, and primitives.\n",
    "\n",
    "See the [documentation](https://docs.quantum.ibm.com/) for more details."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "93d8bb3f-10f2-49b8-a2df-ac17673694da",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "from qiskit import *\n",
    "from qiskit.primitives import StatevectorEstimator, StatevectorSampler\n",
    "from qiskit.quantum_info import Statevector, Pauli, SparsePauliOp"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e170b2eb-4fc0-4451-bbc3-4a0f37ac8b8b",
   "metadata": {},
   "source": [
    "The basic element needed for your first program is the [QuantumCircuit](https://qiskit.org/documentation/stubs/qiskit.circuit.QuantumCircuit.html).  We begin by creating a `QuantumCircuit` comprised of two qubits that start out in the $|00\\rangle$ state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f27f574b-0af9-4a3a-9dce-82952baf59b0",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create a Quantum Circuit acting on a quantum register of two qubits\n",
    "circ = QuantumCircuit(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75e9aac3-8510-4a19-9514-dd9751d704ca",
   "metadata": {},
   "source": [
    "Let's add gates that prepare a Bell-state (i.e. a maximally entangled two-qubit state):\n",
    "\n",
    "$$|\\psi\\rangle = \\left(|00\\rangle+|11\\rangle\\right)/\\sqrt{2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "86ac0a79-d919-48d6-963a-e340a95614ef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<qiskit.circuit.instructionset.InstructionSet at 0x11677ca00>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Add a H gate on qubit 0, putting this qubit in superposition.\n",
    "circ.h(0)\n",
    "# Add a CX (CNOT) gate on control qubit 0 and target qubit 1, putting the qubits in a Bell state.\n",
    "circ.cx(0, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ecb37bd-b81a-49df-a022-a2d7e61bce26",
   "metadata": {},
   "source": [
    "We can visualize the circuit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "05d873c0-2667-4f90-b2a4-3c54af4d3b1f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 287.294x200.667 with 1 Axes>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circ.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5c55df9-2ca5-4c5d-a242-f38f3ed29666",
   "metadata": {},
   "source": [
    "For a full list of all gates that are available, check Qiskit's [circuit library](https://docs.quantum.ibm.com/api/qiskit/circuit_library)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb493210-e410-4b86-888d-15ec6d99e624",
   "metadata": {},
   "source": [
    "We can extract the quantum state vector at the end of the circuit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "7cef7626-afea-4af0-8a37-33ed0b14816f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Statevector([0.70710678+0.j, 0.        +0.j, 0.        +0.j,\n",
      "             0.70710678+0.j],\n",
      "            dims=(2, 2))\n"
     ]
    }
   ],
   "source": [
    "Statevector(circ)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c10314f-aaaa-4a79-878d-d43a392a1a4a",
   "metadata": {},
   "source": [
    "We can compute expectation values of observables using the `Estimator`. If you use the `StatevectorEstimator`, the expectation value is still calculated using the state vector representation of the quantum state."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "d1767dc7-9157-480b-8fc8-895be74a6abe",
   "metadata": {},
   "outputs": [],
   "source": [
    "estimator = StatevectorEstimator()\n",
    "observable = Pauli('XX')\n",
    "result = estimator.run([(circ, observable)]).result()[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "53b70be7-0565-4454-8289-fd75a3bdb847",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Value: 0.9999999999999998\n",
      "Standard error: 0.0\n"
     ]
    }
   ],
   "source": [
    "print(\"Value:\", result.data.evs)\n",
    "print(\"Standard error:\", result.data.stds)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "66094685",
   "metadata": {},
   "source": [
    "You can compute the expectation values of multiple observables by providing a list of them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "2de15307-a8f1-40fd-9bc7-683ea1a841b9",
   "metadata": {},
   "outputs": [],
   "source": [
    "result = estimator.run([(circ, [Pauli('XX'), Pauli('IY')])]).result()[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "ecba83cc-6019-4a2b-b0de-426231e7f486",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Value: [1. 0.]\n",
      "Standard error: [0. 0.]\n"
     ]
    }
   ],
   "source": [
    "print(\"Value:\", result.data.evs)\n",
    "print(\"Standard error:\", result.data.stds)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae6a0a1d-38bb-4781-92c0-c393734ef1e4",
   "metadata": {},
   "source": [
    "In practice on a real quantum computer, we can only measure the qubits in the computational basis (i.e., eigenstates of the Pauli-$\\hat{Z}$ operator) and obtain one of two possible outcomes for every qubit: 0, 1 corresponding to either the eigenvalue +1 (spin up) or eigenvalue -1 (spin down). The expectation values would then be computed by repeatedly running the circuit and measuring the qubits (possibly in different bases) and averaging over the measurement statistics. The more measurements (shots) you take, the more precise the expectation values can be estimated."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "91e5510e-34b6-4b8e-8656-cc8dd8649bac",
   "metadata": {},
   "source": [
    "We can obtain this raw measurement data using the `Sampler`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "d25e4bfa-e6bc-4bc7-b45c-fe1556bd01a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'00': 8, '11': 2}"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circ.measure_all() # we first need to specify which qubits are measured\n",
    "sampler = StatevectorSampler()\n",
    "result = sampler.run([circ], shots=10).result()[0]\n",
    "counts = result.data.meas.get_counts()\n",
    "counts"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab3d981a",
   "metadata": {},
   "source": [
    "Let's visualize the number of times each different bitstring was measured. Note that if you rerun the cell above, the outcome will change each time as the measurement process is inherently probabilistic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "4e9781cd-f88d-4c00-aad1-033e1f9880b2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from qiskit.visualization import plot_histogram\n",
    "plot_histogram(counts)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4b42d48-319d-41a5-bc32-5c5bef6e81f1",
   "metadata": {},
   "source": [
    "### Time evolution of a quantum spin model"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "04134078",
   "metadata": {},
   "source": [
    "Let's simulate the time evolution of the Ising model using qiskit. Recall the previous exercise in which we used the 2nd order Trotter-Suzuki decomposition to approximate the total evolution unitary as\n",
    "\n",
    "$$ \\exp (-i \\hat{H} t) \\approx \\exp(-i \\frac{\\Delta_t}{2}  \\hat H_\\mathrm{Z Z}) \\ldots \\exp(-i \\Delta_t \\hat H_\\mathrm{X})   \\exp(-i \\Delta_t \\hat H_\\mathrm{Z Z}) \\ldots \\exp(-i \\Delta_t \\hat H_\\mathrm{X}) \\exp(-i \\frac{\\Delta_t}{2}  \\hat H_\\mathrm{Z Z}) $$\n",
    "\n",
    "In turn, each of these terms can be expressed as a product of commuting unitaries acting on one or two qubits only, e.g.,\n",
    "\n",
    "$$\\exp(-i \\Delta_t \\hat H_\\mathrm{X}) = \\prod_{i=1}^N \\exp(-i \\Delta_t \\Gamma \\hat{X}_i) = \\prod_{i=1}^N R_{X_i}(2 \\Delta_t \\Gamma ).$$\n",
    "\n",
    "$R_{X_i}$ denotes a rotation of qubit $i$ around the $X$ axis of the Bloch sphere. Note the extra factor of 2 since the definition of the $R_X$ gate in qiskit comes with a factor of 1/2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "31fa1dd1-e96c-4d6e-8dbe-927373311dcd",
   "metadata": {},
   "outputs": [],
   "source": [
    "def time_evol_circuit(N, t, dt, J, gamma):\n",
    "    n_steps = round(t / dt) # number of Suzuki-Trotter steps\n",
    "    circ = QuantumCircuit(N)\n",
    "    for i in range(N-1): # N-1 because we assume open boundary conditions\n",
    "        circ.rzz(2*J*dt/2, i, (i+1)%N)\n",
    "    for step in range(n_steps-1):\n",
    "        for i in range(N):\n",
    "            circ.rx(2*gamma*dt, i)\n",
    "        for i in range(N-1):\n",
    "            circ.rzz(2*J*dt, i, (i+1)%N)\n",
    "    for i in range(N):\n",
    "        circ.rx(2*gamma*dt, i)\n",
    "    for i in range(N-1):\n",
    "        circ.rzz(2*J*dt/2, i, (i+1)%N)\n",
    "    return circ"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "3fadac35-847c-4b31-a9a9-66a27aa47445",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 2210.55x1204 with 1 Axes>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circ = time_evol_circuit(4, 1., 0.1, 1.0, 0.5)\n",
    "circ.draw('mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f90b3c49-bcbc-4515-8bfc-735e51804a2e",
   "metadata": {},
   "source": [
    "Starting from the $|0\\dots 0\\rangle$ initial state, let's measure the average magnetization along z-direction every 100 steps during the time evolution.\n",
    "\n",
    "$$ \\hat{O} = \\frac 1 N \\sum_i \\hat{Z}_i$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "19fb5d1d-70b8-492b-a969-8e5c17e12508",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "SparsePauliOp(['IIIIZ', 'IIIZI', 'IIZII', 'IZIII', 'ZIIII'],\n",
       "              coeffs=[0.2+0.j, 0.2+0.j, 0.2+0.j, 0.2+0.j, 0.2+0.j])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 5 # number of qubits\n",
    "t = 10. # total evolution time\n",
    "dt = 1e-2 # Trotter time step size (Delta t)\n",
    "J = 1.0 # Ising coupling strength\n",
    "gamma = 0.8 # Ising transverse field\n",
    "n_points = 50 # number of time stamps at which we want to measure the expectation value of the observable\n",
    "ts = np.linspace(0, t, n_points+1) # measure at these times\n",
    "\n",
    "# generate 100 different circuits corresponding to 100 different evolution times\n",
    "circuits = [time_evol_circuit(N, t, dt, J, gamma) for t in ts]\n",
    "\n",
    "# define the observable we want to measure\n",
    "obs_terms = [(\"Z\", [i], 1/N) for i in range(N)]\n",
    "observable = SparsePauliOp.from_sparse_list(obs_terms, num_qubits=N)\n",
    "observable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "2bc0e159-c9a1-4dd4-a6b0-6bbbf8885418",
   "metadata": {},
   "outputs": [],
   "source": [
    "circuit_observable_pairs = [(circ, observable) for circ in circuits]\n",
    "results = estimator.run(circuit_observable_pairs).result()\n",
    "exp_values = [results[i].data.evs for i in range(n_points+1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "612b6f29",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(ts, exp_values)\n",
    "plt.xlabel(\"$t$\")\n",
    "plt.ylabel(r\"$\\langle \\psi_t | \\hat{O} | \\psi_t \\rangle$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3fb9e495",
   "metadata": {},
   "source": [
    "### Transpilation to native gates of the device"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eaede193",
   "metadata": {},
   "source": [
    "When writing a quantum circuit you are free to use any quantum gate (unitary operator) that you like, alongside with a collection of non-gate operations such as qubit measurements and reset operations. However, when running a circuit on a real quantum device one no longer has this flexibility. Due to limitations in, for example, the physical interactions between qubits, difficulty in implementing multi-qubit gates, control electronics, etc., a quantum computing device can only natively support a handful of quantum gates and non-gate operations. In the present case of [IBM Q devices](https://quantum-computing.ibm.com/services/resources?tab=systems), the native gate set can be found by querying the devices themselves, and looking for the corresponding attribute in their configuration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "4757f690",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['id', 'rz', 'sx', 'x', 'cx', 'reset']"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from qiskit.providers.fake_provider import Fake5QV1, Fake20QV1\n",
    "\n",
    "Fake5QV1().configuration().basis_gates"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d4d84fcc",
   "metadata": {},
   "source": [
    "Every quantum circuit run on an IBM Q device must be expressed using only these basis gates.\n",
    "\n",
    "Let's take the following circuit as an example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "2bc8b71a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1123.61x367.889 with 1 Axes>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circ = time_evol_circuit(4, 0.1, 0.1, 1.0, 0.5)\n",
    "circ.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ab2f210",
   "metadata": {},
   "source": [
    "We have $R_{ZZ}$, and $R_X$ rotation gates, all of which are not in our devices basis gate set, and must be expanded. We can decompose the circuit to show what it would look like in the native gate set of the IBM Quantum devices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "4c202e52",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/g2/zp6r_4wd2jbb_yn3bzh8tqjh0000gq/T/ipykernel_57190/2126254506.py:6: DeprecationWarning: The `transpile` function will stop supporting inputs of type `BackendV1` ( fake_5q_v1 ) in the `backend` parameter in a future release no earlier than 2.0. `BackendV1` is deprecated and implementations should move to `BackendV2`.\n",
      "  circ_transpiled = transpile(circ, backend)\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1920.89x451.5 with 1 Axes>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from qiskit import transpile\n",
    "\n",
    "# backend = Fake20QV1()\n",
    "backend = Fake5QV1()\n",
    "\n",
    "circ_transpiled = transpile(circ, backend)\n",
    "circ_transpiled.draw(output='mpl')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f3cfe0a0",
   "metadata": {},
   "source": [
    "A few things to highlight. First, the circuit has gotten longer with respect to the initial one. Second, although we had only 12 two-qubit gates in the original circuit, the fact that those were not in the basis set means that, when expanded, it requires more than a single cx gate to implement. All said, unrolling to the basis set of gates leads to an increase in the depth of a quantum circuit and the number of gates. The deeper a circuit, the more noise and the bigger the errors our quantum states will be subject to. Keep this in mind when designing circuits that are run on real quantum devices!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d884df68",
   "metadata": {},
   "source": [
    "In general, [transpilation](https://qiskit.org/documentation/apidoc/transpiler.html) is the process of rewriting a given input circuit to match the topology of a specific quantum device, and/or to optimize the circuit for execution on present day noisy quantum systems.\n",
    "\n",
    "Most circuits must undergo a series of transformations that make them compatible with a given target device, and optimize them to reduce the effects of noise on the resulting outcomes. Rewriting quantum circuits to match hardware constraints and optimizing for performance can be far from trivial. The flow of logic in the rewriting tool chain need not be linear, and can often have iterative sub-loops, conditional branches, and other complex behaviors. The details are not important for you at this stage, just be aware that transpilation is an important but potentially very complex process."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8819ab54",
   "metadata": {},
   "source": [
    "### Problem 1: Noisy expectation values due to sampling"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "caea2eb1",
   "metadata": {},
   "source": [
    "As mentioned earlier, the measurement outcome of a digital quantum computation are always bitstrings. If we want to compute the expectation value of some observable, in practice, we need to measure reapeatedly, collect all occurences of each measured bit(string), and then compute weighted averages over those. Hence, the result will necessary be noisy. In this exercise you will study how the number of measurements (shots) affects the standard error on the estimated expectation value."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "889cd1ec",
   "metadata": {},
   "source": [
    "a) Define a function that given a circuit and a specified number of shots/measurements returns the expectation value of the average z-magnetization\n",
    "\n",
    "$$ \\hat{O} = \\frac 1 N \\sum_i \\hat{Z}_i$$\n",
    "\n",
    "You can only use the `Sampler()` and not the `Estimator()` for this exercise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "70ed960e",
   "metadata": {},
   "outputs": [],
   "source": [
    "def measure_average_sz_magnetization(circ, n_shots):\n",
    "    circ.measure_all()\n",
    "    sampler = StatevectorSampler()\n",
    "    result = sampler.run([circ], shots=n_shots).result()[0]\n",
    "    counts = result.data.meas.get_counts()\n",
    "    # print(counts)\n",
    "\n",
    "    avg_mag = 0\n",
    "    for (bitstring, count) in counts.items():\n",
    "        bits = np.array([int(bit) for bit in bitstring]) # convert characters in string to integer type\n",
    "        szs = (-1)**(bits)\n",
    "        avg_mag += np.mean(szs) * count / n_shots\n",
    "    return avg_mag"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51339fc3",
   "metadata": {},
   "source": [
    "Given a observable that is diagonal in the $Z$-basis: $\\hat{O} = \\sum_\\mathbf{z} f(\\mathbf{z}) |\\mathbf{z}\\rangle\\langle \\mathbf{z}|$ with possible measurement outcomes $f(\\mathbf{z})$, the expectation value $\\langle \\psi |\\hat{O}|\\psi\\rangle$ can be estimated as\n",
    "\n",
    "$$ \\langle \\psi |\\hat{O}|\\psi\\rangle \\sim \\frac{1}{M_s} \\sum_{i=1}^{M_s} f(\\mathbf{z}^{(i)}) ,$$\n",
    "\n",
    "where $M_s$ is the number of measurements (shots) and $f(\\mathbf{z}^{(i)})$ is the measurement outcome for the ith measurement.\n",
    "\n",
    "If you measure the j-th qubit on the quantum device, $\\hat{Z}_j = \\sum_\\mathbf{z} (-1)^{z_j} |\\mathbf{z}\\rangle\\langle \\mathbf{z}|$, so $f(\\mathbf{z})=(-1)^{z_j}$.\n",
    "\n",
    "(Note that in Qiskit 0 corresponds to eigenvalue +1 (spin up) and 1 to eigenvalue -1 (spin down).)\n",
    "\n",
    "For the observable that is an average over all $\\hat{Z}_i$ operators, we get\n",
    "\n",
    "$$ \\hat{O} = \\frac 1 N \\sum_i \\hat{Z}_i = \\sum_\\mathbf{z} \\frac{1}{N} \\sum_i (-1)^{z_i} |\\mathbf{z}\\rangle\\langle \\mathbf{z}|, $$\n",
    "\n",
    "such that $f(\\mathbf{z}) = \\frac 1 N \\sum_i (-1)^{z_i}$.\n",
    "\n",
    "For a single measured bistring $\\mathbf{z}$, the `np.mean((-1)**(bits))` in the code above is computing this average. Now we still have to average over all the collected measurements, i.e. $\\frac{1}{M_s} \\sum_{j=1}^{M_s} f(\\mathbf{z}^{(j)})$. So we have to sum over all bitstrings and if a bitstring appeared multiple times, we have to account for that in the sum (that's why we multiply by `count`). Finally we divide by the total number of shots $M_s$ (`n_shots`)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d80fe511",
   "metadata": {},
   "source": [
    "b) Given the circuit below, compute the expectation value a few times (~20) and then calculate the standard deviation over all individual results you obtained. Set the number of shots to 1000."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "1e2d3490",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.016400048780415254)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "circ = time_evol_circuit(N, 10., dt, J, gamma)\n",
    "result = []\n",
    "for i in range(20):\n",
    "    result.append(measure_average_sz_magnetization(circ, 1000))\n",
    "np.std(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3298d1c8",
   "metadata": {},
   "source": [
    "c) Repeat the steps from above for different number of shots as specified below. Then plot the computed standard deviations against the number of shots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "e1d333bd",
   "metadata": {},
   "outputs": [],
   "source": [
    "n_shots_list = np.logspace(2, 5, 4, base=10, dtype=int)\n",
    "stds = []\n",
    "for n_shots in n_shots_list:\n",
    "    result = []\n",
    "    for i in range(20):\n",
    "        result.append(measure_average_sz_magnetization(circ, n_shots))\n",
    "    stds.append(np.std(result))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "ddce04da",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(n_shots_list, stds)\n",
    "plt.loglog()\n",
    "plt.xlabel(\"number of shots\")\n",
    "plt.ylabel(\"standard deviation\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f03b58ca",
   "metadata": {},
   "source": [
    "d) Can you figure out a simple mathematical relationship of how the standard deviation of the mean scales with the number of shots? What implications does this have on results obtained from a quantum computation?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "85f1fdf7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.50062907, -0.6276524 ])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "curve = np.polyfit(np.log(n_shots_list), np.log(stds), 1)\n",
    "curve"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b742fd77",
   "metadata": {},
   "source": [
    "The slope of the linear regression is roughly 0.5. This means that $\\sigma \\sim 1/\\sqrt{M_{shots}}$. You just rediscovered the general scaling law of the standard error of the sample mean with the sample size. This holds for any observable and circuit.\n",
    "\n",
    "In general, you can estimate the standard error via\n",
    "\n",
    "$$\\sigma_{SE} = \\frac{\\sigma_{\\hat{O}}}{\\sqrt{M_{s}}},$$\n",
    "\n",
    "and $\\sigma_{\\hat{O}}^2 = Var[\\hat{O}] = \\langle \\psi |\\hat{O}^2|\\psi \\rangle - \\langle \\psi |\\hat{O}|\\psi \\rangle^2$ can be estimated via the sample standard deviation\n",
    "\n",
    "$$\\sigma_{\\hat{O}}^2=\\frac{1}{M_s-1} \\sum_{i=1}^{M_s}\\left(f(\\mathbf{z}^{(i)})-\\mu\\right)^2 .  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "b9e17f27",
   "metadata": {},
   "outputs": [],
   "source": [
    "def measure_average_sz_magnetization(circ, n_shots):\n",
    "    circ.measure_all()\n",
    "    sampler = StatevectorSampler()\n",
    "    result = sampler.run([circ], shots=n_shots).result()[0]\n",
    "    counts = result.data.meas.get_counts()\n",
    "\n",
    "    avg_mag = 0\n",
    "    for (bitstring, count) in counts.items():\n",
    "        bits = np.array([int(bit) for bit in bitstring])\n",
    "        szs = (-1)**(bits)\n",
    "        avg_mag += np.mean(szs) * count / n_shots\n",
    "\n",
    "    std_mag = 0\n",
    "    for (bitstring, count) in counts.items():\n",
    "        bits = np.array([int(bit) for bit in bitstring])\n",
    "        szs = (-1)**(bits)\n",
    "        std_mag += (np.mean(szs) - avg_mag)**2 * count / (n_shots - 1)\n",
    "\n",
    "    return avg_mag, std_mag"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "2abf5245",
   "metadata": {},
   "outputs": [],
   "source": [
    "results2 = [measure_average_sz_magnetization(circ, shots) for shots in n_shots_list]\n",
    "results2 = np.array(results2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "7d069aa7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(n_shots_list, stds)\n",
    "plt.plot(n_shots_list, np.sqrt(results2[:,1]/n_shots_list), linestyle='--')\n",
    "plt.loglog()\n",
    "plt.xlabel(\"number of shots\")\n",
    "plt.ylabel(\"standard deviation\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87f10a50",
   "metadata": {},
   "source": [
    "The purpose of this exercise was to show you that statistical uncertainties in measurements are a necessary evil in quantum computation. In practice, any quantum algorithm needs to be designed and understood using random variables, and any results you obtain should always include well-estimated uncertainties."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4a184c7-96b9-40bb-8e43-e60ab26490ca",
   "metadata": {},
   "source": [
    "### Problem 2: Digitial adiabatic state preperation "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45eca8ed-9848-42b0-9835-c081d9373ddf",
   "metadata": {},
   "source": [
    "Let's turn to the problem of ground state preparation on quantum computers. One way of obtaining ground states is by __adiabatic time evolution__. Let's say that we know how to prepare the ground state of some Hamiltonian $H_i$ and our goal is to find the ground state of a different, more complicated Hamiltonian $H_f$. We then define a time-dependent Hamiltonian $H(t)$ that interpolates between the initial Hamiltonian $H_i$ and final Hamiltonian $H_f$, i.e.,\n",
    "\n",
    "$$H(t) = (1-u(t)) H_i + u(t) H_f ,$$\n",
    "\n",
    "where $u(t)$ is a smooth function of time which is 0 at $t=0$ and 1 at $t=T$. For example: $u(t) = t/T$.\n",
    "\n",
    "The [adiabatic theorem](https://en.wikipedia.org/wiki/Adiabatic_theorem) then guarantees that we stay in the instantaneous ground state of $H(t)$ throughout the time evolution as long as we evolve slowly enough and as long as there is a gap between the ground and first excited state.\n",
    "\n",
    "Let's apply the idea of adiabatic ground state preparation to our Ising model example. As an example imagine you want to prepare the ground state of $H_f = J \\sum_i \\hat{X}_i\\hat{X}_{i+1} - \\sum_i \\hat{Z}_i$ for some fixed coupling strength $J$. We know that the initial state on a quantum computer is $|0\\dots 0 \\rangle$ which is exactly the ground state of the transverse-field Hamiltonian, i.e., $H_i = - \\sum_i \\hat{Z}_i$. Hence, we need to adiabatically evolve with the following time-dependent Hamiltonian:\n",
    "\n",
    "$$H(t) = (1-u(t)) H_i + u(t) H_f = u(t) J \\sum_i \\hat{X}_i\\hat{X}_{i+1} - \\sum_i \\hat{Z}_i .$$\n",
    "\n",
    "So we adiabtically turn on the interaction term of the Ising model.\n",
    "\n",
    "We now want to simulate the time evolution of our initial $|0\\dots 0 \\rangle$ state with the Hamiltonian above on a quantum device. Let's figure out how to do this step by step. First, we compute the propagator $G_{\\Delta_t}(t)$ that evolves a quantum state at time $t$ by a small (infinitesimal) time increment $\\Delta_t$:\n",
    "\n",
    "$$ \\ket{\\Psi(t+\\Delta_t)} = G_{\\Delta_t}(t) \\ket{\\Psi(t)} =  e^{-i \\Delta_t \\bar{H}_t} \\ket{\\Psi(t)}.$$\n",
    "\n",
    "a) Determine $\\bar{H}_t$ up to first order in the Magnus expansion for the time-dependent Ising Hamiltonian above using pen and paper (no need to write code). Assume a linear ramp, i.e., $u(t)= t/T$ and open boundary conditions."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d87c50e",
   "metadata": {},
   "source": [
    "Remember from class that the first term in the Magnus expansion is just the time-averaged Hamiltonian, i.e., \n",
    "\n",
    "$$\\bar{H}^{(1)}_t = \\frac{1}{\\Delta_t}\\int_{t}^{t+\\Delta_t} H(t') dt'$$\n",
    "\n",
    "Integrating the Hamiltonian from above then gives:\n",
    "\n",
    "$$ \\bar{H}^{(1)}_t = - \\sum_i \\hat{Z}_i + \\frac{J}{2T} (2t + \\Delta_t) \\sum_i \\hat{X}_i\\hat{X}_{i+1} \\equiv H_Z + H_{XX}(t)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e1d53a66",
   "metadata": {},
   "source": [
    "b) Assuming that $\\Delta_t$ is small, use the first-order Trotter-Suzuki decomposition to split the propagator $G_{\\Delta_t}(t)$ into a product of local terms (again using pen and paper).\n",
    "\n",
    "How does the error of the obtained expression scale w.r.t. the exact time evolution?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "107e15cc",
   "metadata": {},
   "source": [
    "$$ G_{\\Delta_t}(t) = e^{-i\\Delta_t \\bar{H}^{(1)}_t} + \\mathcal{O}\\left(\\Delta_t^2\\right) = e^{-i\\Delta_t H_Z} e^{-i\\Delta_t H_{XX}(t)} + \\mathcal{O}\\left(\\Delta_t^2\\right) = \\prod_i \\exp({i\\Delta_t \\hat{Z}_i}) \\prod_i \\exp({-i \\Delta_t\\frac{ J}{2T}(2t + \\Delta_t) \\hat{X}_i\\hat{X}_{i+1}}) + \\mathcal{O}\\left(\\Delta_t^2\\right)$$\n",
    "\n",
    "\n",
    "The first-order Magnus expansion introduces an error of $\\mathcal{O}\\left(\\Delta_t^2\\right)$ for the propagator. Similarly, the first-order Trotter-Suzuki decomposition introduces an error on the order of $\\Delta_t^2$ so the overall error still scales as $\\Delta_t^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2926ff7",
   "metadata": {},
   "source": [
    "c) Now write down the propagator $G(t)$ for the time evolution from time $t=0$ to $t=n_{steps} \\Delta_t$ using the result above.\n",
    "\n",
    "$$ \\ket{\\Psi(t)} = G(t) \\ket{\\Psi(0)} =  ... \\ket{\\Psi(0)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08f06c8f",
   "metadata": {},
   "source": [
    "$$G(t) = \\prod_{n=0}^{n_{steps}-1}\\left[\\prod_{i=1}^{N}\\exp(i\\Delta_t \\hat{Z}_i) \\prod_{i=1}^{N-1}\\exp(-i \\frac{\\Delta_t^2 J}{2T}(2n+1) \\hat{X}_i\\hat{X}_{i+1})\\right]$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "59f42660",
   "metadata": {},
   "source": [
    "d) Modify the function `time_evol_circuit` from above to the time-dependent case considered here.\n",
    "\n",
    "Note that as before $N$ is the total number of spins/qubits, $t$ is the time up to which we want to evolve, $(T,J)$ are parameters in the Hamiltonian, and `dt` is the step size $\\Delta_t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "4a36deed",
   "metadata": {},
   "outputs": [],
   "source": [
    "def adiabatic_time_evol_circuit(N, t, T, dt, J):\n",
    "    n_steps = round(t / dt) # number of Suzuki-Trotter steps\n",
    "    circ = QuantumCircuit(N)\n",
    "    for step in range(n_steps):\n",
    "        for i in range(N-1):\n",
    "            circ.rxx(2*J*(2*(step) + 1)*dt**2/(2*T), i, (i+1)%N)\n",
    "        for i in range(N):\n",
    "            circ.rz(-2*dt, i)\n",
    "    return circ"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6026e1a8",
   "metadata": {},
   "source": [
    "During the time evolution we want to measure the energy w.r.t. $H(t)$ to see if we actually stay in the instantaneous ground state at all times.\n",
    "\n",
    "e) Define a function that given a time $t$ returns the Hamiltonian $H(t)$ as a `SparsePauliOp`. There are different ways of doing this, any one is fine."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "6b97e546",
   "metadata": {},
   "outputs": [],
   "source": [
    "def hamiltonian(N, t, T, J):\n",
    "    h_terms = [(\"Z\", [i], -1) for i in range(N)]\n",
    "    h_terms += [(\"XX\", [i, (i+1)%N], J*t/T) for i in range(N-1)]\n",
    "\n",
    "    return SparsePauliOp.from_sparse_list(h_terms, num_qubits=N)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d7975976",
   "metadata": {},
   "source": [
    "Now it's time to simulate the evolution and measure the energy of the Hamiltonian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "9ca539a5",
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 5 # number of qubits\n",
    "T = 1. # total evolution time\n",
    "dt = 1e-2 # Trotter time step size (Delta t)\n",
    "J = 0.5 # Ising coupling strength\n",
    "n_points = 50 # number of time stamps at which we want to measure the expectation value of the observable\n",
    "ts = np.linspace(0, T, n_points+1) # measure at these times\n",
    "measured_energies = [] # store measured energies here"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b53039c7",
   "metadata": {},
   "source": [
    "f) Use the `StatevectorEstimator` to run the circuits and measure the corresponding Hamiltonian $H(t)$ at all times `ts`. Plot the expectation values as a function of time $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "51dee96e",
   "metadata": {},
   "outputs": [],
   "source": [
    "estimator = StatevectorEstimator()\n",
    "circuit_observable_pairs = [(adiabatic_time_evol_circuit(N, t, T, dt, J), hamiltonian(N, t, T, J)) for t in ts]\n",
    "results = estimator.run(circuit_observable_pairs).result()\n",
    "measured_energies = [results[i].data.evs for i in range(n_points+1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "99632779",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(ts, measured_energies)\n",
    "plt.xlabel(\"$t$\")\n",
    "plt.ylabel(r\"$\\langle \\psi_t | H(t) | \\psi_t \\rangle $\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f156a5fa",
   "metadata": {},
   "source": [
    "g) To see whether we actually stay in the instantaneous ground state of $H(t)$ compute the true ground state at all times $t$ using exact diagonalization. Plot the results on top of the figure above.\n",
    "\n",
    "Note that you can convert a `SparsePauliOp` to a sparse matrix format using the `to_matrix(sparse=True)` method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "1f4913ad",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.sparse.linalg import eigsh\n",
    "exact_energies = []\n",
    "for t in ts:\n",
    "    H = hamiltonian(N, t, T, J).to_matrix(sparse=True)\n",
    "    E_ed = np.sort(eigsh(H, k=4, which=\"SA\", return_eigenvectors=False))\n",
    "    exact_energies.append(E_ed[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "8de3db7c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(ts, measured_energies)\n",
    "plt.plot(ts, exact_energies, linestyle=\"--\")\n",
    "plt.xlabel(\"$t$\")\n",
    "plt.ylabel(r\"$\\langle \\psi_t | H(t) | \\psi_t \\rangle $\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f41934df",
   "metadata": {},
   "source": [
    "h) Why do the two curves deviate at longer times? What parameters can you change to follow the instantaneous ground state all the way to the final time $t=T$ more accurately? Try it out!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31d3b108",
   "metadata": {},
   "source": [
    "<details><summary>💡Hint</summary>\n",
    "\n",
    "Recall the adiabatic theorem and it's conditions.\n",
    "</details>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e4c2dac7",
   "metadata": {},
   "source": [
    "We stay in the instantaneous ground state only if we evolve slowly, otherwise we excite the system. So increase the total evolution time to, for example, $T=10$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a86d3f50",
   "metadata": {},
   "source": [
    "i) 🚀Advanced: Critical slowing down\n",
    "\n",
    "One can show that the total required evolution time $T$ of an adiabatic protocol has to be larger than the inverse squared energy gap between ground and first excited state at any point during the evolution:\n",
    "\n",
    "$$ T \\geq \\frac{C}{\\Delta^2}, \\qquad \\Delta = E_1 - E_0 ,$$\n",
    "\n",
    "where $C$ is some constant.\n",
    "\n",
    "Now imagine you want to prepare the ground state of the Ising model at the critical point $J/\\Gamma = 1$ for an infinitely-large spin chain using adiabatic evolution. How long would you have to evovle for? You can answer this question by pure reasoning, however, feel free to calculate (analytically or numerically) what happens to $T$ when you increase the number of spins $N$. The previous tutorial might help you in that case.\n",
    "\n",
    "Intuitively, why does adiabatic state preparation break down when it crosses a phase transition?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2e01786",
   "metadata": {},
   "source": [
    "The Ising model has a phase transition at $J/\\Gamma = 1$ which means that, in the thermodynamic limit, $\\Delta \\rightarrow 0$ exactly at the critical point. This means that the total evolution time $T\\rightarrow \\infty$ and thus, we would have to evolve infinitely slowly. This is commonly referred to as \"critical slowing down\". The closer you get to a phase transition (and the larger the system), the longer you need to evolve to stay in the instantaneous ground state."
   ]
  }
 ],
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   "language": "python",
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