{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "dc15734d-8749-49ae-be80-c3392e9699e1",
   "metadata": {
    "raw_mimetype": "text/restructuredtext"
   },
   "source": [
    "# Time Evolution\n",
    "\n",
    "Time evolutions in `quimb` are handled by the class {class}`~quimb.evo.Evolution`, which is initialized with a starting state and hamiltonian.\n",
    "\n",
    "## Basic Usage\n",
    "\n",
    "Set up the {class}`~quimb.evo.Evolution` object with a initial state and hamiltonian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "1330c133-a456-4273-b4e9-0f5edab0040d",
   "metadata": {},
   "outputs": [],
   "source": [
    "from quimb import *\n",
    "\n",
    "p0 = rand_ket(2**10)\n",
    "h = ham_heis(10, sparse=True)\n",
    "evo = Evolution(p0, h)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bad44ff5-2dcc-43e1-912c-aff549e44221",
   "metadata": {},
   "source": [
    "Update it in a single shot to a new time and get the state,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2edbe41e-48c8-4e69-b393-f3dbce606845",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[0.006074+0.011257j]\n",
       " [0.028495+0.016557j]\n",
       " [0.01617 -0.015705j]\n",
       " ...\n",
       " [0.018271+0.026071j]\n",
       " [0.018582-0.016482j]\n",
       " [0.027511+0.048377j]]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evo.update_to(1)\n",
    "evo.pt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5764132-0502-4650-a401-0c83f4cf23d6",
   "metadata": {},
   "source": [
    "Lazily generate the state at multiple times:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "2b138fbe-6ac1-4ceb-beb8-6b5c2e3e0184",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.003244693043705576\n",
      "0.008360765055266119\n",
      "0.009413166876677937\n"
     ]
    }
   ],
   "source": [
    "for pt in evo.at_times([2, 3, 4]):\n",
    "    print(expec(pt, p0))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c8ea54a-ac64-4afd-81ef-7e87608aaaef",
   "metadata": {
    "raw_mimetype": "text/restructuredtext"
   },
   "source": [
    "## Methods of Updating\n",
    "\n",
    "There are three methods of updating the state:\n",
    "\n",
    "> - `Evolution(..., method='integrate')`: use definite integration.\n",
    ">   Get system at each time step, only need action of Hamiltonian on\n",
    ">   state. Generally efficient. For pure and mixed states. The\n",
    ">   additional option `int_small_step={False, True}` determines\n",
    ">   whether a low or high order adaptive stepping scheme is used,\n",
    ">   giving naturally smaller or larger times steps. See\n",
    ">   {class}`scipy.integrate.ode` for details, `False` corresponds\n",
    ">   to `\"dop853\"`, `True` to `\"dopri5\"`.\n",
    "> - `Evolution(..., method='solve')`. Diagonalize the hamiltonian,\n",
    ">   which once done, allows quickly updating to arbitrary times.\n",
    ">   Supports pure and mixed states, recomended for small systems.\n",
    "> - `Evolution(..., method='expm')`: compute the evolved state\n",
    ">   using the action of the matrix exponential in a 'single shot'\n",
    ">   style. Only needs action of Hamiltonian, for very large systems\n",
    ">   can use distributed MPI. Only for pure states.\n",
    "\n",
    "## Computing on the fly\n",
    "\n",
    "Sometimes, if integrating, it is best to just query the state at time-steps chosen dynamically by the adaptive scheme. This is achieved using the `compute` keyword supplied to `Evolution`. It can also just be a convenient way to set up calculations as well:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "0dc964f7-5ac2-44ad-aaeb-320609078861",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|##########| 100/100 [00:00<00:00, 20144.58%/s]\n"
     ]
    }
   ],
   "source": [
    "p0 = rand_product_state(10)\n",
    "h = ham_heis(10, sparse=True)\n",
    "\n",
    "dims = [2] * 10\n",
    "sysa, sysb = (0, 1), (2, 3)\n",
    "\n",
    "def calc_t_and_logneg(t, pt):\n",
    "    ln = logneg_subsys(pt, dims, sysa, sysb)\n",
    "    return t, ln\n",
    "\n",
    "evo = Evolution(p0, h, compute=calc_t_and_logneg, progbar=True)\n",
    "evo.update_to(1)\n",
    "\n",
    "ts, lns = zip(*evo.results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "2387b525-54e8-4dca-ae42-8d5eca0b9916",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0,\n",
       " 0.2494153162899183,\n",
       " 0.4631400312076022,\n",
       " 0.6768647461252861,\n",
       " 0.9475557600771725,\n",
       " 1.0)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "0908d890-675e-449e-aa3c-a82c7d03aab8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0,\n",
       " 0.2589846231770502,\n",
       " 0.4423181326621942,\n",
       " 0.592697104829058,\n",
       " 0.7379099150663264,\n",
       " 0.7601643146969251)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lns"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2636ba38-c42b-4f56-b495-e968c8d8d464",
   "metadata": {
    "raw_mimetype": "text/restructuredtext"
   },
   "source": [
    "If a dict of callables is supplied to `compute`, (each should take two arguments, the time, and the state, as above), `Evolution.results` will itself be a dictionary containing the results of each function at each time step, under the respective key. This can be more convenient:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "55c0a8d7-ad7b-43ff-8b6d-ff60560001f6",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|##########| 100/100 [00:00<00:00, 18631.41%/s]\n"
     ]
    }
   ],
   "source": [
    "def calc_t(t, _):\n",
    "    return t\n",
    "\n",
    "def calc_logneg(_, pt):\n",
    "    return logneg_subsys(pt, [2]*10, 0, 1)\n",
    "\n",
    "evo = Evolution(p0, h, compute={'t': calc_t, 'ln': calc_logneg}, progbar=True)\n",
    "evo.update_to(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "972f485b-64c4-46e5-86b4-16cbd93d81a2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'t': [0.0,\n",
       "  0.2494153162899183,\n",
       "  0.4631400312076022,\n",
       "  0.6768647461252861,\n",
       "  0.9475557600771725,\n",
       "  1.0],\n",
       " 'ln': [0.0,\n",
       "  0.150158101589196,\n",
       "  0.23732495484813082,\n",
       "  0.2922696608578911,\n",
       "  0.3242059171688551,\n",
       "  0.3263498611123527]}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evo.results"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d904d5df-f633-4b96-98cd-234f326e98fa",
   "metadata": {
    "raw_mimetype": "text/restructuredtext"
   },
   "source": [
    "(time-dependent-evolution)=\n",
    "\n",
    "## Time-Dependent Evolutions\n",
    "\n",
    "If you are using `method='integrate'` you can supply a callable to `ham` to evolve the state with a time dependent Hamiltonian. It should take a single argument `t` and return the Hamiltonian at the time. It probably makes sense to use a custom class here to avoid reconstructing as much of the Hamiltonian as possible at each step.\n",
    "\n",
    "Here we'll evolve the Neel state:\n",
    "\n",
    "$$\n",
    "| \\psi(0) \\rangle  = | \\uparrow \\downarrow \\uparrow \\downarrow \\uparrow \\ldots \\rangle\n",
    "$$\n",
    "\n",
    "with the Hamiltonian:\n",
    "\n",
    "$$\n",
    "H(t) = \\sum_{i = 0}^{L - 1} S^Z_{i} S^Z_{i + 1} + \\cos(t) \\sum_{i}^{L} S^X_i\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "c64f70b8-a689-4fc9-ac49-eb90e2c5921c",
   "metadata": {},
   "outputs": [],
   "source": [
    "class MyTimeDepIsingHam:\n",
    "\n",
    "    def __init__(self, L):\n",
    "        self.h_interaction = ham_ising(L, sparse=True, jz=1.0, bx=0.0, cyclic=False)\n",
    "        self.h_field = ham_ising(L, sparse=True, jz=0.0, bx=1.0, cyclic=False)\n",
    "\n",
    "    def __call__(self, t):\n",
    "        return self.h_interaction + cos(t) * self.h_field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "1480b979-aa44-4726-b01d-830f767613e4",
   "metadata": {},
   "outputs": [],
   "source": [
    "L = 16\n",
    "\n",
    "# our initial state\n",
    "psi0 = neel_state(L)\n",
    "\n",
    "# instantiate the ham object, it's __call__ method will be used by Evolution\n",
    "fn_ham_t = MyTimeDepIsingHam(L)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a86b6c6-8a02-44b0-ac25-85f074af69c5",
   "metadata": {},
   "source": [
    "We still want to compute some properties during the evolution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "67a204ea-4f14-435d-b68d-746756ad77e1",
   "metadata": {},
   "outputs": [],
   "source": [
    "compute = {\n",
    "    'time': lambda t, p: t,\n",
    "    'entropy': lambda t, p: entropy_subsys(p, dims=[2] * L, sysa=range(L // 2))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "256af3bb-af91-4139-bfa9-68f6703403c9",
   "metadata": {},
   "source": [
    "Now we set up the evolution object again:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "e2608200-f85e-461f-b347-116b8b464d24",
   "metadata": {},
   "outputs": [],
   "source": [
    "evo = Evolution(psi0, fn_ham_t, progbar=True, compute=compute)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "d26978f4-1fb2-4f8a-b0d6-1338fff320f6",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|##########| 100/100 [00:07<00:00, 12.53%/s]\n"
     ]
    }
   ],
   "source": [
    "evo.update_to(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4735d921-676f-4db6-8dc8-573b42f44f3f",
   "metadata": {},
   "source": [
    "We can plot the half chain entropy that we computed on the fly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "e4d8cb4e-16ab-47dd-8a01-e0c91836b5dc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "with plt.style.context(NEUTRAL_STYLE):\n",
    "    plt.plot(evo.results['time'], evo.results['entropy'])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "59aa8c6a-e5ee-4720-a132-f54966e3a9d7",
   "metadata": {},
   "source": [
    "Or we can use the final state:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "81f8058b-7128-42f5-9e70-56699dc6ff32",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.003302180752068547"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fidelity(psi0, evo.pt)"
   ]
  }
 ],
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   "language": "python",
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    "name": "ipython",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3"
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  "vscode": {
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