{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3d1fdfd5",
   "metadata": {},
   "source": [
    "(operator-basics)=\n",
    "# Basics\n",
    "\n",
    "The [`quimb.operator`](quimb.operator) module provides a unified way of 'symbolically' defining Hamiltonians or operators, including in symmetry sub-sectors, which can then be built into multiple different concrete representations for consumption by either the matrix or tensor network routines in quimb.\n",
    "\n",
    "Matrix representations:\n",
    "\n",
    "- *Dense matrix*\n",
    "- *Sparse matrix*\n",
    "- *Linear operator* (action on a dense vector)\n",
    "\n",
    "Tensor network representations:\n",
    "\n",
    "- *Matrix product operator (MPO)*\n",
    "- *Dict of local dense terms* (for PEPS algorithms)\n",
    "- *Projected entangled pair operator (PEPO)*\n",
    "\n",
    "VMC routines:\n",
    "\n",
    "- *Coupled configurations*\n",
    "\n",
    "There is also support for certain transformations of the terms, such as the [*Jordan-Wigner* transformation](quimb.operator.SparseOperatorBuilder.jordan_wigner_transform) and [pauli decomposition](quimb.operator.SparseOperatorBuilder.pauli_decompose).\n",
    "\n",
    "Here we'll illustrate with a 2D Heisenberg model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "633065b0",
   "metadata": {},
   "outputs": [],
   "source": [
    "%config InlineBackend.figure_formats = ['svg']\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "import quimb as qu\n",
    "import quimb.operator as qop\n",
    "import quimb.tensor as qtn\n",
    "\n",
    "mpl.style.use(qu.NEUTRAL_STYLE)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ad9cb975",
   "metadata": {},
   "source": [
    "## The `HilbertSpace`\n",
    "\n",
    "For full control over the sites in our system and how to order them we first explicitly declare a [`HilbertSpace`](quimb.operator.HilbertSpace) object, here for a 2D square lattice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "e7680077",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "HilbertSpace(nsites=20, total_size=184_756, symmetry=U1, sector=10)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Lx = 4\n",
    "Ly = 5\n",
    "sites = [(i, j) for i in range(Lx) for j in range(Ly)]\n",
    "nsites = len(sites)\n",
    "\n",
    "# define a default symmetry sector for efficiency\n",
    "symmetry = \"U1\"\n",
    "sector = nsites // 2\n",
    "\n",
    "hilbert_space = qop.HilbertSpace(\n",
    "    sites=sites,\n",
    "    symmetry=symmetry,\n",
    "    sector=sector,\n",
    ")\n",
    "hilbert_space"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "17aa23a5",
   "metadata": {},
   "source": [
    "On its own this just has a few helpful tools for enumerating the valid configurations. The terminology used here is:\n",
    "\n",
    "- `site`: the labels of the individual sites in the system, these can be arbtirary hashable and sortable objects, e.g. our 2D coordinates above.\n",
    "- `reg`: in a linear ordering of the sites, the `reg` is the integer index or 'register' of a site. This can be controlled using the `order` argument to the `HilbertSpace` constructor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1fd3f5c4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hilbert_space.site_to_reg((2, 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "21b42b0b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 4)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hilbert_space.reg_to_site(nsites - 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d24804f3",
   "metadata": {},
   "source": [
    "Individual configurations can be referred to in three ways:\n",
    "\n",
    "- `config`: a dict of site to local state.\n",
    "- `flatconfig`: a `numpy.ndarray[uint8]` of local states in register order.\n",
    "- `rank`: an integer between 0 and `hilbert_space.size - 1` indexing the configurations in lexicographically sorted order."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "87f01213",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{(0, 0): np.uint8(0), (0, 1): np.uint8(0), (0, 2): np.uint8(0), (0, 3): np.uint8(0), (0, 4): np.uint8(0), (1, 0): np.uint8(0), (1, 1): np.uint8(0), (1, 2): np.uint8(0), (1, 3): np.uint8(0), (1, 4): np.uint8(0), (2, 0): np.uint8(1), (2, 1): np.uint8(1), (2, 2): np.uint8(1), (2, 3): np.uint8(1), (2, 4): np.uint8(1), (3, 0): np.uint8(1), (3, 1): np.uint8(1), (3, 2): np.uint8(1), (3, 3): np.uint8(1), (3, 4): np.uint8(1)}\n"
     ]
    }
   ],
   "source": [
    "# the first configuration (00..011..1) as a dict:\n",
    "print(hilbert_space.rank_to_config(0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "93a4a737",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{(0, 0): np.uint8(0), (0, 1): np.uint8(0), (0, 2): np.uint8(0), (0, 3): np.uint8(1), (0, 4): np.uint8(1), (1, 0): np.uint8(1), (1, 1): np.uint8(0), (1, 2): np.uint8(0), (1, 3): np.uint8(1), (1, 4): np.uint8(0), (2, 0): np.uint8(1), (2, 1): np.uint8(1), (2, 2): np.uint8(0), (2, 3): np.uint8(0), (2, 4): np.uint8(0), (3, 0): np.uint8(1), (3, 1): np.uint8(1), (3, 2): np.uint8(1), (3, 3): np.uint8(0), (3, 4): np.uint8(1)}\n"
     ]
    }
   ],
   "source": [
    "# get a random configuration\n",
    "config = hilbert_space.rand_config(seed=42)\n",
    "print(config)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "91f0effe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16489"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# find its rank\n",
    "hilbert_space.config_to_rank(config)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e33967d0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n",
       "      dtype=uint8)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the last configuration (11..100..0) as a flat vector\n",
    "hilbert_space.rank_to_flatconfig(hilbert_space.size - 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f831873a",
   "metadata": {},
   "source": [
    "## The `SparseOperatorBuilder`\n",
    "\n",
    "Now we can 'symbolically' define operators for this Hilbert space using the [`SparseOperatorBuilder`](quimb.operator.SparseOperatorBuilder) object. Here we'll define a 2D Heisenberg Hamiltonian:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "91a7ff5e",
   "metadata": {},
   "outputs": [],
   "source": [
    "H = qop.SparseOperatorBuilder(hilbert_space=hilbert_space)\n",
    "\n",
    "# add nearest neighbor Heisenberg interactions\n",
    "for i in range(Lx):\n",
    "    for j in range(Ly):\n",
    "        if i < Lx - 1:\n",
    "            H += 0.5, (\"+\", (i, j)), (\"-\", (i + 1, j))\n",
    "            H += 0.5, (\"-\", (i, j)), (\"+\", (i + 1, j))\n",
    "            H += 1.0, (\"z\", (i, j)), (\"z\", (i + 1, j))\n",
    "        if j < Ly - 1:\n",
    "            H += 0.5, (\"+\", (i, j)), (\"-\", (i, j + 1))\n",
    "            H += 0.5, (\"-\", (i, j)), (\"+\", (i, j + 1))\n",
    "            H += 1.0, (\"z\", (i, j)), (\"z\", (i, j + 1))\n",
    "\n",
    "# and a single site magnetic field to break symmetry\n",
    "H += -0.1, (\"z\", (0, 0))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72c86afd",
   "metadata": {},
   "source": [
    "```{note}\n",
    "If you don't supply a `HilbertSpace` then the operator builder will build the default one for the minimal set of sites in has seen so far.\n",
    "```\n",
    "\n",
    "We can see all the terms visually with the `show` method (again the ordering is given by the register order):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "de1a90f5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SparseOperatorBuilder(nsites=20, nterms=94, locality=2)\n",
      "+ . . . . - . . . . . . . . . . . . . .  +0.5\n",
      "- . . . . + . . . . . . . . . . . . . .  +0.5\n",
      "z . . . . z . . . . . . . . . . . . . .  +1.0\n",
      "+ - . . . . . . . . . . . . . . . . . .  +0.5\n",
      "- + . . . . . . . . . . . . . . . . . .  +0.5\n",
      "z z . . . . . . . . . . . . . . . . . .  +1.0\n",
      ". + . . . . - . . . . . . . . . . . . .  +0.5\n",
      ". - . . . . + . . . . . . . . . . . . .  +0.5\n",
      ". z . . . . z . . . . . . . . . . . . .  +1.0\n",
      ". + - . . . . . . . . . . . . . . . . .  +0.5\n",
      ". - + . . . . . . . . . . . . . . . . .  +0.5\n",
      ". z z . . . . . . . . . . . . . . . . .  +1.0\n",
      ". . + . . . . - . . . . . . . . . . . .  +0.5\n",
      ". . - . . . . + . . . . . . . . . . . .  +0.5\n",
      ". . z . . . . z . . . . . . . . . . . .  +1.0\n",
      ". . + - . . . . . . . . . . . . . . . .  +0.5\n",
      ". . - + . . . . . . . . . . . . . . . .  +0.5\n",
      ". . z z . . . . . . . . . . . . . . . .  +1.0\n",
      ". . . + . . . . - . . . . . . . . . . .  +0.5\n",
      ". . . - . . . . + . . . . . . . . . . .  +0.5\n",
      ". . . z . . . . z . . . . . . . . . . .  +1.0\n",
      ". . . + - . . . . . . . . . . . . . . .  +0.5\n",
      ". . . - + . . . . . . . . . . . . . . .  +0.5\n",
      ". . . z z . . . . . . . . . . . . . . .  +1.0\n",
      ". . . . + . . . . - . . . . . . . . . .  +0.5\n",
      ". . . . - . . . . + . . . . . . . . . .  +0.5\n",
      ". . . . z . . . . z . . . . . . . . . .  +1.0\n",
      ". . . . . + . . . . - . . . . . . . . .  +0.5\n",
      ". . . . . - . . . . + . . . . . . . . .  +0.5\n",
      ". . . . . z . . . . z . . . . . . . . .  +1.0\n",
      ". . . . . + - . . . . . . . . . . . . .  +0.5\n",
      ". . . . . - + . . . . . . . . . . . . .  +0.5\n",
      ". . . . . z z . . . . . . . . . . . . .  +1.0\n",
      ". . . . . . + . . . . - . . . . . . . .  +0.5\n",
      ". . . . . . - . . . . + . . . . . . . .  +0.5\n",
      ". . . . . . z . . . . z . . . . . . . .  +1.0\n",
      ". . . . . . + - . . . . . . . . . . . .  +0.5\n",
      ". . . . . . - + . . . . . . . . . . . .  +0.5\n",
      ". . . . . . z z . . . . . . . . . . . .  +1.0\n",
      ". . . . . . . + . . . . - . . . . . . .  +0.5\n",
      ". . . . . . . - . . . . + . . . . . . .  +0.5\n",
      ". . . . . . . z . . . . z . . . . . . .  +1.0\n",
      ". . . . . . . + - . . . . . . . . . . .  +0.5\n",
      ". . . . . . . - + . . . . . . . . . . .  +0.5\n",
      ". . . . . . . z z . . . . . . . . . . .  +1.0\n",
      ". . . . . . . . + . . . . - . . . . . .  +0.5\n",
      ". . . . . . . . - . . . . + . . . . . .  +0.5\n",
      ". . . . . . . . z . . . . z . . . . . .  +1.0\n",
      ". . . . . . . . + - . . . . . . . . . .  +0.5\n",
      ". . . . . . . . - + . . . . . . . . . .  +0.5\n",
      ". . . . . . . . z z . . . . . . . . . .  +1.0\n",
      ". . . . . . . . . + . . . . - . . . . .  +0.5\n",
      ". . . . . . . . . - . . . . + . . . . .  +0.5\n",
      ". . . . . . . . . z . . . . z . . . . .  +1.0\n",
      ". . . . . . . . . . + . . . . - . . . .  +0.5\n",
      ". . . . . . . . . . - . . . . + . . . .  +0.5\n",
      ". . . . . . . . . . z . . . . z . . . .  +1.0\n",
      ". . . . . . . . . . + - . . . . . . . .  +0.5\n",
      ". . . . . . . . . . - + . . . . . . . .  +0.5\n",
      ". . . . . . . . . . z z . . . . . . . .  +1.0\n",
      ". . . . . . . . . . . + . . . . - . . .  +0.5\n",
      ". . . . . . . . . . . - . . . . + . . .  +0.5\n",
      ". . . . . . . . . . . z . . . . z . . .  +1.0\n",
      ". . . . . . . . . . . + - . . . . . . .  +0.5\n",
      ". . . . . . . . . . . - + . . . . . . .  +0.5\n",
      ". . . . . . . . . . . z z . . . . . . .  +1.0\n",
      ". . . . . . . . . . . . + . . . . - . .  +0.5\n",
      ". . . . . . . . . . . . - . . . . + . .  +0.5\n",
      ". . . . . . . . . . . . z . . . . z . .  +1.0\n",
      ". . . . . . . . . . . . + - . . . . . .  +0.5\n",
      ". . . . . . . . . . . . - + . . . . . .  +0.5\n",
      ". . . . . . . . . . . . z z . . . . . .  +1.0\n",
      ". . . . . . . . . . . . . + . . . . - .  +0.5\n",
      ". . . . . . . . . . . . . - . . . . + .  +0.5\n",
      ". . . . . . . . . . . . . z . . . . z .  +1.0\n",
      ". . . . . . . . . . . . . + - . . . . .  +0.5\n",
      ". . . . . . . . . . . . . - + . . . . .  +0.5\n",
      ". . . . . . . . . . . . . z z . . . . .  +1.0\n",
      ". . . . . . . . . . . . . . + . . . . -  +0.5\n",
      ". . . . . . . . . . . . . . - . . . . +  +0.5\n",
      ". . . . . . . . . . . . . . z . . . . z  +1.0\n",
      ". . . . . . . . . . . . . . . + - . . .  +0.5\n",
      ". . . . . . . . . . . . . . . - + . . .  +0.5\n",
      ". . . . . . . . . . . . . . . z z . . .  +1.0\n",
      ". . . . . . . . . . . . . . . . + - . .  +0.5\n",
      ". . . . . . . . . . . . . . . . - + . .  +0.5\n",
      ". . . . . . . . . . . . . . . . z z . .  +1.0\n",
      ". . . . . . . . . . . . . . . . . + - .  +0.5\n",
      ". . . . . . . . . . . . . . . . . - + .  +0.5\n",
      ". . . . . . . . . . . . . . . . . z z .  +1.0\n",
      ". . . . . . . . . . . . . . . . . . + -  +0.5\n",
      ". . . . . . . . . . . . . . . . . . - +  +0.5\n",
      ". . . . . . . . . . . . . . . . . . z z  +1.0\n",
      "z . . . . . . . . . . . . . . . . . . .  -0.1\n"
     ]
    }
   ],
   "source": [
    "H.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69225d21",
   "metadata": {},
   "source": [
    "Currently the supported local operators are:\n",
    "```python\n",
    "{\n",
    "    \"I\": {0: (0, 1.0), 1: (1, 1.0)},\n",
    "    # pauli matrices\n",
    "    \"x\": {0: (1, 1.0), 1: (0, 1.0)},\n",
    "    \"y\": {0: (1, 1.0j), 1: (0, -1.0j)},\n",
    "    \"z\": {0: (0, 1.0), 1: (1, -1.0)},\n",
    "    # ZX=iY: 'real Y'\n",
    "    \"⧖\": {0: (1, -1.0), 1: (0, 1.0)},\n",
    "    # spin 1/2 matrices (scaled paulis)\n",
    "    \"sx\": {0: (1, 0.5), 1: (0, 0.5)},\n",
    "    \"sy\": {0: (1, 0.5j), 1: (0, -0.5j)},\n",
    "    \"sz\": {0: (0, 0.5), 1: (1, -0.5)},\n",
    "    # creation / annihilation operators\n",
    "    \"+\": {0: (1, 1.0)},\n",
    "    \"-\": {1: (0, 1.0)},\n",
    "    # number, symmetric number, and hole operators\n",
    "    \"n\": {1: (1, 1.0)},\n",
    "    \"sn\": {0: (0, -0.5), 1: (1, 0.5)},\n",
    "    \"h\": {0: (0, 1.0)},\n",
    "}\n",
    "```\n",
    "\n",
    "### Building matrices\n",
    "\n",
    "Now we can build out concrete realizations of the operator. For example the sparse matrix representation:\n",
    "\n",
    "```{hint}\n",
    "By default this will pick up the default `sector` and `symmetry` from the hilbert space object, or you can explicitly override it here.\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "e479c67f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 992 ms, sys: 180 ms, total: 1.17 s\n",
      "Wall time: 1.18 s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<Compressed Sparse Row sparse matrix of dtype 'float64'\n",
       "\twith 3199196 stored elements and shape (184756, 184756)>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%%time\n",
    "H_sparse = H.build_sparse_matrix()\n",
    "H_sparse"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0531fd27",
   "metadata": {},
   "source": [
    "This allows us to find the full exact groundstate and energy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "c089e9f5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 5.01 s, sys: 82.5 ms, total: 5.09 s\n",
      "Wall time: 694 ms\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "energy, psi = qu.eigh(H_sparse, k=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "548e9b17",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-32.01392954])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "energy"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d27a50da",
   "metadata": {},
   "source": [
    "Now we can define some other operators that act on the same hilbert space, and measure their expectation, here the Z magnetization acting on every site:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "129284b7",
   "metadata": {},
   "outputs": [],
   "source": [
    "zijs = [[None for j in range(Ly)] for i in range(Lx)]\n",
    "\n",
    "for i in range(Lx):\n",
    "    for j in range(Ly):\n",
    "        zop_ij = qop.SparseOperatorBuilder(\n",
    "            # we can supply all the terms at construction\n",
    "            terms=[(1.0, (\"z\", (i, j)))],\n",
    "            # because we only act on 1 site of many, we do\n",
    "            # need to explicitly provide the hilbert space\n",
    "            hilbert_space=hilbert_space,\n",
    "        )\n",
    "\n",
    "        # build the sparse matrix\n",
    "        zsparse_ij = zop_ij.build_sparse_matrix()\n",
    "\n",
    "        # evaluate <psi|Z_ij|psi>\n",
    "        zijs[i][j] = qu.expec(psi, zsparse_ij)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "1744f587",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x731f57a8dfd0>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(zijs)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc3dc240",
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   "source": [
    "If this was a smaller space we could also build the full dense matrix with\n",
    "[`build_dense_matrix`](quimb.operator.SparseOperatorBuilder.build_dense_matrix).\n",
    "\n",
    "The [`SparseOperatorBuilder`](quimb.operator.SparseOperatorBuilder) also has a\n",
    "[`matvec`](quimb.operator.SparseOperatorBuilder.matvec) method which computes the action on a dense vector without forming the explicit matrix. You can use this directly or call [`.aslinearoperator`](quimb.operator.SparseOperatorBuilder.aslinearoperator) to get a [`scipy.sparse.linalg.LinearOperator`](scipy.sparse.linalg.LinearOperator) object which can be used with [`scipy.sparse.linalg`](scipy.sparse.linalg) routines.\n",
    "\n",
    "```{warning}\n",
    "This is not a very efficient implementation yet, you may want to consider for example [`quspin`](https://quspin.github.io/QuSpin/) if performance is critical.\n",
    "```\n",
    "\n",
    "### Building tensor network representations\n",
    "\n",
    "#### Building MPOs for MPS algorithms\n",
    "\n",
    "We can also build out concrete representations of the operator for consumption by various tensor network algorithms in [`quimb.tensor`](quimb.tensor).\n",
    "\n",
    "For example as a matrix product operator (MPO) using [`build_mpo`](quimb.operator.SparseOperatorBuilder.build_mpo):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "5d26ba8a",
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   "outputs": [
    {
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       "        [ 0.,  1.]],\n",
       "\n",
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       "        [ 0.,  0.]],\n",
       "\n",
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      ],
      "text/plain": [
       "MatrixProductOperator(tensors=20, indices=59, L=20, max_bond=17)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H_mpo = H.build_mpo()\n",
    "H_mpo"
   ]
  },
  {
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   "metadata": {},
   "source": [
    "The algorithm to build the MPO is a greedy algorithm that works well for local Hamiltonians, but will not be optimal for complicated long range models. You can check the underyling finite state machine representatino with [`draw_state_machine`](quimb.operator.SparseOperatorBuilder.draw_state_machine):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "e39f002b",
   "metadata": {},
   "outputs": [
    {
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      ],
      "text/plain": [
       "<Figure size 2000x850 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "H.draw_state_machine()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69108cf9",
   "metadata": {},
   "source": [
    "We can use the MPO to find the groundstate with [`DMRG`](quimb.tensor.tn1d.dmrg.DMRG):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "90ba887c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1, R, max_bond=(8/8), cutoff:1e-08\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|##########################################| 19/19 [00:00<00:00, 306.08it/s]"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Energy: -31.656832428772546 ... not converged.\n",
      "2, R, max_bond=(8/16), cutoff:1e-08\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "100%|##########################################| 19/19 [00:00<00:00, 362.23it/s]"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Energy: -31.813407135272556 ... not converged.\n",
      "3, R, max_bond=(16/32), cutoff:1e-08\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "100%|##########################################| 19/19 [00:00<00:00, 279.30it/s]"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Energy: -31.818448856710024 ... not converged.\n",
      "4, R, max_bond=(29/64), cutoff:1e-08\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "100%|##########################################| 19/19 [00:00<00:00, 192.93it/s]"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Energy: -31.81932769634487 ... not converged.\n",
      "5, R, max_bond=(39/128), cutoff:1e-08\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "100%|##########################################| 19/19 [00:00<00:00, 152.65it/s]"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Energy: -31.81949636476171 ... not converged.\n",
      "6, R, max_bond=(42/256), cutoff:1e-08\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n",
      "100%|##########################################| 19/19 [00:00<00:00, 210.71it/s]"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Energy: -31.819509858329468 ... converged!\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "np.True_"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dmrg = qtn.DMRG2(H_mpo)\n",
    "dmrg.solve(verbosity=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5876a193",
   "metadata": {},
   "source": [
    "Then we can check the relative error in the groundstate energy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "963d2dc6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.006072971495561852)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - dmrg.energy / energy.item()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5bab5a3e",
   "metadata": {},
   "source": [
    "#### Building local terms for PEPS\n",
    "\n",
    "For projected entangled pair states (PEPS) algorithms we often need the local terms of the Hamiltonian as dense matrices. We can build these with [`build_local_ham`](quimb.operator.SparseOperatorBuilder.build_local_ham):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "a69e2370",
   "metadata": {},
   "outputs": [
    {
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      ],
      "text/plain": [
       "<Figure size 547.214x547.214 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "(<Figure size 547.214x547.214 with 1 Axes>, <Axes: >)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H_local = H.build_local_ham()\n",
    "H_local.draw()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "31439a8f",
   "metadata": {},
   "source": [
    "We can then feed this into algorithms such as [`simple update`](quimb.tensor.tnag.tebd.SimpleUpdateGen):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "96952e00",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "n=100, D=4, tau=0.1, energy≈-31.9678: 100%|##########| 100/100 [00:02<00:00, 35.70it/s]\n"
     ]
    }
   ],
   "source": [
    "peps = qtn.PEPS.rand(Lx, Ly, 4, seed=42)\n",
    "su = qtn.SimpleUpdate(peps, H_local)\n",
    "su.evolve(100, tau=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "24b9e53c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0014423055073261581"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - su.energy / energy.item()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "121566f2",
   "metadata": {},
   "source": [
    "### Monte carlo usage"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85f30db1",
   "metadata": {},
   "source": [
    "For variational monte carlo (VMC) or other monte carlo routines one needs to the compute the 'local energy'. This is encapsulated in the [`config_coupling`](quimb.operator.SparseOperatorBuilder.config_coupling) and [`flatconfig_coupling`](quimb.operator.SparseOperatorBuilder.flatconfig_coupling) methods. These return a tuple with A) the coupled configurations and B) the corresponding coefficients.\n",
    "\n",
    "Mathematically, for a given configuration $| x \\rangle$, this returns the non-zero coupled configurations: $\\{| y \\rangle\\} $, as well as the coefficients $\\{ \\langle x | H | y \\rangle \\}$ such that the 'local energy' is given by:\n",
    "\n",
    "$$\n",
    "E_{\\text{loc}}(x) = \\sum_j \\dfrac{ \\langle x | H | y \\rangle \\langle y | \\psi \\rangle } { \\langle x | \\psi \\rangle }\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "04e72917",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1],\n",
       "      dtype=uint8)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "flatconfig = hilbert_space.rand_flatconfig(seed=42)\n",
    "flatconfig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "f74fff2d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1],\n",
       "        [0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1],\n",
       "        [0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0],\n",
       "        [0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1],\n",
       "        [0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0]],\n",
       "       dtype=uint8),\n",
       " array([0.5, 4.9, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,\n",
       "        0.5]))"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coupled_flatconfigs, coupled_coeffs = H.flatconfig_coupling(flatconfig)\n",
    "coupled_flatconfigs, coupled_coeffs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "be825435",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x731f54488050>"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(coupled_flatconfigs)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e3512021",
   "metadata": {},
   "source": [
    "### Transforming operators\n",
    "\n",
    "The [`SparseOperatorBuilder`](quimb.operator.SparseOperatorBuilder) also has some methods for transforming the terms.\n",
    "\n",
    "#### Jordan-Wigner transformation\n",
    "\n",
    "For example the [*Jordan-Wigner* transformation](quimb.operator.SparseOperatorBuilder.jordan_wigner_transform) which maps spin-1/2 operators to fermionic ones:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "bb1e5075",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "HilbertSpace(nsites=24, total_size=853_776, symmetry=U1U1, sector=((12, 6), (12, 6)))"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Lx = 4\n",
    "Ly = 3\n",
    "sites = [(s, i, j) for s in (\"u\", \"d\") for i in range(Lx) for j in range(Ly)]\n",
    "nsites = len(sites)\n",
    "\n",
    "# define half filling in each species\n",
    "symmetry = \"U1U1\"\n",
    "sector = (nsites // 2, nsites // 4), (nsites // 2, nsites // 4)\n",
    "\n",
    "hilbert_space = qop.HilbertSpace(\n",
    "    sites=sites,\n",
    "    symmetry=symmetry,\n",
    "    sector=sector,\n",
    ")\n",
    "hilbert_space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "ba9afbdf",
   "metadata": {},
   "outputs": [],
   "source": [
    "H = qop.SparseOperatorBuilder(hilbert_space=hilbert_space)\n",
    "\n",
    "t = 1.0\n",
    "U = 8.0\n",
    "\n",
    "# add nearest neighbor Heisenberg interactions\n",
    "for i in range(Lx):\n",
    "    for j in range(Ly):\n",
    "        for s in (\"u\", \"d\"):\n",
    "            # hopping\n",
    "            if i < Lx - 1:\n",
    "                # the relative ordering within the term now matters too\n",
    "                H.add_term(-t, (\"+\", (s, i, j)), (\"-\", (s, i + 1, j)))\n",
    "                H.add_term(-t, (\"+\", (s, i + 1, j)), (\"-\", (s, i, j)))\n",
    "            if j < Ly - 1:\n",
    "                H.add_term(-t, (\"+\", (s, i, j)), (\"-\", (s, i, j + 1)))\n",
    "                H.add_term(-t, (\"+\", (s, i, j + 1)), (\"-\", (s, i, j)))\n",
    "\n",
    "        # interaction\n",
    "        H.add_term(U, (\"n\", (\"u\", i, j)), (\"n\", (\"d\", i, j)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9029ce5c",
   "metadata": {},
   "source": [
    "```{note}\n",
    "You can either call the method\n",
    "[`jordan_wigner_transform`](quimb.operator.SparseOperatorBuilder.jordan_wigner_transform)\n",
    "to toggle the transformation, or supply `jordan_wigner=True` to the\n",
    "[`SparseOperatorBuilder`](quimb.operator.SparseOperatorBuilder) constructor.\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "81b674c6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SparseOperatorBuilder(nsites=24, nterms=80, locality=4, jordan_wigner=True)\n",
      "+ z z - . . . . . . . . . . . . . . . . . . . .  -1.0\n",
      "- z z + . . . . . . . . . . . . . . . . . . . .  -1.0\n",
      "+ - . . . . . . . . . . . . . . . . . . . . . .  -1.0\n",
      "- + . . . . . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . + z z - . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . - z z + . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . + - . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . - + . . . . . . . . . .  -1.0\n",
      "n . . . . . . . . . . . n . . . . . . . . . . .  +8.0\n",
      ". + z z - . . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". - z z + . . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". + - . . . . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". - + . . . . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . + z z - . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . - z z + . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . + - . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . - + . . . . . . . . .  -1.0\n",
      ". n . . . . . . . . . . . n . . . . . . . . . .  +8.0\n",
      ". . + z z - . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . - z z + . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . + z z - . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . - z z + . . . . . .  -1.0\n",
      ". . n . . . . . . . . . . . n . . . . . . . . .  +8.0\n",
      ". . . + z z - . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . - z z + . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . + - . . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . - + . . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . + z z - . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . - z z + . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . + - . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . - + . . . . . . .  -1.0\n",
      ". . . n . . . . . . . . . . . n . . . . . . . .  +8.0\n",
      ". . . . + z z - . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . - z z + . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . + - . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . - + . . . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . + z z - . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . - z z + . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . + - . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . - + . . . . . .  -1.0\n",
      ". . . . n . . . . . . . . . . . n . . . . . . .  +8.0\n",
      ". . . . . + z z - . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . - z z + . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . + z z - . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . - z z + . . .  -1.0\n",
      ". . . . . n . . . . . . . . . . . n . . . . . .  +8.0\n",
      ". . . . . . + z z - . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . - z z + . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . + - . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . - + . . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . + z z - . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . - z z + . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . + - . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . - + . . . .  -1.0\n",
      ". . . . . . n . . . . . . . . . . . n . . . . .  +8.0\n",
      ". . . . . . . + z z - . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . - z z + . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . + - . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . - + . . . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . + z z - .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . - z z + .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . + - . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . - + . . .  -1.0\n",
      ". . . . . . . n . . . . . . . . . . . n . . . .  +8.0\n",
      ". . . . . . . . + z z - . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . - z z + . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . . + z z -  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . . - z z +  -1.0\n",
      ". . . . . . . . n . . . . . . . . . . . n . . .  +8.0\n",
      ". . . . . . . . . + - . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . - + . . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . . . + - .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . . . - + .  -1.0\n",
      ". . . . . . . . . n . . . . . . . . . . . n . .  +8.0\n",
      ". . . . . . . . . . + - . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . - + . . . . . . . . . . . .  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . . . . + -  -1.0\n",
      ". . . . . . . . . . . . . . . . . . . . . . - +  -1.0\n",
      ". . . . . . . . . . n . . . . . . . . . . . n .  +8.0\n",
      ". . . . . . . . . . . n . . . . . . . . . . . n  +8.0\n"
     ]
    }
   ],
   "source": [
    "H.jordan_wigner_transform()\n",
    "H.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "25508d06",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Compressed Sparse Row sparse matrix of dtype 'float64'\n",
       "\twith 16686516 stored elements and shape (853776, 853776)>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "H_sparse = H.build_sparse_matrix()\n",
    "H_sparse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "f07317fd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.40943827])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "energy, psi = qu.eigh(H_sparse, k=1)\n",
    "energy / (Lx * Ly)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d81a5951",
   "metadata": {},
   "source": [
    "#### Pauli Decomposition\n",
    "\n",
    "You can also decompose an operator into strings of Pauli operators using the\n",
    "[`pauli_decompose`](quimb.operator.SparseOperatorBuilder.pauli_decompose) method,\n",
    "or by supplying `pauli_decompose=True` to the\n",
    "[`SparseOperatorBuilder`](quimb.operator.SparseOperatorBuilder) constructor.\n",
    "The `use_zx` argument enables decomposing Y operators into the real combination of\n",
    "Z and X (i.e. ⴵ = ZX = iY) to avoid complex coefficients:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "7e18b8de",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SparseOperatorBuilder(nsites=24, nterms=105, locality=4, jordan_wigner=True, pauli_decompose=zx)\n",
      "x z z x . . . . . . . . . . . . . . . . . . . .  -0.5\n",
      "ⴵ z z ⴵ . . . . . . . . . . . . . . . . . . . .  +0.5\n",
      "x x . . . . . . . . . . . . . . . . . . . . . .  -0.5\n",
      "ⴵ ⴵ . . . . . . . . . . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . x z z x . . . . . . . .  -0.5\n",
      ". . . . . . . . . . . . ⴵ z z ⴵ . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . x x . . . . . . . . . .  -0.5\n",
      ". . . . . . . . . . . . ⴵ ⴵ . . . . . . . . . .  +0.5\n",
      "z . . . . . . . . . . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . z . . . . . . . . . . .  -2.0\n",
      "z . . . . . . . . . . . z . . . . . . . . . . .  +2.0\n",
      ". x z z x . . . . . . . . . . . . . . . . . . .  -0.5\n",
      ". ⴵ z z ⴵ . . . . . . . . . . . . . . . . . . .  +0.5\n",
      ". x x . . . . . . . . . . . . . . . . . . . . .  -0.5\n",
      ". ⴵ ⴵ . . . . . . . . . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . x z z x . . . . . . .  -0.5\n",
      ". . . . . . . . . . . . . ⴵ z z ⴵ . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . x x . . . . . . . . .  -0.5\n",
      ". . . . . . . . . . . . . ⴵ ⴵ . . . . . . . . .  +0.5\n",
      ". z . . . . . . . . . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . z . . . . . . . . . .  -2.0\n",
      ". z . . . . . . . . . . . z . . . . . . . . . .  +2.0\n",
      ". . x z z x . . . . . . . . . . . . . . . . . .  -0.5\n",
      ". . ⴵ z z ⴵ . . . . . . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . x z z x . . . . . .  -0.5\n",
      ". . . . . . . . . . . . . . ⴵ z z ⴵ . . . . . .  +0.5\n",
      ". . z . . . . . . . . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . z . . . . . . . . .  -2.0\n",
      ". . z . . . . . . . . . . . z . . . . . . . . .  +2.0\n",
      ". . . x z z x . . . . . . . . . . . . . . . . .  -0.5\n",
      ". . . ⴵ z z ⴵ . . . . . . . . . . . . . . . . .  +0.5\n",
      ". . . x x . . . . . . . . . . . . . . . . . . .  -0.5\n",
      ". . . ⴵ ⴵ . . . . . . . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . x z z x . . . . .  -0.5\n",
      ". . . . . . . . . . . . . . . ⴵ z z ⴵ . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . x x . . . . . . .  -0.5\n",
      ". . . . . . . . . . . . . . . ⴵ ⴵ . . . . . . .  +0.5\n",
      ". . . z . . . . . . . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . . z . . . . . . . .  -2.0\n",
      ". . . z . . . . . . . . . . . z . . . . . . . .  +2.0\n",
      ". . . . x z z x . . . . . . . . . . . . . . . .  -0.5\n",
      ". . . . ⴵ z z ⴵ . . . . . . . . . . . . . . . .  +0.5\n",
      ". . . . x x . . . . . . . . . . . . . . . . . .  -0.5\n",
      ". . . . ⴵ ⴵ . . . . . . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . . x z z x . . . .  -0.5\n",
      ". . . . . . . . . . . . . . . . ⴵ z z ⴵ . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . . x x . . . . . .  -0.5\n",
      ". . . . . . . . . . . . . . . . ⴵ ⴵ . . . . . .  +0.5\n",
      ". . . . z . . . . . . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . . . z . . . . . . .  -2.0\n",
      ". . . . z . . . . . . . . . . . z . . . . . . .  +2.0\n",
      ". . . . . x z z x . . . . . . . . . . . . . . .  -0.5\n",
      ". . . . . ⴵ z z ⴵ . . . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . . . x z z x . . .  -0.5\n",
      ". . . . . . . . . . . . . . . . . ⴵ z z ⴵ . . .  +0.5\n",
      ". . . . . z . . . . . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . . . . z . . . . . .  -2.0\n",
      ". . . . . z . . . . . . . . . . . z . . . . . .  +2.0\n",
      ". . . . . . x z z x . . . . . . . . . . . . . .  -0.5\n",
      ". . . . . . ⴵ z z ⴵ . . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . x x . . . . . . . . . . . . . . . .  -0.5\n",
      ". . . . . . ⴵ ⴵ . . . . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . . . . x z z x . .  -0.5\n",
      ". . . . . . . . . . . . . . . . . . ⴵ z z ⴵ . .  +0.5\n",
      ". . . . . . . . . . . . . . . . . . x x . . . .  -0.5\n",
      ". . . . . . . . . . . . . . . . . . ⴵ ⴵ . . . .  +0.5\n",
      ". . . . . . z . . . . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . . . . . z . . . . .  -2.0\n",
      ". . . . . . z . . . . . . . . . . . z . . . . .  +2.0\n",
      ". . . . . . . x z z x . . . . . . . . . . . . .  -0.5\n",
      ". . . . . . . ⴵ z z ⴵ . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . x x . . . . . . . . . . . . . . .  -0.5\n",
      ". . . . . . . ⴵ ⴵ . . . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . . . . . x z z x .  -0.5\n",
      ". . . . . . . . . . . . . . . . . . . ⴵ z z ⴵ .  +0.5\n",
      ". . . . . . . . . . . . . . . . . . . x x . . .  -0.5\n",
      ". . . . . . . . . . . . . . . . . . . ⴵ ⴵ . . .  +0.5\n",
      ". . . . . . . z . . . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . . . . . . z . . . .  -2.0\n",
      ". . . . . . . z . . . . . . . . . . . z . . . .  +2.0\n",
      ". . . . . . . . x z z x . . . . . . . . . . . .  -0.5\n",
      ". . . . . . . . ⴵ z z ⴵ . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . . . . . . x z z x  -0.5\n",
      ". . . . . . . . . . . . . . . . . . . . ⴵ z z ⴵ  +0.5\n",
      ". . . . . . . . z . . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . . . . . . . z . . .  -2.0\n",
      ". . . . . . . . z . . . . . . . . . . . z . . .  +2.0\n",
      ". . . . . . . . . x x . . . . . . . . . . . . .  -0.5\n",
      ". . . . . . . . . ⴵ ⴵ . . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . . . . . . . x x .  -0.5\n",
      ". . . . . . . . . . . . . . . . . . . . . ⴵ ⴵ .  +0.5\n",
      ". . . . . . . . . z . . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . . . . . . . . z . .  -2.0\n",
      ". . . . . . . . . z . . . . . . . . . . . z . .  +2.0\n",
      ". . . . . . . . . . x x . . . . . . . . . . . .  -0.5\n",
      ". . . . . . . . . . ⴵ ⴵ . . . . . . . . . . . .  +0.5\n",
      ". . . . . . . . . . . . . . . . . . . . . . x x  -0.5\n",
      ". . . . . . . . . . . . . . . . . . . . . . ⴵ ⴵ  +0.5\n",
      ". . . . . . . . . . z . . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . . . . . . . . . z .  -2.0\n",
      ". . . . . . . . . . z . . . . . . . . . . . z .  +2.0\n",
      ". . . . . . . . . . . . . . . . . . . . . . . .  +24.0\n",
      ". . . . . . . . . . . z . . . . . . . . . . . .  -2.0\n",
      ". . . . . . . . . . . . . . . . . . . . . . . z  -2.0\n",
      ". . . . . . . . . . . z . . . . . . . . . . . z  +2.0\n"
     ]
    }
   ],
   "source": [
    "H.pauli_decompose(use_zx=True)\n",
    "H.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35bb2145",
   "metadata": {},
   "source": [
    "### Builtin operators\n",
    "\n",
    "The following hamiltonians are built in:\n",
    "\n",
    "- [`heisenberg_from_edges`](quimb.operator.heisenberg_from_edges): the Heisenberg model on an arbitrary graph defined by `edges`, includes the transverse field ising model as a special case of the coupling and magnetic field terms.\n",
    "- [`fermi_hubbard_from_edges`](quimb.operator.fermi_hubbard_from_edges): the Fermi-Hubbard model on an arbitrary graph defined by `edges`.\n",
    "- [`fermi_hubbard_spinless_from_edges`](quimb.operator.fermi_hubbard_spinless_from_edges): the spinless Fermi-Hubbard model (aka t-V model) on an arbitrary graph defined by `edges`."
   ]
  }
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