Update documentation

README.html

@@ -192,12 +192,17 @@
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Product Formula</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="grouping.html">Grouping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ordering.html">Ordering</a></li>
+<li class="toctree-l1"><a class="reference internal" href="grouping.html">Grouping</a></li>
 <li class="toctree-l1"><a class="reference internal" href="interference.html">Error interference</a></li>
 <li class="toctree-l1"><a class="reference internal" href="localization.html">Error localization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="average.html">Average error</a></li>
 </ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Classical simulation</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="path_integral.html">Pauli Path Integral</a></li>
+<li class="toctree-l1"><a class="reference internal" href="tensor_network.html">Tensor network</a></li>
+</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="ground.html">Ground state</a></li>

_sources/fermion.ipynb

@@ -19,20 +19,39 @@
     "from quantum_simulation_recipe import spin_ham, fermion_ham"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "1514eb4a",
+   "metadata": {},
+   "source": [
+    "## Basics"
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "c19aaffa",
    "metadata": {},
    "source": [
-    "### Jordan-Wigner transformation"
+    "### Fermion anti-commutation relation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f8592aaf",
+   "metadata": {},
+   "source": [
+    "### Jordan-Wigner transformation: convert between fermion and spin (qubit)"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "9dd417bb",
    "metadata": {},
    "source": [
-    "## Chemical molecules"
+    "## Chemical molecules\n",
+    "```{math}\n",
+    "H=H_0 + \\sum_{pq}h_{pq} a_p^\\dagger a_q + \\frac{1}{2}\\sum_{pqrs} h_{pqrs} a_p^\\dagger a_q^\\dagger a_r a_s,\n",
+    "```"
    ]
   },
   {
@@ -79,7 +98,11 @@
    "id": "60709827",
    "metadata": {},
    "source": [
-    "## SYK model"
+    "## SYK model\n",
+    "$$\n",
+    "H=\\frac{1}{4\\cdot 4!} \\sum_{i,j,k,l=1}^n J_{ijkl}\\; \\chi_i \\chi_j\\chi_k \\chi_l,\n",
+    "$$\n",
+    "where $\\chi_i$ are majorana fermions with $\\{\\chi_i,\\chi_j\\}=2\\delta_{ij}$ and the coefficients $J_{ijkl}$ are sampled from a Gaussian distribution $N(0,\\frac{3!}{n^3}J^2)$ with zero mean and variance.."
    ]
   },
   {

_sources/field.ipynb

@@ -15,7 +15,19 @@
    "source": [
     "## 1+1 (QED) Lattice field\n",
     "\n",
-    "Schwinger model"
+    "The lattice Schwinger model of 1+1D quantum electrodynamics (QED) can be described by the Hamiltonian (with spin system) $H_{sm}=H_{\\pm}+H_Z$ where\n",
+    "\n",
+    "$$\n",
+    "H_{\\pm} = \\sum_{l=1}^{n-1} H_{\\pm}^l, \\; H_{\\pm}^l = \\frac{w}{2} [X_l X_{l+1}+ Y_l Y_{l+1}]\n",
+    "$$\n",
+    "\n",
+    "and \n",
+    "\n",
+    "$$\n",
+    "H_Z = \\frac{m}{2} \\sum_{l=1}^n (-1)^l Z_l + J \\sum_{l=1}^n L_l^2, \\; L_n = \\frac{1}{2} [Z_l + (-1)^l]\n",
+    "$$\n",
+    "\n",
+    "Here, $m$ is the rest mass of the fermionic particles and anti-particles, and $w$ describes their kinetic energy. The term proportional to $J$ is the energy of the U(1) gauge fields."
    ]
   },
   {
@@ -29,6 +41,15 @@
    },
    "outputs": [],
    "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f60190e1",
+   "metadata": {},
+   "source": [
+    "## QCD \n",
+    "todo"
+   ]
   }
  ],
  "metadata": {

_sources/interference.ipynb

@@ -13,8 +13,9 @@
    "id": "ad7989ef",
    "metadata": {},
    "source": [
-    "## First-Order Trotter Error from a Second-Order Perspective\n",
-    "http://arxiv.org/abs/2107.08032"
+    "- Destructive Error Interference in Product-Formula Lattice Simulation http://arxiv.org/abs/1912.11047\n",
+    "- First-Order Trotter Error from a Second-Order Perspective http://arxiv.org/abs/2107.08032\n",
+    "- Self-healing of Trotter error in digital adiabatic state preparation http://arxiv.org/abs/2209.06242"
    ]
   },
   {

_sources/jax.ipynb

@@ -0,0 +1,51 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "8b24f5d3",
+   "metadata": {},
+   "source": [
+    "# JAX and multi-processing"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "2763848a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from qiskit.quantum_info import SparsePauliOp, Statevector\n",
+    "\n",
+    "import quantum_simulation_recipe as qsr\n",
+    "from quantum_simulation_recipe import spin_ham"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3.8.8 ('base')",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.14"
+  },
+  "vscode": {
+   "interpreter": {
+    "hash": "4e8ef2f9fcac0817bca9a7ca376f64f20b4df5ea3bf7af756a50bda7d3557ea6"
+   }
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

_sources/ordering.ipynb

@@ -35,6 +35,14 @@
     "H.ham"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "b45a921c",
+   "metadata": {},
+   "source": [
+    "## Lie-Trotter-Suzuki product formula"
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "d2c55840",

_sources/path_integral.ipynb

@@ -0,0 +1,42 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "8b24f5d3",
+   "metadata": {},
+   "source": [
+    "# Pauli Path Integral\n",
+    "\n",
+    "Main Reference:\n",
+    "- A Polynomial-Time Classical Algorithm for Noisy Random Circuit Sampling https://arxiv.org/abs/2211.03999\n",
+    "- A polynomial-time classical algorithm for noisy quantum circuits http://arxiv.org/abs/2407.12768"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3.8.8 ('base')",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.14"
+  },
+  "vscode": {
+   "interpreter": {
+    "hash": "4e8ef2f9fcac0817bca9a7ca376f64f20b4df5ea3bf7af756a50bda7d3557ea6"
+   }
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

_sources/quantities.ipynb

@@ -10,12 +10,13 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 4,
    "id": "098ce6c5",
    "metadata": {},
    "outputs": [],
    "source": [
     "from qiskit.quantum_info import SparsePauliOp, commutator, anti_commutator, entropy, Statevector, random_statevector, random_density_matrix, partial_trace\n",
+    "import numpy as np\n",
     "from numpy.linalg import norm  "
    ]
   },
@@ -201,6 +202,14 @@
     "## Different norms"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "1316370b",
+   "metadata": {},
+   "source": [
+    "### Vector induced norm"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 4,
@@ -227,6 +236,37 @@
     "print('Frobenius norm: ', norm(op, ord='fro'))\n",
     "# print(norm(SparsePauliOp.from_list([(\"XYZI\", 1)]), ord='nuc'))"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a987b947",
+   "metadata": {},
+   "source": [
+    "### Schatten p norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "c3aa71a2",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "3.605551275463989"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "test = SparsePauliOp.from_list([('XX', 2), ('YX', 3)])\n",
+    "np.linalg.norm(test.to_matrix(), ord='nuc')\n",
+    "np.linalg.norm(test.to_matrix(), ord=1)"
+   ]
   }
  ],
  "metadata": {

_sources/spin.ipynb

@@ -26,12 +26,12 @@
    "id": "0ee355f6",
    "metadata": {},
    "source": [
-    "## Nearest-neighbor interaction\n",
+    "## 1D Nearest-neighbor interaction\n",
     "\n",
     "```{math}\n",
-    "H=\\sum_{j=1}^{n-1} J_j Z_j Z_{j+1} + \\sum_{j=1}^{n} h_j X_j,\n",
+    "H=\\sum_{j=1}^{n-1} J_j^x X_j X_{j+1} + J_j^y Y_j Y_{j+1} + J_j^z Z_j Z_{j+1} + \\sum_{j=1}^{n} h_j^x X_j + h_j^y Y_j + h_j^z Z_j,\n",
     "```\n",
-    "where $J$ is the interaction strength and $h_j$ is the strength of the external field."
+    "where $J$ is the interaction strength and $h_j$ is the strength of the external field (random disorder)."
    ]
   },
   {
@@ -63,7 +63,10 @@
    "id": "efde70d6",
    "metadata": {},
    "source": [
-    "## Power-law interaction"
+    "## Power-law interaction\n",
+    "```{math}\n",
+    "H=\\sum_{j=1}^{n} \\sum_{k=j+1}^n \\frac{1}{(j-k)^\\alpha} (J^x X_j X_{k} + J^y Y_j Y_{k} + J^z Z_j Z_{k}) + \\sum_{j=1}^{n} h_j^x X_j + h_j^y Y_j + h_j^z Z_j,\n",
+    "```"
    ]
   },
   {
@@ -99,7 +102,9 @@
    "id": "14bd5369",
    "metadata": {},
    "source": [
-    "## Higher dimension lattice"
+    "## Higher dimension lattice\n",
+    "\n",
+    "todo"
    ]
   }
  ],