From 111356702e5d8460b3d1d25f80f4e55c4cd7ee8e Mon Sep 17 00:00:00 2001
From: Jue Xu <xujuexujue@hotmail.com>
Date: Thu, 1 Aug 2024 09:34:15 +0800
Subject: [PATCH] pauli algebra

---
 quantum_simulation_recipe/test.ipynb | 114 ++++++++++++++++++++++++++-
 1 file changed, 112 insertions(+), 2 deletions(-)

diff --git a/quantum_simulation_recipe/test.ipynb b/quantum_simulation_recipe/test.ipynb
index eac674c..fdcec52 100644
--- a/quantum_simulation_recipe/test.ipynb
+++ b/quantum_simulation_recipe/test.ipynb
@@ -26,13 +26,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 12,
    "id": "8c03d3e4",
    "metadata": {},
    "outputs": [],
    "source": [
     "from spin import *\n",
-    "from fermion import *"
+    "from fermion import *\n",
+    "from bounds import norm, tight_bound, commutator"
    ]
   },
   {
@@ -348,6 +349,115 @@
     "from bounds import *"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "9d9b79a5",
+   "metadata": {},
+   "source": [
+    "## Pauli algebra"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "809c7242",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Parity grouping: \n",
+      " [SparsePauliOp(['IIIIZZ', 'IIZZII', 'ZZIIII', 'IIIIIX', 'IIIXII', 'IXIIII'],\n",
+      "              coeffs=[1. +0.j, 1. +0.j, 1. +0.j, 0.2+0.j, 0.2+0.j, 0.2+0.j]), SparsePauliOp(['IIIZZI', 'IZZIII', 'ZIIIIZ', 'IIIIXI', 'IIXIII', 'XIIIII'],\n",
+      "              coeffs=[1. +0.j, 1. +0.j, 1. +0.j, 0.2+0.j, 0.2+0.j, 0.2+0.j])]\n",
+      "XYZ grouping: \n",
+      " [SparsePauliOp(['IIIIIX', 'IIIIXI', 'IIIXII', 'IIXIII', 'IXIIII', 'XIIIII'],\n",
+      "              coeffs=[0.2+0.j, 0.2+0.j, 0.2+0.j, 0.2+0.j, 0.2+0.j, 0.2+0.j]), SparsePauliOp(['IIIIZZ', 'IIIZZI', 'IIZZII', 'IZZIII', 'ZZIIII', 'ZIIIIZ'],\n",
+      "              coeffs=[1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j])]\n"
+     ]
+    }
+   ],
+   "source": [
+    "n = 6\n",
+    "J, h = 1, 0.2 \n",
+    "tfI = Nearest_Neighbour_1d(n=n, Jz=J, hx=h, pbc=True)\n",
+    "print('Parity grouping: \\n', tfI.ham_par)\n",
+    "print('XYZ grouping: \\n', tfI.ham_xyz)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "f730b002",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "commutator of xyz grouping: \n",
+      " SparsePauliOp(['IIIIZY', 'ZIIIIY', 'IIIIYZ', 'IIIZYI', 'IIIYZI', 'IIZYII', 'IIYZII', 'IZYIII', 'IYZIII', 'ZYIIII', 'YZIIII', 'YIIIIZ'],\n",
+      "              coeffs=[0.-0.4j, 0.-0.4j, 0.-0.4j, 0.-0.4j, 0.-0.4j, 0.-0.4j, 0.-0.4j, 0.-0.4j,\n",
+      " 0.-0.4j, 0.-0.4j, 0.-0.4j, 0.-0.4j])\n"
+     ]
+    }
+   ],
+   "source": [
+    "cmm_xyz = commutator(tfI.ham_xyz[0], tfI.ham_xyz[1]).simplify()\n",
+    "print('commutator of xyz grouping: \\n', cmm_xyz)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "92245f95",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "SparsePauliOp(['IIIIIY', 'IIIIYI', 'IIIYII', 'IIYIII', 'IYIIII', 'YIIIII'],\n",
+       "              coeffs=[0.-0.4j, 0.-0.4j, 0.-0.4j, 0.-0.4j, 0.-0.4j, 0.-0.4j])"
+      ]
+     },
+     "execution_count": 17,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from qiskit.quantum_info import SparsePauliOp\n",
+    "magn_op = SparsePauliOp.from_sparse_list([('Z', [i], 1) for i in range(0, n)], n)\n",
+    "corr_op = SparsePauliOp.from_sparse_list([('ZZ', [i,i+1], 1/(n-1)) for i in range(0, n-1)], n)\n",
+    "cmm_H_magn = commutator(tfI.ham_xyz[0], magn_op).simplify()\n",
+    "cmm_H_magn"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "464ec272",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "spectral norm:  3.2000000000000006\n",
+      "induced 1-norm:  4.800000000000001\n",
+      "Frobenius norm:  11.085125168440818\n",
+      "trace (nuclear) norm:  71.65537550532244\n"
+     ]
+    }
+   ],
+   "source": [
+    "print('spectral norm: ', norm(cmm_xyz, ord=2))\n",
+    "print('induced 1-norm: ', norm(cmm_xyz.coeffs, ord=1))\n",
+    "print('Frobenius norm: ', norm(cmm_xyz, ord='fro'))\n",
+    "print('trace (nuclear) norm: ', norm(cmm_xyz, ord='nuc'))"
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "f981124d",