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diff --git a/_build/.doctrees/environment.pickle b/_build/.doctrees/environment.pickle
diff --git a/_build/.doctrees/field.doctree b/_build/.doctrees/field.doctree
diff --git a/_build/.doctrees/localization.doctree b/_build/.doctrees/localization.doctree
diff --git a/_build/.doctrees/path_integral.doctree b/_build/.doctrees/path_integral.doctree
diff --git a/_build/.jupyter_cache/executed/e8553bb183871fdef21bdaf542679280/base.ipynb b/_build/.jupyter_cache/executed/e8553bb183871fdef21bdaf542679280/base.ipynb
diff --git a/_build/.jupyter_cache/global.db b/_build/.jupyter_cache/global.db
diff --git a/_build/html/README.html b/_build/html/README.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/...ml/_images/ee3d33e4b986b720e6db11bbc583efe8aa63781f14a7e02853055d470e5ff977.png b/...ml/_images/ee3d33e4b986b720e6db11bbc583efe8aa63781f14a7e02853055d470e5ff977.png
diff --git a/_build/html/_sources/field.ipynb b/_build/html/_sources/field.ipynb
diff --git a/_build/html/_sources/localization.ipynb b/_build/html/_sources/localization.ipynb
@@ -7,7 +7,7 @@
    "source": [
     "# Error localization \n",
     "\n",
-    "Quantum localization bounds Trotter errors in digital quantum simulation (http://arxiv.org/abs/1806.11123)"
+    "[Quantum localization bounds Trotter errors in digital quantum simulation](http://arxiv.org/abs/1806.11123)"
    ]
   },
   {

diff --git a/_build/html/_sources/path_integral.ipynb b/_build/html/_sources/path_integral.ipynb
@@ -5,12 +5,20 @@
    "id": "8b24f5d3",
    "metadata": {},
    "source": [
-    "# Pauli Path Integral\n",
+    "# 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"
+    "- [A polynomial-time classical algorithm for noisy quantum circuits](http://arxiv.org/abs/2407.12768)\n",
+    "- [A Polynomial-Time Classical Algorithm for Noisy Random Circuit Sampling](https://arxiv.org/abs/2211.03999)"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "be9b368c",
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {

diff --git a/_build/html/adaptive.html b/_build/html/adaptive.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/anticommute.html b/_build/html/anticommute.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/applications/misc.html b/_build/html/applications/misc.html
@@ -199,7 +199,7 @@
 </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="../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>

diff --git a/_build/html/average.html b/_build/html/average.html
@@ -61,7 +61,7 @@
     <script>DOCUMENTATION_OPTIONS.pagename = 'average';</script>
     <link rel="index" title="Index" href="genindex.html" />
     <link rel="search" title="Search" href="search.html" />
-    <link rel="next" title="Pauli Path Integral" href="path_integral.html" />
+    <link rel="next" title="Pauli path integral" href="path_integral.html" />
     <link rel="prev" title="Error localization" href="localization.html" />
   <meta name="viewport" content="width=device-width, initial-scale=1"/>
   <meta name="docsearch:language" content="en"/>
@@ -199,7 +199,7 @@
 </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="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>
@@ -491,7 +491,7 @@ <h2>Observable (quantum mean value)<a class="headerlink" href="#observable-quant
        title="next page">
       <div class="prev-next-info">
         <p class="prev-next-subtitle">next</p>
-        <p class="prev-next-title">Pauli Path Integral</p>
+        <p class="prev-next-title">Pauli path integral</p>
       </div>
       <i class="fa-solid fa-angle-right"></i>
     </a>

diff --git a/_build/html/basic.html b/_build/html/basic.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/bound.html b/_build/html/bound.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/channel.html b/_build/html/channel.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/chem.html b/_build/html/chem.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/debug.html b/_build/html/debug.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/fermion.html b/_build/html/fermion.html
@@ -201,7 +201,7 @@
 </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="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>

diff --git a/_build/html/field.html b/_build/html/field.html
@@ -201,7 +201,7 @@
 </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="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>
@@ -447,15 +447,14 @@ <h2>1+1 (QED) Lattice field<a class="headerlink" href="#qed-lattice-field" title
 <span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">)))</span>
 
 <span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">bottom</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="mf">0.8</span><span class="p">,</span> <span class="n">top</span><span class="o">=</span><span class="mf">0.9</span><span class="p">)</span>
-<span class="n">cax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">((</span><span class="mf">0.85</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.075</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">))</span>
-<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">cax</span><span class="o">=</span><span class="n">cax</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">cax</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">((</span><span class="mf">0.85</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.075</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">)))</span>
 
 <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
 </pre></div>
 </div>
 </div>
 <div class="cell_output docutils container">
-<a class="reference internal image-reference" href="_images/6514999a95c43d661803d77c72115609bac274bde150f6a1af8a5f96af1b3233.png"><img alt="_images/6514999a95c43d661803d77c72115609bac274bde150f6a1af8a5f96af1b3233.png" src="_images/6514999a95c43d661803d77c72115609bac274bde150f6a1af8a5f96af1b3233.png" style="width: 567px; height: 431px;" /></a>
+<a class="reference internal image-reference" href="_images/ee3d33e4b986b720e6db11bbc583efe8aa63781f14a7e02853055d470e5ff977.png"><img alt="_images/ee3d33e4b986b720e6db11bbc583efe8aa63781f14a7e02853055d470e5ff977.png" src="_images/ee3d33e4b986b720e6db11bbc583efe8aa63781f14a7e02853055d470e5ff977.png" style="width: 567px; height: 431px;" /></a>
 </div>
 </div>
 </section>

diff --git a/_build/html/genindex.html b/_build/html/genindex.html
@@ -198,7 +198,7 @@
 </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="path_integral.html">Pauli path integral</a></li>
 <li class="toctree-l1"><a class="reference internal" href="tensor_network.html">Tensor network</a></li>
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 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>

diff --git a/_build/html/ground.html b/_build/html/ground.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/grouping.html b/_build/html/grouping.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/interference.html b/_build/html/interference.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/intro.html b/_build/html/intro.html
@@ -202,7 +202,7 @@
 </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="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>
@@ -443,7 +443,7 @@ <h1>Quantum Simulation Cookbook<a class="headerlink" href="#quantum-simulation-c
 <div class="toctree-wrapper compound">
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Classical simulation</span></p>
 <ul>
-<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="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>
 </div>

diff --git a/_build/html/jax.html b/_build/html/jax.html
@@ -199,7 +199,7 @@
 </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="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>

diff --git a/_build/html/localization.html b/_build/html/localization.html
@@ -199,7 +199,7 @@
 </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="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>
@@ -416,7 +416,7 @@ <h2> Contents </h2>
 
   <section class="tex2jax_ignore mathjax_ignore" id="error-localization">
 <h1>Error localization<a class="headerlink" href="#error-localization" title="Link to this heading">#</a></h1>
-<p>Quantum localization bounds Trotter errors in digital quantum simulation (<a class="reference external" href="http://arxiv.org/abs/1806.11123">http://arxiv.org/abs/1806.11123</a>)</p>
+<p><a class="reference external" href="http://arxiv.org/abs/1806.11123">Quantum localization bounds Trotter errors in digital quantum simulation</a></p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">qiskit.quantum_info</span> <span class="kn">import</span> <span class="n">Statevector</span><span class="p">,</span> <span class="n">SparsePauliOp</span><span class="p">,</span> <span class="n">Operator</span><span class="p">,</span> <span class="n">partial_trace</span><span class="p">,</span> <span class="n">entropy</span><span class="p">,</span> <span class="n">DensityMatrix</span>

diff --git a/_build/html/objects.inv b/_build/html/objects.inv
diff --git a/_build/html/open.html b/_build/html/open.html
@@ -198,7 +198,7 @@
 </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="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>

diff --git a/_build/html/operator.html b/_build/html/operator.html
@@ -201,7 +201,7 @@
 </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="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>

diff --git a/_build/html/ordering.html b/_build/html/ordering.html
@@ -199,7 +199,7 @@
 </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="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>