Reviews Addressed and changed notebook location

Author: virajd98

…lock_encoding-ND_Laplacian/1D_Lap_BE.png …lock_encoding-ND_Laplacian/1D_Lap_BE.pngapplications/Block_encoding-ND_Laplacian/1D_Lap_BE.png renamed to community/paper_implementation_project/Block_encoding-ND_Laplacian/1D_Lap_BE.png applications/Block_encoding-ND_Laplacian/1D_Lap_BE.png renamed to community/paper_implementation_project/Block_encoding-ND_Laplacian/1D_Lap_BE.png

@@ -23,7 +23,7 @@
     "Block encoding is a well known technique in quantum computing used to embed non-unitary matrices on a quantum computer that only allows for unitary evolution. \n",
     "\n",
     "***Definition:***  \n",
-    "    Let $ a, n, m \\in \\mathbb{N} $ such that $ m = a + n $. A $ m $ -qubit unitary $ U $  is said to be an $(\\alpha, a)$ -block-encoding of an $ n $ -qubit operator $ A $  if  \n",
+    "    Let $ a, n, m \\in \\mathbb{N} $ such that $ m = a + n $. An $ m $ -qubit unitary $ U $  is said to be an $(\\alpha, a)$ -block-encoding of an $ n $ -qubit operator $ A $  if  \n",
     "\n",
     "$$\n",
     "\\tilde{A} = \\left( \\langle 0 |^{\\otimes a} \\otimes I_n \\right) U \\left( |0 \\rangle^{\\otimes a} \\otimes I_n \\right)\n",
@@ -32,10 +32,10 @@
     "\n",
     "where,  $A = \\alpha \\tilde{A}$ . The parameters $(\\alpha, a)$  represent the *subnormalization factor* (which adjusts for encoding matrices of any norm), and the *number of ancilla qubits* used in the block-encoding scheme respectively. \n",
     "\n",
-    "Efficiently block encoding arbitrary matrices is a very difficult problem and this task is not trivial even for well structured and sparse matrices. The paper by Sturm et al. 2025 (https://arxiv.org/abs/2509.02429), provides for efficient quantum circuits for block encoding $N-$ dimensional Laplacians, with periodic boundary conditions along all dimensions. This is useful in many applications, especially ones involving problems of linear algebra. Moreover, given an efficient block encoding of a matrix $\\tilde{A}$, its possible to efficiently construct a block encoding of certain polynomials of $\\tilde{A}$ through quantum singular value transformations (QSVT). \n",
+    "Efficiently block encoding arbitrary matrices is a very difficult problem and this task is not trivial even for well structured and sparse matrices. The paper by Sturm et al. 2025 (https://arxiv.org/abs/2509.02429), provides efficient quantum circuits for block encoding $N-$ dimensional Laplacians, with periodic boundary conditions along all dimensions. This is useful in many applications, especially ones involving linear algebra problems. Moreover, given an efficient block encoding of a matrix $\\tilde{A}$, it's possible to efficiently construct a block encoding of certain polynomials of $\\tilde{A}$ through quantum singular value transformations (QSVT). \n",
     "\n",
     "### **Notebook contents**\n",
-    "- ##### Block encoding circuits for D- dimensional Laplacian matrix with periodic boundary conditions along all dimnesions.\n",
+    "- ##### Block encoding circuits for D-dimensional Laplacian matrix with periodic boundary conditions along all dimensions.\n",
     "___"
    ]
   },
@@ -62,16 +62,17 @@
     "\\tag{1}\n",
     "\\end{equation}\n",
     "\n",
-    "We impose **periodic boundary conditions**, meaning the function repeats itself across opposite sides of the domain:\n",
+    "Consider a function $u$ on which the operator $L_D$ acts. We impose **periodic boundary conditions**, meaning the function repeats itself across opposite sides of the domain:\n",
     "$$\n",
-    "v(x^{(0)},\\ldots,0,\\ldots,x^{(D-1)}) \n",
+    "u(x^{(0)},\\ldots,0,\\ldots,x^{(D-1)}) \n",
     "= \n",
-    "v(x^{(0)},\\ldots,1,\\ldots,x^{(D-1)}).\n",
+    "u(x^{(0)},\\ldots,1,\\ldots,x^{(D-1)}).\n",
     "$$\n",
     "\n",
     "This ensures the domain “wraps around” in **every dimension**.\n",
     "\n",
     "\n",
+    "\n",
     "To approximate the operator on a computer, we replace each interval with a **uniform grid** (*we assume equidistant grid points as per the paper*).  \n",
     "\n",
     "For one dimension, choose\n",
@@ -109,6 +110,32 @@
     "j = \\ket{j^{(D-1)}}\\otimes \\cdots \\otimes\\ket{j^{(1)}}\\otimes\\ket{j^{(0)}},\n",
     "$$\n",
     "\n",
+    "- ***Example***\n",
+    "\n",
+    "  If $n = 3$ (so $N = 2^n = 8$), the index $j^{(d)} = 5$ is represented in the computational basis as\n",
+    "  $\n",
+    "  \\ket{5} = \\ket{101},\n",
+    "  $\n",
+    "  corresponding to the binary representation of the integer $5$.\n",
+    "  Explicitly, the basis state $\\lvert 5 \\rangle$ corresponds to the $2^n$-dimensional column vector\n",
+    "  $$\n",
+    "  \\lvert 5 \\rangle\n",
+    "  =\n",
+    "  \\begin{pmatrix}\n",
+    "  0 \\\\\n",
+    "  0 \\\\\n",
+    "  0 \\\\\n",
+    "  0 \\\\\n",
+    "  0 \\\\\n",
+    "  1 \\\\\n",
+    "  0 \\\\\n",
+    "  0\n",
+    "  \\end{pmatrix},\n",
+    "  $$\n",
+    "  where the single nonzero entry appears at position $5$, following zero-based indexing.\n",
+    "\n",
+    "\n",
+    "\n",
     "This ordering turns the entire grid $\\Omega_{D,h}$ into a vector of $\\mathbf{\\Omega_{D,h}}$ size $N^D= 2^{nD}$ where,\n",
     "\n",
     "$$\n",
@@ -583,7 +610,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 3,
    "id": "11",
    "metadata": {},
    "outputs": [],