Minor tutorial updates (#24)

Author: caleb-johnson

docs/tutorials/01_chemistry_hamiltonian.ipynb

@@ -317,9 +317,9 @@
    "source": [
     "First, we will transform the counts into a bitstring matrix and probability array for post-processing.\n",
     "\n",
-    "Each row in the matrix represents one unique bitstring. Since qubits are normally indexed from the right of a bitstring, column ``0`` represents qubit ``N-1``, and column ``N-1`` represents qubit ``0``, where ``N`` is the number of qubits.\n",
+    "Each row in the matrix represents one unique bitstring. Since qubits are normally indexed from the right of a bitstring, column ``0`` represents qubit $N-1$, and column $N-1$ represents qubit ``0``, where $N$ is the number of qubits.\n",
     "\n",
-    "The alpha particles are represented in the column range ``[N / 2, N]``, and the beta particles are represented in the column range ``[0, N / 2)``."
+    "The alpha particles are represented in the column range $[N / 2, N]$, and the beta particles are represented in the column range $[0, N / 2)$."
    ]
   },
   {

docs/tutorials/02_qubit_hamiltonian.ipynb

@@ -4,9 +4,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Ground state energy estimation of a spin (qubit) Hamiltonian\n",
+    "# Minimum eigenvalue estimation of a spin (qubit) Hamiltonian\n",
     "\n",
-    "In this tutorial we implement a [Qiskit pattern](https://docs.quantum.ibm.com/guides/intro-to-patterns) showing how to post-process quantum samples to compute an approximation to the ground state of a ``22``-site XX-Z spin-1/2 chain and two-point correlators. We will follow a sample-based quantum diagonalization approach[[1]](https://arxiv.org/abs/2405.05068).\n",
+    "In this tutorial we implement a [Qiskit pattern](https://docs.quantum.ibm.com/guides/intro-to-patterns) showing how to post-process quantum samples to approximate the minimum eigenvalue and spin-spin correlators for a ``22``-site XX-Z spin-1/2 chain. We will follow a sample-based quantum diagonalization approach [[1]](https://arxiv.org/abs/2405.05068).\n",
     "\n",
     "While a Qiskit pattern typically involves 4 steps, the aim of this tutorial is to focus on the post-processing of the samples obtained from a quantum circuit whose support coincides with that of the ground state. Consequently, we generate a synthetic set of bitstrings to define the subspace and do not design an ansatz nor sample from a quantum circuit in this tutorial.\n",
     "\n",
@@ -20,7 +20,8 @@
     "    - N/A: Will generate synthetic quantum samples\n",
     "4. **Step 4: Post-process results**\n",
     "   - Project the Hamiltonian onto the subspace spanned by the samples\n",
-    "   - Diagonalize the Hamiltonian in the subspace to approximate the ground state"
+    "   - Diagonalize the Hamiltonian in the subspace to approximate the ground state\n",
+    "   - Calculate spin-spin correlators for each site, $l$"
    ]
   },
   {
@@ -39,7 +40,7 @@
     "of many-body Hamiltonians can be written as the linear combination of polynomially-many \n",
     "Pauli strings, including interacting-electron Hamiltonians, spin Hamiltonians, etc.\n",
     "\n",
-    "In particular, we consider the ground state properties of the antiferromagnetic XX-Z spin-1/2 chain\n",
+    "In particular, we consider the properties of the antiferromagnetic XX-Z spin-1/2 chain\n",
     "with $L = 22$ sites:\n",
     "$$\n",
     "H = \\sum_{\\langle i, j \\rangle} J_{xy}\\left( \\sigma^x_i\\sigma^x_j + \\sigma^y_i\\sigma^y_j \\right) + \\sigma^z_i\\sigma^z_j.\n",
@@ -231,9 +232,9 @@
     "from qiskit_addon_sqd.qubit import solve_qubit\n",
     "\n",
     "scipy_kwargs = {\"k\": 4, \"which\": \"SA\"}\n",
-    "energies, eigenstates = solve_qubit(bitstring_matrix, hamiltonian, verbose=True, **scipy_kwargs)\n",
+    "eigenvals, eigenstates = solve_qubit(bitstring_matrix, hamiltonian, verbose=True, **scipy_kwargs)\n",
     "\n",
-    "ground_state = eigenstates[:, 0]"
+    "min_eval = eigenstates[:, 0]"
    ]
   },
   {
@@ -250,7 +251,7 @@
     }
    ],
    "source": [
-    "print(energies)"
+    "print(eigenvals)"
    ]
   },
   {
@@ -296,21 +297,21 @@
     "    pstr[i] = \"X\"\n",
     "    pauli_op = SparsePauliOp(\"\".join(pstr))\n",
     "    sparse_op = project_operator_to_subspace(bitstring_matrix, pauli_op)\n",
-    "    s_x[i] += np.real(np.conjugate(ground_state).T @ sparse_op @ ground_state)\n",
+    "    s_x[i] += np.real(np.conjugate(min_eval).T @ sparse_op @ min_eval)\n",
     "\n",
     "    # Sigma_y\n",
     "    pstr = [\"I\" for i in range(num_spins)]\n",
     "    pstr[i] = \"Y\"\n",
     "    pauli_op = SparsePauliOp(\"\".join(pstr))\n",
     "    sparse_op = project_operator_to_subspace(bitstring_matrix, pauli_op)\n",
-    "    s_y[i] += np.real(np.conjugate(ground_state).T @ sparse_op @ ground_state)\n",
+    "    s_y[i] += np.real(np.conjugate(min_eval).T @ sparse_op @ min_eval)\n",
     "\n",
     "    # Sigma_z\n",
     "    pstr = [\"I\" for i in range(num_spins)]\n",
     "    pstr[i] = \"Z\"\n",
     "    pauli_op = SparsePauliOp(\"\".join(pstr))\n",
     "    sparse_op = project_operator_to_subspace(bitstring_matrix, pauli_op)\n",
-    "    s_z[i] += np.real(np.conjugate(ground_state).T @ sparse_op @ ground_state)"
+    "    s_z[i] += np.real(np.conjugate(min_eval).T @ sparse_op @ min_eval)"
    ]
   },
   {
@@ -377,7 +378,7 @@
     "        pauli_op = SparsePauliOp(\"\".join(pstr))\n",
     "        sparse_op = project_operator_to_subspace(bitstring_matrix, pauli_op)\n",
     "        c_x[distance - 1] += (\n",
-    "            np.real(np.conjugate(ground_state).T @ sparse_op @ ground_state) - s_x[i] * s_x[j_wrap]\n",
+    "            np.real(np.conjugate(min_eval).T @ sparse_op @ min_eval) - s_x[i] * s_x[j_wrap]\n",
     "        )\n",
     "\n",
     "        # Sigma_y Sigma_y\n",
@@ -387,7 +388,7 @@
     "        pauli_op = SparsePauliOp(\"\".join(pstr))\n",
     "        sparse_op = project_operator_to_subspace(bitstring_matrix, pauli_op)\n",
     "        c_y[distance - 1] += (\n",
-    "            np.real(np.conjugate(ground_state).T @ sparse_op @ ground_state) - s_y[i] * s_y[j_wrap]\n",
+    "            np.real(np.conjugate(min_eval).T @ sparse_op @ min_eval) - s_y[i] * s_y[j_wrap]\n",
     "        )\n",
     "\n",
     "        # Sigma_z Sigma_z\n",
@@ -397,7 +398,7 @@
     "        pauli_op = SparsePauliOp(\"\".join(pstr))\n",
     "        sparse_op = project_operator_to_subspace(bitstring_matrix, pauli_op)\n",
     "        c_z[distance - 1] += (\n",
-    "            np.real(np.conjugate(ground_state).T @ sparse_op @ ground_state) - s_z[i] * s_z[j_wrap]\n",
+    "            np.real(np.conjugate(min_eval).T @ sparse_op @ min_eval) - s_z[i] * s_z[j_wrap]\n",
     "        )\n",
     "\n",
     "        distance_counts[distance - 1] += 1\n",