add spaces to notebook

Author: kevinsung

docs/mzm_generation/tutorial.ipynb

@@ -165,6 +165,7 @@
     "\n",
     "All quantities of interest can be\n",
     "determined from the correlation matrix, which is defined as the block matrix\n",
+    "\n",
     "$$\n",
     "\\begin{align}\n",
     "    \\Gamma =\n",
@@ -174,7 +175,9 @@
     "    \\end{pmatrix}\n",
     "\\end{align}\n",
     "$$\n",
+    "\n",
     "where\n",
+    "\n",
     "$$\n",
     "\\begin{align}\n",
     "    T_{ij} &= \\langle a^\\dagger_i a_j \\rangle \\\\\n",
@@ -183,6 +186,7 @@
     "$$\n",
     "\n",
     "The correlation matrix is obtained by measuring the operators\n",
+    "\n",
     "$$\n",
     "\\begin{align}\n",
     "    a_j^\\dagger a_k + a_k^\\dagger a_j \\\\\n",
@@ -193,6 +197,7 @@
     "$$\n",
     "\n",
     "By using virtual fermionic swaps, only measurements between neighboring modes are needed. For neighboring modes, the Jordan-Wigner transformation of these operators is\n",
+    "\n",
     "$$\n",
     "    \\begin{align}\n",
     "        a_j^\\dagger a_k + a_k^\\dagger a_j &\\mapsto \\frac{X_j X_k + Y_j Y_k}{2} \\\\\n",
@@ -201,7 +206,9 @@
     "        -i(a_j^\\dagger a_k^\\dagger - a_k a_j) &\\mapsto \\frac{-X_j Y_k - Y_j X_k}{2}\n",
     "    \\end{align}.\n",
     "$$\n",
+    "\n",
     "While these operators can be measured straighforwardly by measuring the individual Pauli terms, we take a different approach. Because the fermionic Gaussian state has a fixed parity, we would like to perform the measurement in a way that does not disturb the parity. That way, we can be sure that any measured bitstrings with the incorrect parity must have resulted from an incorrect execution, so we can discard them as an error mitigation technique. These measurements can be accomplished using the two-qubit gates with the following matrices:\n",
+    "\n",
     "$$\n",
     "U_1 =\n",
     "\\begin{pmatrix}\n",
@@ -235,6 +242,7 @@
     "    \\frac{1}{\\sqrt{2}}i & 0 & 0 & \\frac{1}{\\sqrt{2}} \\\\\n",
     "\\end{pmatrix}\n",
     "$$\n",
+    "\n",
     "These gates can be implemented using `XXPlusYYGate` and `XXMinusYYGate`.\n",
     "\n",
     "Once the state is prepared, the operators between neighboring modes can immediately be measured. This requires two separate circuits, one to measure the operator between pairs of qubits where the first qubit is even-indexed, and another circuit to measure pairs of qubits where the first qubit is odd-indexed. In order to measure operators between qubits that are not neighbors, a naive approach would use fermionic swap gates to permute the modes. However, the same effect can be achieved by permuting the transformation matrix used to construct the circuit, which amounts to a relabeling of the fermionic modes. This enables all the required operators to be measured using circuits with the same number of gates."