Adjustments to notebook guides (#16) (#19)

(cherry picked from commit 68f0e4b)

Co-authored-by: Caleb Johnson <caleb.johnson@ibm.com>

docs/how_tos/choose_trotter_steps.ipynb

@@ -34,11 +34,11 @@
     "\n",
     "In principle, any $k_j$ values can be chosen but some choices of Trotter steps will lead to a larger noise amplification on real devices than others.\n",
     "Thus, it is important that one tries to find _\"good\"_ values of $k_j$.\n",
-    "In the following we will discuss some heuristics to help you guide in doing so.\n",
+    "In the following we will discuss some heuristics to help guide you in doing so.\n",
     "\n",
-    "1. In practice, our **largest** $k_j$ ($k_{\\text{max}}$) value is simply bound by what we can afford to execute.\n",
+    "1. In practice, our **largest** $k_j$ ($k_{\\text{max}}$) value is simply bound by the depth of quantum circuit we can reliably execute.\n",
     "\n",
-    "2. Since the Trotter error is only well behaved for $\\text{d}t = t/k \\lt 1$ (see also our explanations page on [Understanding the stability of MPFs](https://qiskit.github.io/qiskit-addon-mpf/explanations/mpf_stability.html), our **smallest** $k_j$ ($k_{\\text{min}}$) should satisfy: $t/k_{\\text{min}} \\lt 1$ (Note: empirically $\\text{d}t \\leq 1$ seems to work fine, too).\n",
+    "2. Since the Trotter error is only well behaved for $\\text{d}t = t/k \\lt 1$ (see [Understanding the stability of MPFs](https://qiskit.github.io/qiskit-addon-mpf/explanations/mpf_stability.html)), our **smallest** $k_j$ ($k_{\\text{min}}$) should satisfy: $t/k_{\\text{min}} \\lt 1$ (Note: empirically $\\text{d}t \\leq 1$ seems to work fine, too).\n",
     "\n",
     "3. None of our $x_j$ values should be close to $0$, as this would imply that we might as well exclude the corresponding $k_j$ from the expansion.\n",
     "\n",
@@ -289,7 +289,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.9"
+   "version": "3.12.4"
   }
  },
  "nbformat": 4,

docs/tutorials/01_getting_started.ipynb

@@ -243,12 +243,12 @@
     "\n",
     "where $x_j$ are our weighting coefficients, $\\rho^{k_j}_j$ is the density matrix corresponding to the pure state obtained by evolving the initial state with the product formula, $S^{k_j}$, involving $k_j$ Trotter steps, and $j$ indexes the number of PFs that make up the MPF.\n",
     "\n",
-    "The key here is, that the remaining Trotter error is smaller than the Trotter error that one would obtain by simply using the largest $k_j$ value!\n",
+    "The key here is that the remaining Trotter error is smaller than the Trotter error that one would obtain by simply using the largest $k_j$ value!\n",
     "\n",
-    "You can view the usefulness of this from two perspectives:\n",
+    "You can view the usefulness of MPF from two perspectives:\n",
     "\n",
     "1. For a fixed budget of Trotter steps that you are able to execute, you can obtain results with a Trotter error that is smaller in total.\n",
-    "2. Given some target number of Trotter steps that is too large to execute, you can use the MPF to find a shorter-depth circuit to run which results in a similar Trotter error."
+    "2. For a number of Trotter steps which results in deep circuits, you can use MPF to find several shorter-depth circuits to run which result in a similar Trotter error."
    ]
   },
   {
@@ -269,7 +269,7 @@
     "\n",
     "Determining the static MPF coefficients for a given set of $k_j$ values amounts to solving a linear system of equations:\n",
     "$Ax=b$, where $x$ are our coefficients of interest, $A$ is a matrix depending on $k_j$ and the type of PF we use ($S$), and $b$ is a vector of constraints.\n",
-    "For brevity, we are not going to go into more detail here and instead refer you to the documentation of [LSE](https://qiskit.github.io/qiskit-addon-mpf/stubs/qiskit_addon_mpf.static.LSE.html)\n",
+    "For brevity, we are not going to go into more detail here and instead refer you to the documentation of [LSE](https://qiskit.github.io/qiskit-addon-mpf/stubs/qiskit_addon_mpf.static.LSE.html).\n",
     "\n",
     "We can find a solution for $x$ analytically as $x = A^{-1}b$, see e.g. [Carrera Vazquez et al., 2023] or [Zhuk et al., 2023].\n",
     "However, this exact solution can be _\"ill-conditioned\"_ resulting in very large L1-norms of our coefficients, $x$, which can lead to bad performance of the MPF.\n",
@@ -336,7 +336,7 @@
     "#### Setting up the LSE\n",
     "\n",
     "Now that we have chosen our $k_j$s, we must first construct the LSE, $Ax=b$ as explained above.\n",
-    "The matrix $A$ depends not only on $k_j$ but also our choice of PF, in particular its _order_.\n",
+    "The matrix $A$ depends not only on $k_j$ but also our choice of product formula (PF) -- in particular its _order_.\n",
     "Additionally, one may take into account whether the PF is symmetric or not (see [Carrera Vazquez et al., 2023]), by setting `symmetric=True`.\n",
     "However, this is not required as shown by [Zhuk et al., 2023].\n",
     "\n",
@@ -435,7 +435,7 @@
     "\n",
     "Alternatively to computing $x=A^{-1}b$, you can also use [setup_exact_model](https://qiskit.github.io/qiskit-addon-mpf/stubs/qiskit_addon_mpf.static.setup_exact_model.html) to construct a [cvxpy.Problem](https://www.cvxpy.org/api_reference/cvxpy.problems.html#cvxpy.Problem) instance which uses the LSE as constraints and whose optimal solution will yield $x$.\n",
     "\n",
-    "It will become clear in the next section, why this interface exists, too."
+    "In the next section, it will be clear why this interface exists."
    ]
   },
   {
@@ -588,8 +588,8 @@
    "source": [
     "### 1c: Setting up the Trotter circuits\n",
     "\n",
-    "At this point, we have found our expansion coefficients, $x$, and all that is left to do, is to generate the Trotterized quantum circuits.\n",
-    "Once again, the [qiskit_addon_utils.problem_generators](https://qiskit.github.io/qiskit-addon-utils/stubs/qiskit_addon_utils.problem_generators.html) module comes to the resuce with a handy function do just that:"
+    "At this point, we have found our expansion coefficients, $x$, and all that is left to do is to generate the Trotterized quantum circuits.\n",
+    "Once again, the [qiskit_addon_utils.problem_generators](https://qiskit.github.io/qiskit-addon-utils/stubs/qiskit_addon_utils.problem_generators.html) module comes to the rescue with a handy function do just that:"
    ]
   },
   {
@@ -717,7 +717,7 @@
     "## Step 2: Optimize the problem\n",
     "\n",
     "Normally, this is the step in the pattern during which you optimize your circuits for execution on hardware.\n",
-    "Here, since we only use a noiseless simulator, we simple transpile our circuit for a [GenericBackendV2](https://docs.quantum.ibm.com/api/qiskit/qiskit.providers.fake_provider.GenericBackendV2)."
+    "Here, since we only use a noiseless simulator, we simply transpile our circuit for a [GenericBackendV2](https://docs.quantum.ibm.com/api/qiskit/qiskit.providers.fake_provider.GenericBackendV2)."
    ]
   },
   {
@@ -946,7 +946,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.9"
+   "version": "3.12.4"
   }
  },
  "nbformat": 4,