expand PEA description (#2537)

closes #2536

---------

Co-authored-by: Kaelyn Ferris <[email protected]>

docs/guides/error-mitigation-and-suppression-techniques.ipynb

@@ -246,11 +246,19 @@
    "source": [
     "## Probabilistic Error Amplification (PEA)\n",
     "\n",
-    "One of the main challenges in ZNE is to accurately amplify the noise affecting the target circuit. Gate folding provides an easy way to perform this amplification, but is potentially inaccurate and might lead to incorrect results.  See the article [\"Scalable error mitigation for noisy quantum circuits produces competitive expectation values\"](https://www.nature.com/articles/s41567-022-01914-3), and specifically page 4 of the [supplementary material](https://arxiv.org/pdf/2108.09197) for details.\n",
+    "One of the main challenges in ZNE is to accurately amplify the noise affecting the target circuit. Gate folding provides an easy way to perform this amplification, but is potentially inaccurate and might lead to incorrect results.  See the article [\"Scalable error mitigation for noisy quantum circuits produces competitive expectation values\"](https://arxiv.org/pdf/2108.09197), and specifically page 4 of the supplementary information for details. Probabilistic error amplification provides a more accurate approach to error amplification through noise learning.\n",
     "\n",
-    "PEA is a more sophisticated technique that performs preliminary experiments to reconstruct the noise and then uses this information to perform an accurate amplification. For utility-scale experiments, it is often the best choice.\n",
+    "PEA is a more sophisticated technique that performs preliminary experiments to reconstruct the noise and then uses this information to perform an accurate amplification. It starts by learning the twirled noise model of each layer of entangling gates in the circuit before they are run (see [LayerNoiseLearningOptions](/api/qiskit-ibm-runtime/qiskit_ibm_runtime.options.LayerNoiseLearningOptions) for relevant learning options). After the learning phase, the circuits are executed at each noise factor, where every entangling layer of the circuits is amplified by probabilistically injecting single-qubit noise proportional to the corresponding learned noise model. See the article [\"Evidence for the utility of quantum computing before fault tolerance\"](https://www.nature.com/articles/s41586-023-06096-3) for more details.\n",
     "\n",
-    "Because PEA is a ZNE noise amplification technique, you also need to enable ZNE by setting `resilience.zne_mitigation = True`. Other `resilience.zne` options can be used in addition to set extrapolators, amplification levels, and so on. PEA requires a noise model, which is automatically generated when using primitives.\n",
+    "PEA consists of three stages:\n",
+    "1. _Learning_: The twirled noise model of each layer of entangling gates in the circuit is learned.\n",
+    "1. _Noise amplification_: The original quantum circuit is executed multiple times at different noise factors.\n",
+    "2. _Extrapolation_: The ideal result is estimated by extrapolating the noisy expectation value results to the zero-noise limit.\n",
+    "\n",
+    "\n",
+    "For utility-scale experiments, PEA is often the best choice.\n",
+    "\n",
+    "Because PEA is a ZNE noise amplification technique, you also need to enable ZNE by setting `resilience.zne_mitigation = True`. Other [`resilience.zne`](/api/qiskit-ibm-runtime/qiskit_ibm_runtime.options.ZneOptions) options can additionally be used to set extrapolators, amplification levels, and so on. PEA requires a noise model, which is automatically generated when using primitives.\n",
     "\n",
     "The following snippet provides an example where PEA is used to mitigate the result of an Estimator job:"
    ]