docs(frontend-python): add statistics tutorial

docs/SUMMARY.md

@@ -28,6 +28,7 @@
 * [Multi Parameters](tutorial/multi\_parameters.md)
 * [Simulation](tutorial/simulation.md)
 * [Direct Circuits](tutorial/direct\_circuits.md)
+* [Statistics](tutorial/statistics.md)
 
 ## Application Tutorials
 

docs/tutorial/statistics.md

@@ -0,0 +1,124 @@
+# Statistics
+
+Concrete analyzes all compiled circuits and calculates some statistics. These statistics can be used to find bottlenecks and compare circuits. Statistics are calculated in terms of basic operations. There are 6 basic operations in Concrete:
+
+- **clear addition:** x + y where x is encrypted and y is clear
+- **encrypted addition:** x + y where both x and y are encrypted
+- **clear multiplication:** x * y where x is encrypted and y is clear
+- **encrypted negation:** -x where x is encrypted
+- **key switch:** building block for table lookups
+- **packing key switch:** building block for table lookups
+- **programmable bootstrapping:** building block for table lookups
+
+You can print all statistics using `show_statistics` configuration option:
+
+```python
+from concrete import fhe
+
+@fhe.compiler({"x": "encrypted"})
+def f(x):
+    return (x**2) + (2*x) + 4
+
+inputset = range(2**2)
+circuit = f.compile(inputset, show_statistics=True)
+```
+
+This code will print:
+
+```
+Statistics
+--------------------------------------------------------------------------------
+size_of_secret_keys: 22648
+size_of_bootstrap_keys: 51274176
+size_of_keyswitch_keys: 64092720
+size_of_inputs: 16392
+size_of_outputs: 16392
+p_error: 9.627450598589458e-06
+global_p_error: 9.627450598589458e-06
+complexity: 99198195.0
+programmable_bootstrap_count: 1
+programmable_bootstrap_count_per_parameter: {
+    BootstrapKeyParam(polynomial_size=2048, glwe_dimension=1, input_lwe_dimension=781, level=1, base_log=23, variance=9.940977002694397e-32): 1
+}
+key_switch_count: 1
+key_switch_count_per_parameter: {
+    KeyswitchKeyParam(level=5, base_log=3, variance=1.939836732335308e-11): 1
+}
+packing_key_switch_count: 0
+clear_addition_count: 1
+clear_addition_count_per_parameter: {
+    LweSecretKeyParam(dimension=2048): 1
+}
+encrypted_addition_count: 1
+encrypted_addition_count_per_parameter: {
+    LweSecretKeyParam(dimension=2048): 1
+}
+clear_multiplication_count: 1
+clear_multiplication_count_per_parameter: {
+    LweSecretKeyParam(dimension=2048): 1
+}
+encrypted_negation_count: 0
+--------------------------------------------------------------------------------
+```
+
+{% hint style="info" %}
+Each of these properties can be directly accessed on the circuit (e.g., `circuit.programmable_bootstrap_count`).
+{% endhint %}
+
+## Tags
+
+Circuit analysis also considers [tags](../tutorial/tagging.md)!
+
+Imagine you have a neural network with 10 layers, each of them tagged. You can easily see the amount of additions and multiplications required for matrix multiplications per layer:
+
+```
+Statistics
+--------------------------------------------------------------------------------
+clear_multiplication_count_per_tag: {
+    /model/model: 53342
+    /model/model.0/Gemm: 14720
+    /model/model.0/Gemm.matmul: 14720
+    /model/model.2/Gemm: 11730
+    /model/model.2/Gemm.matmul: 11730
+    /model/model.4/Gemm: 9078
+    /model/model.4/Gemm.matmul: 9078
+    /model/model.6/Gemm: 6764
+    /model/model.6/Gemm.matmul: 6764
+    /model/model.8/Gemm: 4788
+    /model/model.8/Gemm.matmul: 4788
+    /model/model.10/Gemm: 3150
+    /model/model.10/Gemm.matmul: 3150
+    /model/model.12/Gemm: 1850
+    /model/model.12/Gemm.matmul: 1850
+    /model/model.14/Gemm: 888
+    /model/model.14/Gemm.matmul: 888
+    /model/model.16/Gemm: 264
+    /model/model.16/Gemm.matmul: 264
+    /model/model.18/Gemm: 110
+    /model/model.18/Gemm.matmul: 110
+}
+encrypted_addition_count_per_tag: {
+    /model/model: 53342
+    /model/model.0/Gemm: 14720
+    /model/model.0/Gemm.matmul: 14720
+    /model/model.2/Gemm: 11730
+    /model/model.2/Gemm.matmul: 11730
+    /model/model.4/Gemm: 9078
+    /model/model.4/Gemm.matmul: 9078
+    /model/model.6/Gemm: 6764
+    /model/model.6/Gemm.matmul: 6764
+    /model/model.8/Gemm: 4788
+    /model/model.8/Gemm.matmul: 4788
+    /model/model.10/Gemm: 3150
+    /model/model.10/Gemm.matmul: 3150
+    /model/model.12/Gemm: 1850
+    /model/model.12/Gemm.matmul: 1850
+    /model/model.14/Gemm: 888
+    /model/model.14/Gemm.matmul: 888
+    /model/model.16/Gemm: 264
+    /model/model.16/Gemm.matmul: 264
+    /model/model.18/Gemm: 110
+    /model/model.18/Gemm.matmul: 110
+}
+--------------------------------------------------------------------------------
+```