Reorganize API index page into sections (#12333)

* Reorganize API index page into sections

* Sort alphabetically within each section

* Clarify ordering expectation

(cherry picked from commit 8c8c78a)

docs/apidoc/index.rst

@@ -1,42 +1,89 @@
 .. module:: qiskit
+..
+   Within each section, the modules should be ordered alphabetically by
+   module name (not RST filename).
 
 =============
 API Reference
 =============
 
+Circuit construction:
+
 .. toctree::
    :maxdepth: 1
 
    circuit
-   circuit_library
    circuit_classical
-   circuit_singleton
-   compiler
-   visualization
    classicalfunction
+   circuit_library
+   circuit_singleton
+
+Quantum information:
+
+.. toctree::
+   :maxdepth: 1
+
+   quantum_info
+
+Transpilation:
+
+.. toctree::
+   :maxdepth: 1
+
    converters
-   assembler
    dagcircuit
    passmanager
+   synthesis
+   qiskit.synthesis.unitary.aqc
+   transpiler
+   transpiler_passes
+   transpiler_synthesis_plugins
+   transpiler_preset
+   transpiler_plugins
+
+Primitives and providers:
+
+.. toctree::
+   :maxdepth: 1
+
+   primitives
    providers
    providers_basic_provider
    providers_fake_provider
    providers_models
-   pulse
-   scheduler
-   synthesis
-   qiskit.synthesis.unitary.aqc
-   primitives
+
+Results and visualizations:
+
+.. toctree::
+   :maxdepth: 1
+
+   result
+   visualization
+
+Serialization:
+
+.. toctree::
+   :maxdepth: 1
+
    qasm2
    qasm3
-   qobj
    qpy
-   quantum_info
-   result
-   transpiler
-   transpiler_passes
-   transpiler_preset
-   transpiler_plugins
-   transpiler_synthesis_plugins
-   utils
+
+Pulse-level programming:
+
+.. toctree::
+   :maxdepth: 1
+
+   pulse
+   scheduler
+
+Other:
+
+.. toctree::
+   :maxdepth: 1
+
+   assembler
+   compiler
    exceptions
+   qobj
+   utils