[labs.dla 5] All Cartan involutions + Cartan subalgebra

doc/releases/changelog-dev.md

@@ -17,41 +17,41 @@
 * Developers of plugin devices now have the option of providing a TOML-formatted configuration file
   to declare the capabilities of the device. See [Device Capabilities](https://docs.pennylane.ai/en/latest/development/plugins.html#device-capabilities) for details.
 
-  * An internal module `pennylane.devices.capabilities` is added that defines a new `DeviceCapabilites`
-    data class, as well as functions that load and parse the TOML-formatted configuration files.
-    [(#6407)](https://github.com/PennyLaneAI/pennylane/pull/6407)
+* An internal module `pennylane.devices.capabilities` is added that defines a new `DeviceCapabilites`
+  data class, as well as functions that load and parse the TOML-formatted configuration files.
+  [(#6407)](https://github.com/PennyLaneAI/pennylane/pull/6407)
 
-    ```pycon
-      >>> from pennylane.devices.capabilities import DeviceCapabilities
-      >>> capabilities = DeviceCapabilities.from_toml_file("my_device.toml")
-      >>> isinstance(capabilities, DeviceCapabilities)
-      True
-    ```
+  ```pycon
+    >>> from pennylane.devices.capabilities import DeviceCapabilities
+    >>> capabilities = DeviceCapabilities.from_toml_file("my_device.toml")
+    >>> isinstance(capabilities, DeviceCapabilities)
+    True
+  ```
 
-  * Devices that extends `qml.devices.Device` now has an optional class attribute `capabilities`
-    that is an instance of the `DeviceCapabilities` data class, constructed from the configuration
-    file if it exists. Otherwise, it is set to `None`.
-    [(#6433)](https://github.com/PennyLaneAI/pennylane/pull/6433)
+* Devices that extends `qml.devices.Device` now has an optional class attribute `capabilities`
+  that is an instance of the `DeviceCapabilities` data class, constructed from the configuration
+  file if it exists. Otherwise, it is set to `None`.
+  [(#6433)](https://github.com/PennyLaneAI/pennylane/pull/6433)
 
-    ```python
-    from pennylane.devices import Device
+  ```python
+  from pennylane.devices import Device
 
-    class MyDevice(Device):
+  class MyDevice(Device):
 
-        config_filepath = "path/to/config.toml"
+      config_filepath = "path/to/config.toml"
 
-        ...
-    ```
-    ```pycon
-    >>> isinstance(MyDevice.capabilities, DeviceCapabilities)
-    True
-    ```
+      ...
+  ```
+  ```pycon
+  >>> isinstance(MyDevice.capabilities, DeviceCapabilities)
+  True
+  ```
 
-  * Default implementations of `Device.setup_execution_config` and `Device.preprocess_transforms`
-    are added to the device API for devices that provides a TOML configuration file and thus have
-    a `capabilities` property.
-    [(#6632)](https://github.com/PennyLaneAI/pennylane/pull/6632)
-    [(#6653)](https://github.com/PennyLaneAI/pennylane/pull/6653)
+* Default implementations of `Device.setup_execution_config` and `Device.preprocess_transforms`
+  are added to the device API for devices that provides a TOML configuration file and thus have
+  a `capabilities` property.
+  [(#6632)](https://github.com/PennyLaneAI/pennylane/pull/6632)
+  [(#6653)](https://github.com/PennyLaneAI/pennylane/pull/6653)
 
 * Support is added for `if`/`else` statements and `for` and `while` loops in circuits executed with `qml.capture.enabled`, via Autograph.
   Autograph conversion is now used by default in `make_plxpr`, but can be skipped with the keyword arg `autograph=False`.
@@ -60,10 +60,20 @@
   [(#6426)](https://github.com/PennyLaneAI/pennylane/pull/6426)
   [(#6645)](https://github.com/PennyLaneAI/pennylane/pull/6645)
 
-  * New `qml.GQSP` template has been added to perform Generalized Quantum Signal Processing (GQSP).
+* New `qml.GQSP` template has been added to perform Generalized Quantum Signal Processing (GQSP).
     The functionality `qml.poly_to_angles` has been also extended to support GQSP.
     [(#6565)](https://github.com/PennyLaneAI/pennylane/pull/6565)
 
+* Added `unary_mapping()` function to map `BoseWord` and `BoseSentence` to qubit operators, using unary mapping.
+  [(#6576)](https://github.com/PennyLaneAI/pennylane/pull/6576)
+
+* Added `binary_mapping()` function to map `BoseWord` and `BoseSentence` to qubit operators, using standard-binary mapping.
+  [(#6564)](https://github.com/PennyLaneAI/pennylane/pull/6564)
+
+* New functionality to calculate angles for QSP and QSVT has been added. This includes the function `qml.poly_to_angles`
+  to obtain angles directly and the function `qml.transform_angles` to convert angles from one subroutine to another.
+  [(#6483)](https://github.com/PennyLaneAI/pennylane/pull/6483)
+
 * Added a function `qml.trotterize` to generalize the Suzuki-Trotter product to arbitrary quantum functions.
   [(#6627)](https://github.com/PennyLaneAI/pennylane/pull/6627)
 
@@ -96,32 +106,31 @@
   b: ──RY(-0.17)─╰X─╰X──RY(-0.17)─┤  State
   ```
 
-<h4>New `labs` module `dla` for handling dynamical Lie algebras (DLAs)</h4>
+<h4>New `pennylane.labs.dla` module for handling (dynamical) Lie algebras (DLAs)</h4>
 
 * Added a dense implementation of computing the Lie closure in a new function
   `lie_closure_dense` in `pennylane.labs.dla`.
   [(#6371)](https://github.com/PennyLaneAI/pennylane/pull/6371)
   [(#6695)](https://github.com/PennyLaneAI/pennylane/pull/6695)
 
-* New functionality to calculate angles for QSP and QSVT has been added. This includes the function `qml.poly_to_angles`
-  to obtain angles directly and the function `qml.transform_angles` to convert angles from one subroutine to another.
-  [(#6483)](https://github.com/PennyLaneAI/pennylane/pull/6483)
-
 * Added a dense implementation of computing the structure constants in a new function
   `structure_constants_dense` in `pennylane.labs.dla`.
   [(#6376)](https://github.com/PennyLaneAI/pennylane/pull/6376)
 
-* Added utility functions for handling dense matrices in the Lie theory context.
+* Added utility functions for handling dense matrices and advanced functionality in the Lie theory context.
   [(#6563)](https://github.com/PennyLaneAI/pennylane/pull/6563)
   [(#6392)](https://github.com/PennyLaneAI/pennylane/pull/6392)
+  [(#6396)](https://github.com/PennyLaneAI/pennylane/pull/6396)
 
 * Added a ``cartan_decomp`` function along with two standard involutions ``even_odd_involution`` and ``concurrence_involution``.
   [(#6392)](https://github.com/PennyLaneAI/pennylane/pull/6392)
 
-* Added `unary_mapping()` function to map `BoseWord` and `BoseSentence` to qubit operators, using unary mapping
-  [(#6576)](https://github.com/PennyLaneAI/pennylane/pull/6576); 
-added `binary_mapping()` function to map `BoseWord` and `BoseSentence` to qubit operators, using standard-binary mapping.
-  [(#6564)](https://github.com/PennyLaneAI/pennylane/pull/6564)
+* Added a `recursive_cartan_decomp` function and all canonical Cartan involutions.
+  [(#6396)](https://github.com/PennyLaneAI/pennylane/pull/6396)
+
+* Added a `cartan_subalgebra` function to compute the (horizontal) Cartan subalgebra of a Cartan decomposition.
+  [(#6403)](https://github.com/PennyLaneAI/pennylane/pull/6403)
+  [(#6396)](https://github.com/PennyLaneAI/pennylane/pull/6396)
 
 
 <h4>New API for Qubit Mixed</h4>

pennylane/labs/dla/__init__.py

@@ -23,6 +23,8 @@
     ~lie_closure_dense
     ~structure_constants_dense
     ~cartan_decomp
+    ~recursive_cartan_decomp
+    ~cartan_subalgebra
 
 
 Utility functions
@@ -34,44 +36,80 @@
     :toctree: api
 
     ~adjvec_to_op
-    ~change_basis_ad_rep
-    ~check_orthonormal
-    ~check_commutation
-    ~check_all_commuting
-    ~check_cartan_decomp
     ~op_to_adjvec
+    ~trace_inner_product
     ~orthonormalize
     ~pauli_coefficients
     ~batched_pauli_decompose
-    ~trace_inner_product
+    ~check_orthonormal
+    ~check_commutation
+    ~check_all_commuting
+    ~check_cartan_decomp
+    ~change_basis_ad_rep
 
 Involutions
 ~~~~~~~~~~~
 
+A map :math:`\theta: \mathfrak{g} \rightarrow \mathfrak{g}` from the Lie algebra :math:`\mathfrak{g}` to itself is called an involution
+when it fulfills :math:`\theta(\theta(g)) = g \ \forall g \in \mathfrak{g}` and is compatible with commutators,
+:math:`[\theta(g), \theta(g')]=\theta([g, g']).` Involutions are used to construct a :func:`~cartan_decomp`. There are seven canonical
+Cartan involutions of real simple Lie algebras (``AI, AII, AIII, BDI, CI, CII, DIII``),
+see `Wikipedia <https://en.wikipedia.org/wiki/Symmetric_space#Classification_result>`__.
+In addition, there is a canonical Cartan involution for real semisimple algebras that consist of
+two isomorphic simple components (``ClassB``), see `here <https://en.wikipedia.org/wiki/Symmetric_space#Classification_scheme>`__.
+
 .. currentmodule:: pennylane.labs.dla
 
 .. autosummary::
     :toctree: api
 
     ~even_odd_involution
     ~concurrence_involution
+    ~khaneja_glaser_involution
+    ~AI
+    ~AII
+    ~AIII
+    ~BDI
+    ~CI
+    ~CII
+    ~DIII
+    ~ClassB
 
 
 """
 
 from .lie_closure_dense import lie_closure_dense
 from .structure_constants_dense import structure_constants_dense
-from .cartan import cartan_decomp, even_odd_involution, concurrence_involution
+from .cartan import (
+    cartan_decomp,
+    recursive_cartan_decomp,
+)
 from .dense_util import (
+    adjvec_to_op,
     change_basis_ad_rep,
+    check_all_commuting,
+    check_cartan_decomp,
+    check_commutation,
+    check_orthonormal,
     pauli_coefficients,
     batched_pauli_decompose,
-    orthonormalize,
-    check_orthonormal,
     trace_inner_product,
-    adjvec_to_op,
     op_to_adjvec,
-    check_commutation,
-    check_all_commuting,
-    check_cartan_decomp,
+    orthonormalize,
 )
+
+from .involutions import (
+    khaneja_glaser_involution,
+    even_odd_involution,
+    concurrence_involution,
+    AI,
+    AII,
+    AIII,
+    BDI,
+    CI,
+    CII,
+    DIII,
+    ClassB,
+)
+
+from .cartan_subalgebra import cartan_subalgebra