[docs] Restore the docs for trajectory noisy simulators (#2648)

Ported from PR #2466
Follow-up to PR #2481

Signed-off-by: Pradnya Khalate <[email protected]>

Author: khalatepradnya

docs/sphinx/using/backends/sims/noisy.rst

@@ -1,7 +1,106 @@
-
-Density Matrix Simulators
+Noisy Simulators
 ==================================
 
+Trajectory Noisy Simulation
+++++++++++++++++++++++++++++++++++
+
+The :code:`nvidia` target supports noisy quantum circuit simulations using 
+quantum trajectory method across all configurations: single GPU, multi-node 
+multi-GPU, and with host memory. When simulating many trajectories with small 
+state vectors, the simulation is batched for optimal performance.
+
+When a :code:`noise_model` is provided to CUDA-Q, the :code:`nvidia` target 
+will incorporate quantum noise into the quantum circuit simulation according 
+to the noise model specified.
+
+
+.. tab:: Python
+
+    .. literalinclude:: ../../../snippets/python/using/backends/trajectory.py
+        :language: python
+        :start-after: [Begin Docs]
+
+    .. code:: bash 
+        
+        python3 program.py
+        { 00:15 01:92 10:81 11:812 }
+
+.. tab:: C++
+
+    .. literalinclude:: ../../../snippets/cpp/using/backends/trajectory.cpp
+        :language: cpp
+        :start-after: [Begin Documentation]
+
+    .. code:: bash 
+
+        nvq++ --target nvidia program.cpp [...] -o program.x
+        ./program.x
+        { 00:15 01:92 10:81 11:812 }
+
+
+In the case of bit-string measurement sampling as in the above example, each measurement 'shot' is executed as a trajectory, whereby Kraus operators specified in the noise model are sampled.
+
+For observable expectation value estimation, the statistical error scales asymptotically as :math:`1/\sqrt{N_{trajectories}}`, where :math:`N_{trajectories}` is the number of trajectories.
+Hence, depending on the required level of accuracy, the number of trajectories can be specified accordingly.
+
+.. tab:: Python
+
+    .. literalinclude:: ../../../snippets/python/using/backends/trajectory_observe.py
+        :language: python
+        :start-after: [Begin Docs]
+
+    .. code:: bash 
+        
+        python3 program.py
+        Noisy <Z> with 1024 trajectories = -0.810546875
+        Noisy <Z> with 8192 trajectories = -0.800048828125
+
+.. tab:: C++
+
+    .. literalinclude:: ../../../snippets/cpp/using/backends/trajectory_observe.cpp
+        :language: cpp
+        :start-after: [Begin Documentation]
+
+    .. code:: bash 
+
+        nvq++ --target nvidia program.cpp [...] -o program.x
+        ./program.x
+        Noisy <Z> with 1024 trajectories = -0.810547
+        Noisy <Z> with 8192 trajectories = -0.800049
+
+
+The following environment variable options are applicable to the :code:`nvidia` target for trajectory noisy simulation. Any environment variables must be set
+prior to setting the target.
+
+.. list-table:: **Additional environment variable options for trajectory simulation**
+  :widths: 20 30 50
+
+  * - Option
+    - Value
+    - Description
+  * - ``CUDAQ_OBSERVE_NUM_TRAJECTORIES``
+    - positive integer
+    - The default number of trajectories for observe simulation if none was provided in the `observe` call. The default value is 1000.
+  * - ``CUDAQ_BATCH_SIZE``
+    - positive integer or `NONE`
+    - The number of state vectors in the batched mode. If `NONE`, the batch size will be calculated based on the available device memory. Default is `NONE`.
+  * - ``CUDAQ_BATCHED_SIM_MAX_BRANCHES``
+    - positive integer
+    - The number of trajectory branches to be tracked simultaneously in the gate fusion. Default is 16. 
+  * - ``CUDAQ_BATCHED_SIM_MAX_QUBITS``
+    - positive integer
+    - The max number of qubits for batching. If the qubit count in the circuit is more than this value, batched trajectory simulation will be disabled. The default value is 20.
+  * - ``CUDAQ_BATCHED_SIM_MIN_BATCH_SIZE``
+    - positive integer
+    - The minimum number of trajectories for batching. If the number of trajectories is less than this value, batched trajectory simulation will be disabled. Default value is 4.
+
+.. note::
+    
+    Batched trajectory simulation is only available on the single-GPU execution mode of the :code:`nvidia` target. 
+    
+    If batched trajectory simulation is not activated, e.g., due to problem size, number of trajectories, or the nature of the circuit (dynamic circuits with mid-circuit measurements and conditional branching), the required number of trajectories will be executed sequentially.  
+
+
 
 Density Matrix 
 ++++++++++++++++

docs/sphinx/using/backends/simulators.rst

@@ -28,7 +28,7 @@ technical details and code examples for using each circuit simulator.
      - < 28
    * - `nvidia`
      - State Vector
-     - General purpose (default)
+     - General purpose (default); Trajectory simulation for noisy circuits
      - Single GPU
      - single / double
      - < 33 / 32 (64 GB)