Computing Expectation Values
-----------------------------

CUDA Quantum provides generic library functions enabling one to compute expectation values 
of quantum spin operators with respect to a parameterized CUDA Quantum kernel. Let's take a look 
at an example of this:

.. tab:: Python

   .. literalinclude:: ../../examples/python/expectation_values.py
      :language: python

   Here we define a parameterized CUDA Quantum kernel that takes an angle, `theta`, as a single
   input. This angle becomes the argument of a single `ry` rotation.

   We define a Hamiltonian operator via the CUDA Quantum `cudaq.SpinOperator` type.

   CUDA Quantum provides a generic function `cudaq.observe`. This function takes as input three
   arguments. The first two argument are a parameterized kernel and the `cudaq.SpinOperator` whose
   expectation value we wish to compute. The final arguments are the runtime parameters at which we
   evaluate the parameterized kernel.


.. tab:: C++

   .. literalinclude:: ../../examples/cpp/basics/expectation_values.cpp
      :language: cpp

   Here we define a parameterized CUDA Quantum kernel, a callable type named :code:`ansatz` that takes as 
   input a single angle :code:`theta`. This angle becomes the argument of a single :code:`ry` rotation. 

   In host code, we define a Hamiltonian operator via the CUDA Quantum :code:`spin_op` type. 
   CUDA Quantum provides a generic function :code:`cudaq::observe`. This function takes as input three terms. 
   The first two terms are a parameterized kernel and the :code:`spin_op` whose expectation value we wish to compute.
   The last term contains the runtime parameters at which we evaluate the parameterized kernel. 

   The return type of this function is an :code:`cudaq::observe_result` which contains all the data 
   from the execution, but is trivially convertible to a double, resulting in the expectation value we are interested in. 

   To compile and execute this code, we run the following:

   .. code:: bash 

      nvq++ expectation_values.cpp -o exp_vals.x 
      ./exp_vals.x