Adding support for observables in LRE executors

[Shivansh20128]

This PR aims to add support for observables in the LRE executors.
#Resolves #2513

Description

1. Added required parameters and code changes in lre.py.
2. Added tests in the test file to check observable support.

---

License

• I license this contribution under the terms of the GNU GPL, version 3 and grant Unitary Fund the right to provide additional permissions as described in section 7 of the GNU GPL, version 3.

Before opening the PR, please ensure you have completed the following where appropriate.

• I added unit tests for new code.
• I used type hints in function signatures.
• I used Google-style docstrings for functions.
• I updated the documentation where relevant.
• Added myself / the copyright holder to the AUTHORS file

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 98.67%. Comparing base (34eb6ce) to head (be35300).
Report is 1 commits behind head on main.

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #2681   +/-   ##
=======================================
  Coverage   98.67%   98.67%           
=======================================
  Files          93       93           
  Lines        4223     4223           
=======================================
  Hits         4167     4167           
  Misses         56       56           

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[Shivansh20128]

Hi! I tried adding the test for a batched executor as well but could not prepare an executor that would support an observable as discussed here.
If its desirable, can someone suggest how I can proceed.

I was trying something like this:

@pytest.mark.parametrize(
    "degree, fold_multiplier, observable",
    [(2, 2, Observable[PauliString(spec="X")])],
)
def test_lre_batched_executor_with_observable(degree, fold_multiplier, observable):
    """Verify LRE batch executor with observable works as expected."""
    test_batched_executor = Executor(batched_executor, max_batch_size=200)
    lre_exp_val_batched = execute_with_lre(
        test_cirq,
        test_batched_executor,
        degree=degree,
        fold_multiplier=fold_multiplier,
        observable=observable,
    )
    assert isinstance(lre_exp_val_batched, float)

    assert test_batched_executor.calls_to_executor == 1
    assert (
        test_batched_executor.executed_circuits
        == multivariate_layer_scaling(test_cirq, degree, fold_multiplier)
    )

You can find the batched_executor here.

The problem is that I was unable to formulate a batched_executor that supports observables.

[bdg221]

If you check test_executor.py you can see a lot of examples of executors. I would recommend using executor_density_matrix_typed and executor_density_matrix_batched.
You should then be able to do:

test_batched_executor = Executor(executor_density_matrix_batched, max_batch_size=200)

[bdg221]

@Shivansh20128 Let me know if you have any more questions about the batched executors with observables. Hopefully, the test_exectuor.py examples are helpful.

[bdg221]

This looks to be the exact same as test_lre_exp_value_with_observable. I would recommend moving the assert lines to that test and removing this one.