Add pauli_rep property to Adjoint when possible

[albi3ro]

Context:

The Operator.pauli_rep property is a provides an efficient representation of an operator as a linear combination of pauli's, if the operator is an obvious linear combination of pauli's.

>>> (qml.X(0) + 2 * qml.Y(0)).pauli_rep
1.0 * X(0)
+ 2.0 * Y(0)
>>> (qml.X(0) ** 3).pauli_rep
1.0 * X(0)
>>> qml.RX(0.5, 0).pauli_rep is None
True

The returned object is a special dictionary mapping pauli words to coefficients, the PauliSentence

Currently, the pauli_rep for any Adjoint class is None.

>>> qml.adjoint(qml.X(0)).pauli_rep  is None
True

But if the target has a pauli rep, we should be able to provide one on the Adjoint class. We only need to take the conjugate of all the coefficients. You can see similar "modify pauli rep of target" behavior in our SProd (scalar product) class:

pennylane/pennylane/ops/op_math/sprod.py

Line 151 in 8a12fa5

pr = {pw: qnp.dot(coeff, scalar) for pw, coeff in base_pauli_rep.items()}

The task

This issue would require initializing the _pauli_rep during initialization of all Adjoint operators to be the adjoint of the target's pauli rep.

The PR should also:

• Add unit tests to tests/ops/op_math/test_adjoint.py
• Obey black, pylint, and isort
• Include a changelog entry

[vivek-kumar9696]

Hi! I would like to take up this issue. Is there anyone else working on it ?
This is my first time trying to contribute to open source codebase and I wanted to know if issues are only assigned after a PR has been raised.

Thanks!!

[albi3ro]

Thanks for reaching out @vivek-kumar9696 . I've assigned you the issue.

If you have any questions, feel free to either reach out here, or open up a draft PR to start a discussion there.

[vivek-kumar9696]

@albi3ro Thanks for the reply. I will provide biweekly updates regularly.
Excited to work on this and close my first issue!!