General methods for second quantized operators

What is the expected enhancement?

There is some general functionality which would be nice to have for all types of second-quantized ops:

1. Checks for hermiticity of an operator (Is the operator equivalent to its adjoint)?
2. A direct way to compute commutators, and checks for commutativity.
3. I think it would be nice to also have a .to_matrix method for the FermionicOp as is already in place for the SpinOp.
4. For the .to_matrix() methods it would be nice to return a sparse datatype such as one of the scipy.sparse formats like csc. Maybe this could be its own method .to_sparse_matrix().

Points 1. and 2. should be straightforward to implement with the current arithmetics already in place.

For 3. one needs to make a choice whether to use the full Fock space (with varying particle number) as a basis or only a certain particle-number (and fixed spin?) subspace. It might be nice to have this be specified by the user.

[LaurinFischer]

Okay so in summary, I think what we want to implement is:

• Hermiticity checks for all SecondQuantizedOps SecondQuantizedOp.is_hermite() -> bool

• A way to compute commutators such as commutator(op1: SecondQuantizedOp, op2: SecondQuantizedOp).is_zero()

• .to_matrix()-method for FermionicOp over the full Fock basis

• scipy.sparse representations of the matrices for FermionicOp and SpinOp, e.g. with parameter to_matrix(sparse: bool)

After an offline discussion with @ikkoham I think I could take care of these (this would also be a good learning opportunity for me to get into the workflow). Should I open a new issue for each of these points or can we use this issue to keep track of the changes?

[ikkoham]

Thank you for nice suggestion.

1. Checks for hermiticity of an operator (Is the operator equivalent to its adjoint)?

SecondQuantizedOp.is_hermite() -> bool is nice!

1. A direct way to compute commutators, and checks for commutativity.

We have two choices.

1. SecondQuantizedOp.is_commutative(op: SecondQuantizedOp) -> bool
2. commutator(op1: SecondQuantizedOp, op2: SecondQuantizedOp).is_zero(). I’ve made this in opflow.

1. I think it would be nice to also have a .to_matrix method for the FermionicOp as is already in place for the SpinOp.

The basis ofFermionicOp.to_matrix() isn’t trivial as you said. Which basis is convenient for chemistry users?

1. For the .to_matrix() methods it would be nice to return a sparse datatype such as one of the scipy.sparse formats like csc. Maybe this could be its own method .to_sparse_matrix().

Nice. See https://github.com/Qiskit/qiskit-nature/blob/master/qiskit_nature/operators/second_quantization/spin_op.py#L439. This parameter is same with quantum_info.Pauli.