Add matrix and sparse matrix to `Permute` template

[albi3ro]

PennyLane has a Permute template that decomposes into SWAP operations to reorder wires.

qml.Permute([4, 2, 0, 1, 3], wires=[0, 1, 2, 3, 4])

We should be able to directly construct the matrix for this template. Since the matrix is so sparse, we should add a sparse matrix as well, though that can added separately in a later PR.

A successful PR completely this task would:

1. Add a matrix or compute_matrix method to qml.Permute (and/or sparse_matrix)
2. Add tests to tests/templates/test_subroutines/test_permute.py
3. Add a changelog entry

[Gopal-Dahale]

@albi3ro I would like to work on this issue. If possible assign it to me.

Question: What should be the arguments for this function? Also, the docs state that qml.Permute has a compute_matrix function (I think it is of the super class i.e. Operation). Do you want to override it?

[albi3ro]

Thanks @Gopal-Dahale !

For some added information, Operator defines the default behaviour for compute_matrix and matrix. This default behaviour is raising a MatrixNotImplementedError. We want to override that behavior.

We have two options for defining the matrix:

1. Permute.compute_matrix(*parameters, **hyperparameters). This static method accepts op.parameters and op.hyperparameters. Since the existing hyperparameters does not provide enough information to calculate the matrix, we would need to add wires to the hyperparameters dictionary.

2. Permute.matrix(self, wire_order=None). If we override this instead, we need to manually set the property Permute.has_matrix=True. We would also need to use qml.math.expand_matrix to convert from the default wire order to any arbitrary wire order.

See the source code for the GroverOperator and the QFT template for examples of templates that define compute_matrix.

[Gopal-Dahale]

@albi3ro I have created a draft pr. I have some questions:

1. The compute_matrix method's signature is *params, **hyperparams. I need the number of wires (n_wires) and the permutation for computing the matrix. So the call to the function will look like this (for 3 wires)

P_matrix = qml.templates.Permute([1,2,0], wires=wires).compute_matrix(3, [1,2,0])

Passing the permutation in the compute_matrix seems redundant as it is given in Permute but I do not see any other alternative to this. As compute_matrix is a static method, I cannot access self.hyperparameters dict. 2. Passing permutation as the second argument to compute_matrix method as a 'dict' gives an error

TypeError: unhashable type: 'dict'

I am not sure what exactly the signature of the compute_matrix method should be.

[Gopal-Dahale]

@albi3ro Need some help here.

[albi3ro]

@Gopal-Dahale , I left some clarifications on the PR itself here: https://github.com/PennyLaneAI/pennylane/pull/3173#issuecomment-1281300771 .

Hope that helps clear things up a little.