[Feature] unroll_block_with_scalling does not always return proper decomposition of pauli terms with coeffs

[jpmoutinho]

Porting an issue from Qadence related to the unroll_block_with_scaling. This is a function in Qadence that returns a tuple of blocks and their scaling. E.g.

block = add(4*kron(Z(0), Z(1)), 2*Z(0))
unroll_block_with_scaling(block)
> [(4, kron(Z(0), Z(1))), (2, Z(0))]

This is a routine used in a few different places, and I have also discussed it with @kaosmicadei and we concluded that we should have pretty equivalent functionality in the new block system.

Porting here the issue below opened by Raja:

obs = add(2*kron(2*Z(0), Z(1)), 4 * Z(0))
terms =[pauli[0] for pauli in unroll_block_with_scaling(obs)]
coeffs = torch.tensor(
    [pauli[1] for pauli in unroll_block_with_scaling(obs)], dtype=float
)
print("coeffs:", coeffs)
> coeffs: tensor([2., 4.])

The issue is that he expected the first coefficient to be 4, but it is 2. This is because unroll_block_with_scaling does not simplify anything inside the kron. I suspect this will not be an issue in the new version.

[kaosmicadei]

Indeed, the distributive property is now applied by default, so 2 * (2 * Z(0) * Z(1) + 4 * Z(0)) is converted straight to 4 * Z(0) * Z(1) + 8 * Z(0), and an expression like (1 + cos(phi)) * Z(0) * Z(1) will be expanded into Z(0) * Z(1) + cos(phi) * Z(0) * Z(1).

The unroll_block_with_scaling is now called collect_operators and returns a dictionary of operator-coefficient.

expr = 4 * Z(0) * Z(1) + 8 * Z(0)
collect_operators(expr)
# -> {Z(0) * Z(1): 4, Z(0): 8}

expr = Z(0) * Z(1) + cos(phi) * Z(0) * Z(1)
collect_operators(expr)
# -> {Z(0) * Z(1): 1 + cos(phi)}

[RolandMacDoland]

Indeed, the distributive property is now applied by default, so 2 * (2 * Z(0) * Z(1) + 4 * Z(0)) is converted straight to 4 * Z(0) * Z(1) + 8 * Z(0), and an expression like (1 + cos(phi)) * Z(0) * Z(1) will be expanded into Z(0) * Z(1) + cos(phi) * Z(0) * Z(1).

The unroll_block_with_scaling is now called collect_operators and returns a dictionary of operator-coefficient.

expr = 4 * Z(0) * Z(1) + 8 * Z(0)
collect_operators(expr)
# -> {Z(0) * Z(1): 4, Z(0): 8}

expr = Z(0) * Z(1) + cos(phi) * Z(0) * Z(1)
collect_operators(expr)
# -> {Z(0) * Z(1): 1 + cos(phi)}

What I think would be good is if we could search by Pauli words in the returned dict. Ie. searching for all operators that contain Z(0) or of a certain pattern like Z(0) * Z(1)