[labs.dla 5] [WIP] (Almost) All Cartan involutions

[Qottmann]

Providing dense implementations for all Cartan involutions listed in appendix C of https://arxiv.org/abs/2406.04418

Not sure if we should make this pennylane.labs functionality yet, but required for research purposes atm

Iterative Cartan decomposition:

from pennylane.labs.dla import AI, AII, AIII, BDI, CI, CII, DIII, cartan_decomposition, check_cartan_decomp

g = list(qml.pauli.pauli_group(3)) # su(8)
g = [qml.matrix(_, wire_order=range(3)) for _ in g]
k0, m0 = cartan_decomposition(g, AII)
print(f"First iteration: {len(k0)}, {len(m0)}")
assert check_cartan_decomp(k0, m0)

k1, m1 = cartan_decomposition(k0, CI)
assert check_cartan_decomp(k1, m1)
print(f"Second iteration: {len(k1)}, {len(m1)}")

First iteration: 36, 28
Second iteration: 16, 20

[sc-76116]

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