[labs] [WIP] Variational KHK decomposition

[Qottmann]

Variational KHK decomposition using the adjoint representation

Based off https://arxiv.org/pdf/2104.00728 but with some important modifications, in particular using the adjoint representation which is faster and enables working with arbitrary bases

TODO

• Compatible with dense representations
• more extensive testing

n = 2

gens = [X(i) @ X(i+1) for i in range(n-1)]
gens += [Z(i) for i in range(n)]
H = qml.sum(*gens)

g = qml.lie_closure(gens)
g = [op.pauli_rep for op in g]

involution = concurrence_involution

assert not involution(H)
k, m = cartan_decomposition(g, involution=involution)
assert check_cartan_decomp(k, m)

g = k + m
adj = qml.structure_constants(g)

g, k, mtilde, h, adj = cartan_subalgebra(g, k, m, adj, tol=1e-14, start_idx=0)
len(g), len(k), len(mtilde), len(h), adj.shape
dims = (len(k), len(mtilde), len(h))
khk_res = variational_khk(H, g, dims, adj, verbose=True)
validate_khk(H, k, mtilde+h, khk_res, n, 1e-6)

[sc-74997]