code: def _extract_su2su2_prefactors(U, V):
      r"""This function is used for the case of 2 CNOTs and 3 CNOTs. It does something
      similar as the 1-CNOT case, but there is no special form for one of the
      SO(4) operations.
  
      Suppose U, V are SU(4) matrices for which there exists A, B, C, D such that
      (A \otimes B) V (C \otimes D) = U. The problem is to find A, B, C, D in SU(2)
      in an analytic and fully differentiable manner.
  
      This decomposition is possible when U and V are in the same double coset of
      SU(4), meaning there exists G, H in SO(4) s.t. G (Edag V E) H = (Edag U
      E). This is guaranteed here by how V was constructed in both the
      _decomposition_2_cnots and _decomposition_3_cnots methods.
  
      Then, we can use the fact that E SO(4) Edag gives us something in SU(2) x
      SU(2) to give A, B, C, D.
      """
  
      # A lot of the work here happens in the magic basis. Essentially, we
      # don't look explicitly at some U = G V H, but rather at
      #     E^\dagger U E = G E^\dagger V E H
      # so that we can recover
      #     U = (E G E^\dagger) V (E H E^\dagger) = (A \otimes B) V (C \otimes D).
      # There is some math in the paper explaining how when we define U in this way,
      # we can simultaneously diagonalize functions of U and V to ensure they are
      # in the same coset and recover the decomposition.
      u = math.dot(
          math.cast_like(Edag, V), math.dot(U, math.cast_like(E, V))
      )
      v = math.dot(
          math.cast_like(Edag, V), math.dot(V, math.cast_like(E, V))
      )
  
      uuT = math.dot(u, math.T(u))
      vvT = math.dot(v, math.T(v))
  
      # Get the p and q in SO(4) that diagonalize uuT and vvT respectively (and
      # their eigenvalues). We are looking for a simultaneous diagonalization,
      # which we know exists because of how U and V were constructed. Furthermore,
      # The way we will do this is by noting that, since uuT/vvT are complex and
      # symmetric, so both their real and imaginary parts share a set of
      # real-valued eigenvectors, which are also eigenvectors of uuT/vvT
      # themselves. So we can use eigh, which orders the eigenvectors, and so we
      # are guaranteed that the p and q returned will be "in the same order".
      _, p = math.linalg.eigh(math.real(uuT) + math.imag(uuT))
      _, q = math.linalg.eigh(math.real(vvT) + math.imag(vvT))
  
      # If determinant of p/q is not 1, it is in O(4) but not SO(4), and has determinant
      # We can transform it to SO(4) by simply negating one of the columns.
      p = math.dot(
          p, math.diag([1, 1, 1, math.sign(math.linalg.det(p))])
      )
      q = math.dot(
          q, math.diag([1, 1, 1, math.sign(math.linalg.det(q))])
      )
  
      # Now, we should have p, q in SO(4) such that p^T u u^T p = q^T v v^T q.
      # Then (v^\dag q p^T u)(v^\dag q p^T u)^T = I.
      # So we can set G = p q^T, H = v^\dag q p^T u to obtain G v H = u.
      G = math.dot(math.cast_like(p, 1j), math.T(q))
      H = math.dot(math.conj(math.T(v)), math.dot(math.T(G), u))
  
      # These are still in SO(4) though - we want to convert things into SU(2) x SU(2)
      # so use the entangler. Since u = E^\dagger U E and v = E^\dagger V E where U, V
      # are the target matrices, we can reshuffle as in the docstring above,
      #     U = (E G E^\dagger) V (E H E^\dagger) = (A \otimes B) V (C \otimes D)
      # where A, B, C, D are in SU(2) x SU(2).
      AB = math.dot(
          math.cast_like(E, G), math.dot(G, math.cast_like(Edag, G))
      )
      CD = math.dot(
          math.cast_like(E, H), math.dot(H, math.cast_like(Edag, H))
      )
  
      # Now, we just need to extract the constituent tensor products.
      A, B = _su2su2_to_tensor_products(AB)
      C, D = _su2su2_to_tensor_products(CD)
  
      return A, B, C, D
  
file_path: pennylane/pennylane/ops/op_math/decompositions/two_qubit_unitary.py
imports: import numpy
  import pennylane
  from pennylane import math
  from .single_qubit_unitary import one_qubit_decomposition
label_desc: Use image node_id 4 for calling a global function with example usage: _extract_su2su2_prefactors(U, V) and returns: A,  B,  C,  D