A schematic circuit for the controlled unknown qubit operation. An ancillary photon C with two degrees of freedom (DoFs) in the state |H〉₁ will be used in this scheme. {|H〉_i, |V〉_i} is a polarization basis of the photon C from the spatial mode i, while |H〉 and |V〉 denote the horizonal polarization and vertical polarization respectively. U is an unknown qubit operation on the polarization state. Quantum fusion is an invertible transformation or equivalent swapping operations on special systems, i.e., the quantum information of the qubits A and B are swapped to the spatial DoF and polarization DoF of the ancillary photon, respectively. Quantum splitting is the inverse of quantum fusion.