A schematic circuit for the controlled unknown qubit operation with an ancillary high-dimensional system. U is an unknown qubit operation. C is an ancillary 4D system. Controlled-S denotes a general controlled swapping gate defined by |0〉〈0|⊗ (|0〉〈2|+|2〉〈0|+|1〉〈3|+|3〉〈1|)+|1〉〈1|⊗ I₄, where {|0〉, |1〉, |2〉, |3〉} is the basis of the system C. SWAP denotes the swapping gate on a two-qubit system. H denotes the Hadamard operation.