The fixed ancilla-register interaction. (a) Decomposition into local and non-local parts. (b) The two-qubit entangling gate is implemented on the register qubits j and k via the sequence of Eq. (13) with an ancilla initialised in the state , where the form of is given in Eq. (14). This two-qubit gate is not symmetric with respect to the exchange of j and k. (c) The single-qubit gate is applied to a register qubit j by two applications of L with an ancilla initialised in the state , .