Figure 2

Two-qubit gates in the first minimal control model with interactions of the form${\mathit{K}}_{\mathit{a}}^{\mathit{j}}$. This gate sequence has no overall effect on the ‘entangling’ ancilla qubit but implements the entangling unitary M, given by Eq. (9) and which is locally equivalent to CZ, on the register qubits. 2n additional ancillas with appropriate initial states in the computational basis are required to implement the unitary $u_0^{†}$ (up to arbitrary accuracy) on each register qubit and these extra interactions are shown in the blue dashed box. The number of these additional ancillas depends on the form of $u_0$ and $u_1$.