Fig. 3: Implementation and verification of multi-qubit gates.

a Gate sequence to implement a CCZ unitary. The qutrit swap gates (pink) are used to shelve and then retrieve the control state \({\left\vert 11\right\rangle }_{c}\). They sandwich a cross-Kerr gate (green) that induces a Z gate on Q₁ if and only if Q₂ is in \(\left\vert 2\right\rangle\). The final stage (blue) is used to correct the residual ZZ phases between the qubits. b The three-body operation manifests as the phase shift of the target qubit (Q₁) when the control qubits (Q₂ and Q₃) are in \({\left\vert 11\right\rangle }_{c}\). The solid lines represent cosine fits. c Pauli fidelities of the dressed cycle and the reference cycle from cycle benchmarking. The resulting gate fidelity is \({{{{{\mathcal{F}}}}}}_{{{{{\rm{CCZ}}}}}}=96.0(3)\%\). d Gate sequence to implement a CCCZ unitary. A cascade of qutrit swap interactions (pink) is used to shelve and retrieve the respective \({\left\vert 11\right\rangle }_{c}\) states. In the middle of the sequence is a cross-Kerr gate (green) that induces a Z gate on Q₄ if and only if Q₃ is in \(\left\vert 2\right\rangle\). All the residual correlated phases are corrected in the final stage (blue). e The four-body operation is effectively revealed through a π-phase-shift of the target qubit (Q₄) for the control state \({\left\vert 110\right\rangle }_{c}\). The solid lines represent cosine fits. f Truth table of the four-qubit Toffoli gate. The target state is shown to be flipped when the control state is \(\left\vert {{{{{\rm{Q}}}}}}_{3}{{{{{\rm{Q}}}}}}_{2}{{{{{\rm{Q}}}}}}_{1}\right\rangle={\left\vert 110\right\rangle }_{c}\) (\(\left\vert 0110\right\rangle \leftrightarrow \left\vert 1110\right\rangle\)). The corresponding truth table fidelity is \({{{{{\mathcal{F}}}}}}_{{{{{\rm{CCCZ}}}}}}=92(1)\%\).