Extended Data Fig. 9: Time-optimal pulses for multi-qubit controlled phase gates and GHZ-state data.

a, The execution time of a C_NZ blockade gate as a function of the number of qubits. The N-qubit gates are realized by applying a phase flip to the \({\left|0\right\rangle }^{\otimes N}\) state, which is not equivalent to the method of ref. ⁸ for N > 2. For the CCZ gate, applying a π phase to the |111⟩ state (|111⟩ → −|111⟩), while leaving all other basis states invariant, is related by a global bit-flip to applying a relative π phase to the |000⟩ state; however, the two implementations are not equivalent up to a global Z rotation, contrary to the two-qubit case. The time-optimal CCZ gate using the second approach realizes the CCZ gate about 34% faster with (ΩT/2π) = 1.72, as compared with (ΩT/2π) = 2.61 from the first approach. The two approaches are different because the relative phase of π is accumulated between different basis states, which have different rates of phase accumulation. In the case of applying |111⟩ → −|111⟩, the states with the slowest relative rate are |111⟩ and |011⟩, which are driven with the Rabi frequencies of \(\sqrt{3}\Omega \) and \(\sqrt{2}\Omega \), respectively, resulting in the phase accumulation rate proportional to \((\sqrt{3}-\sqrt{2})\Omega \approx 0.32\Omega \). By contrast, when the relative phase is applied on the state |000⟩, the smallest accumulation rate is given by the |001⟩ state, which is driven with the Rabi frequency Ω. In general, an arbitrary global single-qubit rotation at the end of the gate can be included to incorporate all of the above approaches in the optimization procedure. b, Time-optimal phase profiles (without analytic parameterization) for the C_NZ gates up to six qubits realized by applying a phase flip to the \({\left|0\right\rangle }^{\otimes N}\) state. c, Circuit used to generate the GHZ state \((| 000\rangle +| 111\rangle )/\sqrt{(2})\) after two CCZ gates. d, GHZ states measured experimentally on applying this circuit to seven three-qubit groups in parallel, with populations in |000⟩ and |111⟩ of 92.9(3)% and a parity contrast of 89(1)%, giving a raw GHZ-state fidelity of 90.9(6)%.