Extended Data Fig. 1: CZ gate echo, atomic level structure, and typical pulse sequence.

a, The two-qubit gates we apply, in addition to applying a controlled-Z operation between the two qubits, also induce a single-qubit phase Z(ζ) to both qubits, composed of the intrinsic phase of the CZ gate⁵ and additional spurious phases from the 420-nm Rydberg laser and pulsing the traps off. Since we apply all gates in parallel by global pulses of the Rydberg laser, if a qubit is not adjacent to another qubit, it does not perform a CZ gate but still acquires the same Z(ζ) (identical to being adjacent to another qubit in state |0⟩, which is dark to the Rydberg laser). As diagrammed, we cancel the additional, undesired Z(ζ) by applying a π pulse between pairs of CZ gates. This echo procedure removes any need to calibrate the intrinsic phase from the CZ gate, and renders us insensitive to any spurious changes in Z(ζ) slower than ~200 μs. The additional Y(π) propagates in a known way through the CZ gates and multiplies certain stabilizers by a −1 sign, which simply redefines the sign of stabilizers and logical qubits. b, Level diagram showing key ⁸⁷Rb atomic levels used. Our Rydberg excitation scheme from |1⟩ to |r⟩ is composed of a two-photon transition driven by a 420-nm laser and a 1013-nm laser (see ref. ²⁵ for description of laser system). A DC magnetic field of B = 8.5 G is applied throughout this work. c, A typical pulse sequence for running a quantum circuit.